Growth of Accreting Supermassive Black Hole Seeds and Neutrino Radiation

Research ArticleGrowth of Accreting Supermassive Black HoleSeeds and Neutrino Radiation

Gagik Ter-Kazarian

Division of Theoretical Astrophysics Ambartsumian Byurakan Astrophysical Observatory Byurakan 378433 Aragatsotn Armenia

Correspondence should be addressed to Gagik Ter-Kazarian gago 50yahoocom

Received 1 May 2014 Revised 26 June 2014 Accepted 27 June 2014

Academic Editor Gary Wegner

Copyright copy 2015 Gagik Ter-KazarianThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the framework of microscopic theory of black hole (MTBH) which explores the most important processes of rearrangementof vacuum state and spontaneous breaking of gravitation gauge symmetry at huge energies we have undertaken a large series ofnumerical simulations with the goal to trace an evolution of themass assembly history of 377 plausible accreting supermassive blackhole seeds in active galactic nuclei (AGNs) to the present time and examine the observable signatures today Given the redshiftsmasses and luminosities of these black holes at present time collected from the literature we compute the initial redshifts andmasses of the corresponding seed black holes For the present masses119872BH119872⊙

≃ 11 times 106 to 13 times 1010 of 377 black holes the

computed intermediate seedmasses are ranging from119872SeedBH 119872⊙

≃ 264 to 29times105 We also compute the fluxes of ultrahigh energy(UHE) neutrinos produced via simple or modified URCA processes in superdense protomatter nuclei The AGNs are favored aspromising pure UHE neutrino sources because the computed neutrino fluxes are highly beamed along the plane of accretion diskpeaked at high energies and collimated in smaller opening angle (120579 ≪ 1)

1 Introduction

With typical bolometric luminosities sim1045minus48 erg sminus1 theAGNs are amongst the most luminous emitters in the uni-verse particularly at high energies (gamma-rays) and radiowavelengths From its historical development up to currentinterests the efforts in theAGNphysics have evoked the studyof a major unsolved problem of how efficiently such hugeenergies observed can be generatedThis energy scale severelychallenges conventional source models The huge energyrelease from compact regions of AGN requires extremelyhigh efficiency (typically ge10 per cent) of conversion of restmass to other forms of energy This serves as the mainargument in favour of supermassive black holes with massesof millions to billions of times the mass of the Sun as centralengines ofmassive AGNsThe astrophysical black holes comein a wide range of masses from ge3119872

⊙for stellar mass black

holes [1] to sim1010119872⊙for supermassive black holes [2 3]

Demography of local galaxies suggests that most galaxiesharbour quiescent supermassive black holes in their nucleiat the present time and that the mass of the hosted blackhole is correlated with properties of the host bulge The

visible universe should therefore contain at least 100 billionsupermassive black holes A complex study of evolution ofAGNs requires an answer to the key questions such as howdidthe first black holes form how did massive black holes get tothe galaxy centers and how did they grow in accreting massnamely an understanding of the important phenomenon ofmass assembly history of accreting supermassive black holeseeds The observations support the idea that black holesgrow in tandem with their hosts throughout cosmic historystarting from the earliest times While the exact mechanismfor the formation of the first black holes is not currentlyknown there are several prevailing theories [4] Howevereach proposal towards formation and growth of initial seedblack holes has its own advantage and limitations in provingthe whole view of the issue In this report we review the massassembly history of 377 plausible accreting supermassiveblack hole seeds in AGNs and their neutrino radiation inthe framework of gravitation theory which explores themostimportant processes of rearrangement of vacuum state anda spontaneous breaking of gravitation gauge symmetry athuge energies We will proceed according to the followingstructureMost observational theoretical and computational

Hindawi Publishing CorporationJournal of AstrophysicsVolume 2015 Article ID 205367 30 pageshttpdxdoiorg1011552015205367

2 Journal of Astrophysics

aspects of the growth of black hole seeds are summarizedin Section 2 The other important phenomenon of ultrahighenergy cosmic rays in relevance to AGNs is discussed inSection 3 The objectives of suggested approach are outlinedin Section 4 In Section 5 we review the spherical accretionon superdense protomatter nuclei in use In Section 6 wediscuss the growth of the seed black hole at accretion andderive its intermediate mass initial redshift and neutrinopreradiation time (PRT) Section 7 is devoted to the neutrinoradiation produced in superdense protomatter nuclei Thesimulation results of the seed black hole intermediate massesPRTs seed redshifts and neutrino fluxes for 377 AGN blackholes are brought in Section 8 The concluding remarks aregiven in Section 9 We will refrain from providing lengthydetails of the proposed gravitation theory at huge energiesand neutrino flux computations For these the reader isinvited to visit the original papers and appendices of thepresent paper In the latter we also complete the spacetimedeformation theory in the model context of gravitation bynew investigation of building up the complex of distortion(DC) of spacetime continuum and showing how it restoresthe world-deformation tensor which still has been put in byhand Finally note that we regard the considered black holesonly as the potential neutrino sources The obtained resultshowever may suffer if not all live black holes at present residein final stage of their growth driven by the formation ofprotomatter disk at accretion and they radiate neutrino Weoften suppress the indices without notice Unless otherwisestated we take geometrized units throughout this paper

2 A Breakthrough in Observational andComputational Aspects on Growth of BlackHole Seeds

Significant progress has been made in the last few years inunderstanding how supermassive black holes form and growGiven the currentmasses of 106minus9119872

⊙ most black hole growth

happens in the AGN phase A significant fraction of the totalblack hole growth 60 [6] happens in the most luminousAGN quasars In an AGN phase which lasts sim108 years thecentral supermassive black hole can gain up to sim107minus8119872

⊙

so even the most massive galaxies will have only a few ofthese events over their lifetime Aforesaid gathers supportespecially from a breakthroughmade in recent observationaltheoretical and computational efforts in understanding ofevolution of black holes and their host galaxies particularlythrough self-regulated growth and feedback from accretion-powered outflows see for example [4 7ndash18] Whereas themultiwavelength methods are used to trace the growth ofseed BHs the prospects for future observations are reviewedThe observations provide strong support for the existenceof a correlation between supermassive black holes and theirhosts out to the highest redshifts The observations of thequasar luminosity function show that the most supermassiveblack holes get most of their mass at high redshift whileat low redshift only low mass black holes are still growing[19] This is observed in both optical [20] and hard X-ray luminosity functions [19 21] which indicates that this

result is independent of obscuration Natarajan [13] hasreported that the initial black hole seeds form at extremelyhigh redshifts from the direct collapse of pregalactic gasdiscs Populating dark matter halos with seeds formed inthis fashion and using a Monte-Carlo merger tree approachhe has predicted the black hole mass function at highredshifts and at the present time The most aspects of themodels that describe the growth and accretion history ofsupermassive black holes and evolution of this scenario havebeen presented in detail by [9 10] In these models at earlytimes the properties of the assembling black hole seeds aremore tightly coupled to properties of the dark matter haloas their growth is driven by the merger history of halosWhile a clear picture of the history of black hole growth isemerging significant uncertainties still remain [14] and inspite of recent advances [6 13] the origin of the seed blackholes remains an unsolved problem at present The NuSTARdeep high-energy observations will enable obtaining a nearlycomplete AGN survey including heavily obscured Compton-thick sources up to 119911 sim 15 [22] A similar mission ASTRO-H [23] will be launched by Japan in 2014These observationsin combination with observations at longer wavelengths willallow for the detection and identification of most growingsupermassive black holes at 119911 sim 1 The ultradeep X-ray andnear-infrared surveys covering at least sim1 deg2 are requiredto constrain the formation of the first black hole seeds Thiswill likely require the use of the next generation of space-based observatories such as the James Webb Space Telescopeand the International X-ray Observatory The superb spatialresolution and sensitivity of the Atacama Large MillimeterArray (ALMA) [24] will revolutionize our understanding ofgalaxy evolution Combining these new data with existingmultiwavelength information will finally allow astrophysi-cists to pave the way for later efforts by pioneering some ofthe census of supermassive black hole growth in use today

3 UHE Cosmic-Ray Particles

The galactic sources like supernova remnants (SNRs) ormicroquasars are thought to accelerate particles at least upto energies of 3 times 1015 eV The ultrahigh energy cosmic-ray (UHECR) particles with even higher energies have sincebeen detected (comprehensive reviews can be found in [25ndash29]) The accelerated protons or heavier nuclei up to energiesexceeding 1020 eV are firstly observed by [30] The cosmic-ray events with the highest energies so far detected haveenergies of 2 times 1011 GeV [31] and 3 times 1011 GeV [32] Theseenergies are 107 times higher than the most energetic man-made accelerator the LHC at CERN These highest energiesare believed to be reached in extragalactic sources like AGNsor gamma-ray bursts (GRBs) During propagation of suchenergetic particles through the universe the threshold forpion photoproduction on the microwave background is sim2times 1010 GeV and at sim3 times 1011 GeV the energy-loss distance isabout 20Mpc Propagation of cosmic rays over substantiallylarger distances gives rise to a cutoff in the spectrum at sim1011 GeV as was first shown by [33 34] the GZK cutoffThe recent confirmation [35 36] of GZK suppression in the

Journal of Astrophysics 3

cosmic-ray energy spectrum indicates that the cosmic rayswith energies above the GZK cutoff 119864GZK sim 40EeV mostlycome from relatively close (within the GZK radius 119903GZK sim100Mpc) extragalactic sources However despite the detailedmeasurements of the cosmic-ray spectrum the identificationof the sources of the cosmic-ray particles is still an openquestion as they are deflected in the galactic and extragalacticmagnetic fields and hence have lost all information abouttheir originwhen reaching Earth Only at the highest energiesbeyond sim10196 GeV cosmic-ray particles may retain enoughdirectional information to locate their sources The lattermust be powerful enough to sustain the energy density inextragalactic cosmic rays of about 3times10minus19 erg cmminus3 which isequivalent tosim8times 1044 ergMpcminus3 yrminus1Though it has not beenpossible up to now to identify the sources of galactic or extra-galactic cosmic rays general considerations allow limitingpotential source classes For example the existing data on thecosmic-ray spectrum and on the isotropic 100MeV gamma-ray background limit significantly the parameter space inwhich topological defects can generate the flux of the highestenergy cosmic rays and rule out models with the standardX-particle mass of 1016 GeV and higher [37] Eventually theneutrinos will serve as unique astronomical messengers andthey will significantly enhance and extend our knowledgeon galactic and extragalactic sources of the UHE universeIndeed except for oscillations induced by transit in a vacuumHiggs field neutrinos can penetrate cosmological distancesand their trajectories are not deflected by magnetic fieldsas they are neutral providing powerful probes of highenergy astrophysics in ways which no other particle canMoreover the flavor composition of neutrinos originating atastrophysical sources can serve as a probe of new physicsin the electroweak sector Therefore an appealing possibilityamong the various hypotheses of the origin of UHECR isso-called Z-burst scenario [38ndash51] This suggests that if ZeVastrophysical neutrino beam is sufficiently strong it canproduce a large fraction of observed UHECR particles within100Mpc by hitting local light relic neutrinos clustered in darkhalos and form UHECR through the hadronic Z (s-channelproduction) andW-bosons (t-channel production) decays byweak interactions The discovery of UHE neutrino sourceswould also clarify the productionmechanism of the GeV-TeVgamma rays observed on Earth [43 52 53] as TeV photonsare also produced in the up-scattering of photons in reactionsto accelerated electrons (inverse-Compton scattering) Thedirect link between TeV gamma-ray photons and neutrinosthrough the charged and neutral pion production which iswell known from particle physics allows for a quite robustprediction of the expected neutrino fluxes provided thatthe sources are transparent and the observed gamma raysoriginate from pion decay The weakest link in the Z-bursthypothesis is probably both unknown boosting mechanismof the primary neutrinos up to huge energies of hundredsZeV and their large flux required at the resonant energy 119864] ≃119872

2

119885(2119898]) ≃ 42 times 10

21 eV (eV119898]) well above the GZKcutoff Such a flux severely challenges conventional sourcemodels Any concomitant photon flux should not violateexisting upper limits [37 48 49 54] The obvious question is

then raised where in the Cosmos are these neutrinos comingfrom It turns out that currently at energies in excess of10

19 eV there are only two good candidate source classes forUHE neutrinos AGNs and GRBs The AGNs as significantpoint sources of neutrinos were analyzed in [50 55 56]While hard to detect neutrinos have the advantage of repre-senting aforesaid unique fingerprints of hadron interactionsand therefore of the sources of cosmic rays Two basicevent topologies can be distinguished track-like patterns ofdetected Cherenkov light (hits) which originate from muonsproduced in charged-current interactions of muon neutrinos(muon channel) spherical hit patterns which originate fromthe hadronic cascade at the vertex of neutrino interactionsor the electromagnetic cascade of electrons from charged-current interactions of electron neutrinos (cascade channel)If the charged-current interaction happens inside the detectoror in case of charged-current tau-neutrino interactions thesetwo topologies overlap which complicates the reconstruc-tion At the relevant energies the neutrino is approximatelycollinear with the muon and hence the muon channel isthe prime channel for the search for point-like sources ofcosmic neutrinos On the other hand cascades deposit allof their energy inside the detector and therefore allow fora much better energy reconstruction with a resolution of afew 10 Finally numerous reports are available at presentin literature on expected discovery potential and sensitivityof experiments to neutrino point-like sources Currentlyoperating high energy neutrino telescopes attempt to detectUHE neutrinos such as ANTARES [57 58] which is themostsensitive neutrino telescope in theNorthernHemisphere Ice-Cube [35 59ndash64] which is worldwide largest and hence mostsensitive neutrino telescope in the Southern HemisphereBAIKAL [65] as well as the CR extended experiments ofTheTelescope Array [66] Pierre Auger Observatory [67 68] andJEM-EUSO mission [69] The JEM-EUSO mission whichis planned to be launched by a H2B rocket around 2015-2016 is designed to explore the extremes in the universe andfundamental physics through the detection of the extremeenergy (119864 gt 10

20 eV) cosmic rays The possible originsof the soon-to-be famous 28 IceCube neutrino-PeV events[59ndash61] are the first hint for astrophysical neutrino signalAartsen et al have published an observation of two sim1 PeVneutrinos with a 119875 value 28120590 beyond the hypothesis thatthese events were atmospherically generated [59] The anal-ysis revealed an additional 26 neutrino candidates depositingldquoelectromagnetic equivalent energiesrdquo ranging from about30 TeV up to 250 TeV [61] New results were presented at theIceCube Particle Astrophysics Symposium (IPA 2013) [62ndash64] If cosmic neutrinos are primarily of extragalactic originthen the 100GeV gamma ray flux observed by Fermi-LATconstrains the normalization at PeV energies at injectionwhich in turn demands a neutrino spectral index Γ lt 21 [70]

4 MTBH Revisited Preliminaries

For the benefit of the reader a brief outline of the key ideasbehind the microscopic theory of black hole as a guidingprinciple is given in this section to make the rest of the


paper understandable There is a general belief reinforced bystatements in textbooks that according to general relativity(GR) a long-standing standard phenomenological black holemodel (PBHM)mdashnamely the most general Kerr-Newmanblack hole model with parameters of mass (119872) angularmomentum (119869) and charge (119876) still has to be put in byhandmdashcan describe the growth of accreting supermassiveblack hole seed However such beliefs are suspect and shouldbe critically reexamined The PBHM cannot be currentlyaccepted as convincing model for addressing the afore-mentioned problems because in this framework the verysource of gravitational field of the black hole is a kind ofcurvature singularity at the center of the stationary blackhole A meaningless central singularity develops which ishidden behind the event horizon The theory breaks downinside the event horizon which is causally disconnected fromthe exterior world Either the Kruskal continuation of theSchwarzschild (119869 = 0 119876 = 0) metric or the Kerr (119876 = 0)metric or the Reissner-Nordstrom (119869 = 0) metric showsthat the static observers fail to exist inside the horizonAny object that collapses to form a black hole will go onto collapse to a singularity inside the black hole Therebyany timelike worldline must strike the central singularitywhich wholly absorbs the infalling matter Therefore theultimate fate of collapsing matter once it has crossed theblack hole surface is unknown This in turn disables anyaccumulation of matter in the central part and thus neitherthe growth of black holes nor the increase of their mass-energy density could occur at accretion of outside matteror by means of merger processes As a consequence themass and angular momentum of black holes will not changeover the lifetime of the universe But how can one be surethat some hitherto unknown source of pressure does notbecome important at huge energies and halt the collapse Tofill the voidwhich the standard PBHMpresents one plausibleidea to innovate the solution to alluded key problems wouldappear to be the framework of microscopic theory of blackhole This theory has been originally proposed by [71] andreferences therein and thoroughly discussed in [72ndash75]Here we recount some of the highlights of the MTBHwhich is the extension of PBHM and rather completes it byexploring the most important processes of rearrangementof vacuum state and a spontaneous breaking of gravitationgauge symmetry at huge energies [71 74 76] We will notbe concerned with the actual details of this framework butonly use it as a backdrop to validate the theory with someobservational tests For details the interested reader is invitedto consult the original papers Discussed gravitational theoryis consistent with GR up to the limit of neutron stars Butthis theory manifests its virtues applied to the physics ofinternal structure of galactic nuclei In the latter a significantchange of properties of spacetime continuum so-calledinner distortion (ID) arises simultaneously with the stronggravity at huge energies (see Appendix A) Consequently thematter undergoes phase transition of second kind whichsupplies a powerful pathway to form a stable superdenseprotomatter core (SPC) inside the event horizon Due to thisthe stable equilibrium holds in outward layers too and thusan accumulation of matter is allowed now around the SPC

The black hole models presented in phenomenological andmicroscopic frameworks have been schematically plotted inFigure 1 to guide the eye A crucial point of the MTBH isthat a central singularity cannot occur which is now replacedby finite though unbelievably extreme conditions held in theSPC where the static observers existed The SPC surroundedby the accretion disk presents the microscopic model ofAGNThe SPC accommodates the highest energy scale up tohundreds of ZeV in central protomatter core which accountsfor the spectral distribution of the resulting radiation ofgalactic nuclei External physics of accretion onto the blackhole in earlier part of its lifetime is identical to the processesin Schwarzschildrsquos model However a strong difference inthe model context between the phenomenological black holeand the SPC is arising in the second part of its lifetime(see Section 6) The seed black hole might grow up drivenby the accretion of outside matter when it was gettingmost of its mass An infalling matter with time forms theprotomatter disk around the protomatter core tapering offfaster at reaching out the thin edge of the event horizon Atthis metric singularity inevitably disappears (see appendices)and the neutrinos may escape through vista to outsideworld even after the neutrino trapping We study the growthof protomatter disk and derive the intermediate mass andinitial redshift of seed black hole and examine luminositiesneutrino surfaces for the disk In this framework we havecomputed the fluxes of UHE neutrinos [75] produced in themediumof the SPC via simple (quark and pionic reactions) ormodified URCA processes even after the neutrino trapping(G Gamow was inspired to name the process URCA afterthe name of a casino in Rio de Janeiro when M Schenbergremarked to him that ldquothe energy disappears in the nucleusof the supernova as quickly as the money disappeared at thatroulette tablerdquo) The ldquotrappingrdquo is due to the fact that asthe neutrinos are formed in protomatter core at superhighdensities they experience greater difficulty escaping from theprotomatter core before being dragged along with thematternamely the neutrinos are ldquotrappedrdquo comove with matterThe part of neutrinos annihilates to produce further thesecondary particles of expected ultrahigh energies In thismodel of course a key open question is to enlighten themechanisms that trigger the activity and how a large amountofmatter can be steadily funneled to the central regions to fuelthis activity In high luminosity AGNs the large-scale internalgravitational instabilities drive gas towards the nucleus whichtrigger big starbursts and the coeval compact cluster justformed It seemed they have some connection to the nuclearfueling through mass loss of young stars as well as their tidaldisruption and supernovae Note that we regard the UHECRparticles as a signature of existence of superdence protomattersources in the universe Since neutrino events are expected tobe of sufficient intensity our estimates can be used to guideinvestigations of neutrino detectors for the distant future

5 Spherical Accretion onto SPC

As alluded to above the MTBH framework supports the ideaof accreting supermassive black holes which link to AGNsIn order to compute the mass accretion rate in use it is


AGN

ADAD

EH

infin

(a)

AGN

PDPC

ADAD

EHSPC

(b)

Figure 1 (a) The phenomenological model of AGN with the central stationary black hole The meaningless singularity occurs at the centerinside the black hole (b) The microscopic model of AGN with the central stable SPC In due course the neutrinos of huge energies mayescape through the vista to outside world Accepted notations EH = event horizon AD = accretion disk SPC = superdense protomatter corePC = protomatter core

necessary to study the accretion onto central supermassiveSPC The main features of spherical accretion can be brieflysummed up in the following three idealized models thatillustrate some of the associated physics [72]

51 Free Radial Infall We examine the motion of freelymoving test particle by exploring the external geometry of theSPC with the line element (A7) at 119909 = 0 Let us denote the 4-vector of velocity of test particle V120583 = 119889120583119889 120583 = (119905 )and consider it initially for simplest radial infall V2 = V3 =0 We determine the value of local velocity V lt 0 of theparticle for the moment of crossing the EH sphere as well asat reaching the surface of central stable SPC The equation ofgeodesics is derived from the variational principle 120575 int 119889119878 = 0which is the extremum of the distance along the wordline forthe Lagrangian at hand

2119871 = (1 minus 1199090)2 1199052

minus (1 + 1199090)2 119903

2

minus 2sin2 1205932 minus 2 120579

2

(1)

where 119905 equiv 119889 119889120582 is the 119905-component of 4-momentum and 120582is the affine parameter along the worldline We are using anaffine parametrization (by a rescaling 120582 rarr 120582(120582

1015840)) such that

119871 = const is constant along the curve A static observermakesmeasurements with local orthonormal tetrad

119890=10038161003816100381610038161 minus 1199090

1003816100381610038161003816

minus1

119890119905 119890

= (1 + 119909

0)minus1

119890119903

119890=

minus1119890120579 119890

= ( sin )

minus1

119890120579

(2)

The Euler-Lagrange equations for and can be derivedfrom the variational principle A local measurement of theparticlersquos energy made by a static observer in the equatorialplane gives the time component of the 4-momentum asmeasured in the observerrsquos local orthonormal frame This

is the projection of the 4-momentum along the time basisvector The Euler-Lagrange equations show that if we orientthe coordinate system as initially the particle is moving in theequatorial plane (ie = 1205872 120579 = 0) then the particle alwaysremains in this plane There are two constants of the motioncorresponding to the ignorable coordinates and namelythe 119864-ldquoenergy-at-infinityrdquo and the 119897-angular momentum Weconclude that the free radial infall of a particle from theinfinity up to the moment of crossing the EH sphere aswell as at reaching the surface of central body is absolutelythe same as in the Schwarzschild geometry of black hole(Figure 2(a)) We clear up a general picture of orbits justoutside the event horizon by considering the Euler-Lagrangeequation for radial component with ldquoeffective potentialrdquo Thecircular orbits are stable if119881 is concave up namely at gt 4where is the mass of SPC The binding energy per unitmass of a particle in the last stable circular orbit at = 4is bind = (119898 minus 119864) ≃ 1 minus (2732)

12 Namely this isthe fraction of rest-mass energy released when test particleoriginally at rest at infinity spirals slowly toward the SPC tothe innermost stable circular orbit and then plunges into itThereby one of the important parameters is the capture crosssection for particles falling in from infinity 120590capt = 120587119887

2

maxwhere 119887max is the maximum impact parameter of a particlethat is captured

52 Collisionless Accretion The distribution function for acollisionless gas is determined by the collisionless Boltzmannequation or Vlasov equation For the stationary and sphericalflow we obtain then

(119864 gt 0) = 16120587 (119866)2

120588infinVminus1infin119888minus2 (3)


AGN

SPC

x0 = 1x0 = 2

x0 = 0

t = infin

vr lt 0 vr = 0

r = infin

EH

(a)

1minus 1 1+x0

10

15

25

ntimes10

minus40(g

cmminus

3 )

(b)

Figure 2 (a) The free radial infall of a particle from the infinity to EH sphere (1199090= 1) which is similar to the Schwarzschild geometry of

BH Crossing the EH sphere a particle continues infall reaching finally the surface (1199090= 2) of the stable SPC (b) Approaching the EH sphere

(1199090= 1) the particle concentration increases asymptotically until the threshold value of protomatter Then due to the action of cutoff effect

the metric singularity vanishes and the particles well pass EH sphere

where the particle density 120588infin

is assumed to be uniform atfar from the SPC and the particle speed is V

infin≪ 1 During

the accretion process the particles approaching the EHbecome relativistic Approaching event horizon the particleconcentration increases asymptotically as (()119899

infin)1199090rarr1

asymp

minus(ln 00)2 V

infin up to the ID threshold value

119889()

minus13=

04 fm (Figure 2(b)) Due to the action of cutoff effect themetric singularity then vanishes and the particles well passEH sphere (119909

0= 1) and in the sequel form the protomatter

disk around the protomatter core

53 Hydrodynamic Accretion For real dynamical conditionsfound in considered superdense medium it is expected thatthemean free path for collisionswill bemuch shorter than thecharacteristic length scale that is the accretion of ambientgas onto a stationary nonrotating compact SPC will behydrodynamical in nature For any equation of state obeyingthe causality constraint the sound speed implies 1198862 lt 1 andthe flowmust pass through a critical sonic point 119903

119904outside the

event horizon The locally measured particle velocity readsV = (1 minus

00119864

2) where 119864 = 119864

infin119898 = (

00(1 minus 119906

2))12 and

119864infin

is the energy at infinity of individual particle of the mass119898Thus the proper flow velocity V = 119906 rarr 0 and is subsonicAt = 119877

1198922 the proper velocity equals the speed of light

|V| = 119906 = 1 gt 119886 and the flow is supersonic This conditionis independent of the magnitude of 119906 and is not sufficient byitself to guarantee that the flow passes through a critical pointoutside EH For large ge 119903

119904 it is expected that the particles be

nonrelativistic with 119886 le 119886119904≪ 1 (ie 119879 ≪ 1198981198882119870 = 1013119870)

as they were nonrelativistic at infinity (119886infin≪ 1) Considering

the equation of accretion onto superdense protomatter corewhich is an analogue of Bondi equations for spherical steady-state adiabatic accretion onto the SPC we determine a massaccretion rate

= 212058711989811989911990411990352

119904(ln

00)1015840

119904 (4)

where prime (1015840)119904denotes differentiation with respect to at

the point 119903119904 The gas compression can be estimated as

119899infin

asymp11990352

119904

21199032[(ln

00)1015840

119904

1 + 119903119903()]

12

(5)

The approximate equality between the sound speed and themean particle speed implies that the hydrodynamic accretionrate is larger than the collisionless accretion rate by the largefactor asymp109

6 The Intermediate Mass PRT and InitialRedshift of Seed Black Hole

The key objectives of the MTBH framework are then anincrease of the mass 119872Seed

BH gravitational radius 119877Seed119892

andof the seed black hole BHSeed at accretion of outside matterThereby an infalling matter forms protomatter disk aroundprotomatter core tapering off faster at reaching the thin edgeof event horizon So a practical measure of growth BHSeed

rarr

BH may most usefully be the increase of gravitational radiusor mass of black hole

Δ119877119892= 119877

BH119892minus 119877

Seed119892=2119866

1198882119872

119889=2119866

1198882120588119889119881119889

Δ119872BH = 119872BH minus119872SeedBH = 119872

SeedBHΔ119877

119892

119877Seed119892

(6)

where 119872119889 120588

119889 and 119881

119889 respectively are the total mass

density and volume of protomatter disk At the value BH119892

of gravitational radius when protomatter disk has finallyreached the event horizon of grown-up supermassive black


PC PD120579

1205880

120593Z

Z0Z1

EHBH EHSeed

1205881120588

d120576d = d2Rg asymp 120579

RSPCRBHg

RSeedg

Rd

Figure 3 A schematic cross section of the growth of supermassiveblack hole driven by the formation of protomatter disk at accretionwhen protomatter disk has finally reached the event horizon ofgrown-up supermassive black hole

hole the volume 119889can be calculated in polar coordinates

(120588 119911 120593) from Figure 3

119889= int

BH119892

1205880

119889120588int

2120587

0

120588119889120601int

1199111(120588)

minus1199111(120588)

119889119911

minus int

119877119889

1205880

119889120588int

2120587

0

120588119889120601int

1199110(120588)

minus1199110(120588)

119889119911

(119877119889≪BH119892

)

≃radic2120587

3119877119889(

BH119892)

2

(7)

where 1199111(120588) ≃ 119911

0minus 119911

0(120588 minus 120588

0)(

BH119892minus 120588

0) 119911

0(120588) = radic119877

2

119889minus 1205882

and in approximation119877119889≪

BH119892

we set 1199110(120588

0) ≃ 120588

0≃ 119877

119889radic2

61 The Intermediate Mass of Seed Black Hole From the firstline of (6) by virtue of (7) we obtain

BH119892= 119896(1 plusmn radic1 minus

2

119896119877Seed119892) (8)

where 2119896 = 873 [km]119877119889120588119889119872

⊙ The (8) is valid at (2

119896)119877Seed119892le 1 namely

119877⊙

119877119889

ge 209[km]119877⊙

120588119889

120588⊙

119877Seed119892

119877⊙

(9)

For the values 120588119889= 26times10

16[g cm]minus3 (see below) and119877Seed

119892≃

295 [km](103 to 106) inequality (9) is reduced to 119877⊙119877

119889ge

234 times 108(1 to 103) or [cm]119877

119889ge 034(10

minus2 to 10) Thiscondition is always satisfied because for considered 377 blackholes with the masses 119872BH119872⊙

≃ 11 times 106 to 13 times 1010

we approximately have 119877119889119903OV ≃ 10

minus10 to 10minus7 [71] Notethat Woo and Urry [5] collect and compare all the AGNBHmass and luminosity estimates from the literature Accordingto (6) the intermediate mass of seed black hole reads

119872SeedBH119872

⊙

≃119872BH119872

⊙

(1 minus 168 times 10minus6 119877119889

[cm]119872BH119872

⊙

) (10)

62 PRT The PRT is referred to as a lapse of time 119879BH fromthe birth of black hole till neutrino radiation the earlierpart of the lifetime That is 119879BH = 119872119889

where is theaccretion rate In approximation at hand 119877

119889≪ 119877

119892 the PRT

reads

119879BH = 120588119889119881119889

≃ 933 sdot 10

15[g cmminus3

]

1198771198891198772

119892

(11)

In case of collisionless accretion (3) and (11) give

119879BH ≃ 26 sdot 1016 119877119889

cm10

minus24 g cmminus3

120588infin

Vinfin

10 km 119904minus1yr (12)

In case of hydrodynamic accretion (4) and (11) yield

119879BH ≃ 88 sdot 10381198771198891198772

119892cmminus3

11989911990411990352

119904 (ln11989200)1015840

119904

(13)

Note that the spherical accretion onto black hole in generalis not necessarily an efficient mechanism for converting rest-mass energy into radiation Accretion onto black hole maybe far from spherical accretion because the accreted gaspossesses angular momentum In this case the gas will bethrown into circular orbits about the black hole when cen-trifugal forces will become significant before the gas plungesthrough the event horizon Assuming a typical mass-energyconversion efficiency of about 120598 sim 10 in approximation119877119889≪ 119877

119892 according to (12) and (13) the resulting relationship

of typical PRT versus bolometric luminosity becomes

119879BH ≃ 032119877119889

119903OV(119872BH119872

⊙

)

210

39119882

119871bol[yr] (14)

We supplement this by computing neutrino fluxes in the nextsection

63 Redshift of Seed Black Hole Interpreting the redshiftas a cosmological Doppler effect and that the Hubble lawcould most easily be understood in terms of expansion of theuniverse we are interested in the purely academic questionof principle to ask what could be the initial redshift 119911Seed ofseed black hole if the mass the luminosity and the redshift119911 of black hole at present time are known To follow thehistory of seed black hole to the present time let us placeourselves at the origin of coordinates 119903 = 0 (according tothe Cosmological Principle this is mere convention) andconsider a light traveling to us along the minus119903 direction withangular variables fixed If the light has left a seed black holelocated at 119903

119904 120579

119904 and 120593

119904 at time 119905

119904 and it has to reach us at

a time 1199050 then a power series for the redshift as a function

of the time of flight is 119911Seed = 1198670(1199050minus 119905

119904) + sdot sdot sdot where 119905

0

is the present moment and 1198670is Hubblersquos constant Similar

expression 119911 = 1198670(1199050minus119905

1)+sdot sdot sdot can be written for the current

black hole located at 1199031 120579

1 and 120593

1 at time 119905

1 where 119905

1=

119905119904+ 119879BH as seed black hole is an object at early times Hence

in the first-order approximation byHubblersquos constant wemayobtain the following relation between the redshifts of seed


and present black holes 119911Seed ≃ 119911+1198670119879BHThis relation is in

agreement with the scenario of a general recession of distantgalaxies away from us in all directions the furthest naturallybeing those moving the fastest This relation incorporatingwith (14) for the value 119867

0= 70 [km][sMpc] favored today

yields

119911Seed≃ 119911 + 2292 times 10

28 119877119889

119903OV(119872BH119872

⊙

)

2119882

119871bol (15)

7 UHE Neutrino Fluxes

The flux can be written in terms of luminosity as 119869]120576 =]1205764120587119863

2

119871(119911)(1 + 119911) where 119911 is the redshift and 119863

119871(119911) is the

luminosity distance depending on the cosmological modelThe (1+119911)minus1 is due to the fact that each neutrino with energy1015840

] if observed near the place and time of emission 1199051015840 will bered-shifted to energy ] =

1015840

]119877(1199051)119877(1199050) = 1015840

](1 + 119911)minus1 of

the neutrino observed at time 119905 after its long journey to uswhere 119877(119905) is the cosmic scale factor Computing the UHEneutrino fluxes in the framework of MTBH we choose thecosmological model favored today with a flat universe filledwith matterΩ

119872= 120588

119872120588

119888and vacuum energy densitiesΩ

119881=

120588119881120588

119888 therebyΩ

119881+Ω

119872= 1 where the critical energy density

120588119888= 3119867

2

0(8120587119866

119873) is defined through the Hubble parameter

1198670[77]

119863119871(119911) =

(1 + 119911) 119888

1198670radicΩ

119872

int

1+119911

1

119889119909

radicΩ119881Ω

119872+ 1199093

= 24 times 1028119868 (119911) cm

(16)

Here 119868(119911) = (1+119911) int1+1199111119889119909radic23 + 1199093 we set the values119867

0=

70 kmsMpc Ω119881= 07 andΩ

119872= 03

71 URCA Reactions The neutrino luminosity of SPC ofgiven mass by modified URCA reactions with no muonsreads [75]

URCA]120576 = 38 times 10

50120576119889(119872

⊙

)

175

[erg sminus1] (17)

where 120576119889= 1198892 119877

119892and 119889 is the thickness of the protomatter

disk at the edge of even horizon The resulting total UHEneutrino flux of cooling of the SPC can be obtained as

119869URCA]120576 ≃ 522 times 10

minus8

times120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175

[erg cmminus2 sminus1 srminus1]

(18)

where the neutrino is radiated in a cone with the beamingangle 120579 sim 120576

119889≪ 1 119868(119911) = (1 + 119911) int1+119911

1119889119909radic23 + 1199093 As it is

seen the nucleon modified URCA reactions can contributeefficiently only to extragalactic objects with enough smallredshift 119911 ≪ 1

72 Pionic Reactions The pionic reactions occurring in thesuperdense protomatter medium of SPC allow both thedistorted energy and momentum to be conserved This is theanalogue of the simple URCA processes

120587minus+ 119899 997888rarr 119899 + 119890

minus+ ]

119890 120587

minus+ 119899 997888rarr 119899 + 120583

minus+ ]

120583(19)

and the two inverse processes As in the modified URCAreactions the total rate for all four processes is essentially fourtimes the rate of each reaction alone The muons are alreadypresent when pions appear The neutrino luminosity of theSPC of given mass by pionic reactions reads [75]

120587

]120576 = 578 times 1058120576119889(119872

⊙

)

175

[erg sminus1] (20)

Then the UHE neutrino total flux is

119869120587

]120576 ≃ 791120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(21)

The resulting total energy-loss rate will then be dramaticallylarger due to the pionic reactions (19) rather than themodified URCA processes

73 Quark Reactions In the superdense protomattermediumthe distorted quark Fermi energies are far below the charmedc- t- and b-quark production thresholds Therefore onlyup- down- and strange quarks are presentThe120573 equilibriumis maintained by reactions like

119889 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119889 + ]

119890 (22)

119904 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119904 + ]

119890 (23)

which are 120573 decay and its inverse These reactions constitutesimple URCA processes in which there is a net loss of a ]

119897]119897

pair at nonzero temperatures In this application a sufficientaccuracy is obtained by assuming 120573-equilibrium and thatthe neutrinos are not retained in the medium of Λ-likeprotomatter The quark reactions (22) and (23) proceed atequal rates in 120573 equilibrium where the participating quarksmust reside close to their Fermi surface Hence the totalenergy of flux due to simple URCA processes is rather twicethan that of (22) or (23) alone For example the spectral fluxesof theUHEantineutrinos andneutrinos for different redshiftsfrom quark reactions are plotted respectively in Figures 4and 5 [75] The total flux of UHE neutrino can be written as

119869119902

]120576 ≃ 7068120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(24)

8 Simulation

For simulation we use the data of AGNBH mass andluminosity estimates for 377 black holes presented by [5]These masses are mostly based on the virial assumption forthe broad emission lines with the broad-line region size


5E6

3E7

2E7

25E7

1E7

15E7

0

y2 = E100ZeV y2 = E100ZeV

z = 001

z = 007

z = 002

z = 07

z = 05

z = 01

z = 003

z = 005

10000

20000

30000

40000

50000

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1 srminus

1 )

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1srminus

1 )

00

4 8 12 16 200 4 8 12 16

Figure 4 The spectral fluxes of UHE antineutrinos for different redshifts from quark reactions

5E7

2E8

2E5

1E8

15E8

16E5

12E5

28E5

24E5

00 4 8 12 16


(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

00

4 8 12 16 20

z = 001

z = 002

z = 003

z = 005

40000

80000

z = 007

z = 07

z = 05

z = 01

Figure 5 The spectral fluxes of UHE neutrinos for different redshifts from quark reactions

determined from either reverberation mapping or opticalluminosity Additional black hole mass estimates based onproperties of the host galaxy bulges either using the observedstellar velocity dispersion or using the fundamental planerelation Since the aim is to have more than a thousand ofrealizations each individual run is simplified with a useof previous algorithm of the SPC-configurations [71] as aworking model given in Appendix G Computing the cor-responding PRTs seed black hole intermediate masses andtotal neutrino fluxes a main idea comes to solving an inverseproblem Namely by the numerous reiterating integrations ofthe state equations of SPC-configurationswe determine those

required central values of particle concentration (0) and ID-field119909(0) for which the integrated totalmass of configurationhas to be equal to the black hole mass 119872BH given fromobservations Along with all integral characteristics theradius119877

119889is also computed which is further used in (10) (14)

(15) (18) (21) and (24) for calculating119872SeedBH 119879BH 119911

Seed and119869119894

]120576 respectivelyThe results are summed up in Tables 1 2 3 4and 5 Figure 6 gives the intermediate seed masses119872Seed

BH 119872⊙

versus the present masses 119872BH119872⊙of 337 black holes on

logarithmic scales For the present masses119872BH119872⊙≃ 11 times

106 to 13 times 1010 the computed intermediate seed masses


6 7 8 9 10 11

log (MBHM⊙)

1

2

3

4

5

6

log

(MSe

edBH

M⊙)

Figure 6 The 119872SeedBH 119872⊙

-119872BH119872⊙relation on logarithmic scales

of 337 black holes from [5] The solid line is the best fit to data ofsamples

are ranging from 119872SeedBH 119872⊙

≃ 264 to 29 times 105 Thecomputed neutrino fluxes are ranging from (1) (quarkreactions)mdash119869119902]120576120576119889 [erg cm

minus2 sminus1 srminus1] ≃ 829times10minus16 to 318times10

minus4 with the average 119869119902]120576 ≃ 553times10minus10120576119889[erg cmminus2 sminus1 srminus1]

(2) (pionic reactions)mdash119869120587]120576 ≃ 0112119869119902

]120576 with the average119869120587

]120576 ≃ 366 times 10minus11120576119889[erg cmminus2 sminus1 srminus1] and (3) (modified

URCA processes)mdash119869URCA]120576 ≃ 739times 10minus11119869119902

]120576 with the average119869URCA]120576 ≃ 241 times 10

minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance

the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576

119889= 1198892 119903

119892≪ 1

To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576

119889we may estimate

120576119889≃ 169times10

minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872

⊙(2 119877

119892= 59times10

14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important

topic for another investigation elsewhere

9 Conclusions

The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed

BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed

BH 119872⊙versus the

present masses 119872BH119872⊙of 337 black holes on logarithmic

scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the

119872BH119872⊙≃ 11 times 10

6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon

Appendices

A Outline of the Key Points of ProposedGravitation Theory at Huge Energies

Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the


Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm

minus2 sminus1 srminus1)

Name 119911 Type log 119871bol log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587

NGC 1068 0004 SY2 4498 minus759201 723 25922 802 0006 minus549 minus1562 minus644NGC 4258 0001 SY2 4345 minus798201 762 29822 1033 0148 minus497 minus1510 minus592

Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm



119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587

3C 120 0033 SY1 4534 minus778201 742 27822 804 0034 minus769 minus1782 minus8643C 3903 0056 SY1 4488 minus891201 855 39122 1076 0103 minus1015 minus2028 minus1110Akn 120 0032 SY1 4491 minus863201 827 36322 1017 0055 minus915 minus1928 minus1010F9 0047 SY1 4523 minus827201 791 32722 913 0052 minus887 minus1900 minus982IC 4329A 0016 SY1 4478 minus713201 677 21322 730 0017 minus591 minus1604 minus686Mrk 79 0022 SY1 4457 minus822201 786 32222 969 0041 minus810 minus1823 minus905Mrk 110 0035 SY1 4471 minus718201 682 21822 747 0036 minus669 minus1682 minus764Mrk 335 0026 SY1 4469 minus705201 669 20522 723 0027 minus620 minus1633 minus715Mrk 509 0034 SY1 4503 minus822201 786 32222 923 0041 minus849 minus1862 minus944Mrk 590 0026 SY1 4463 minus756201 720 25622 831 0030 minus709 minus1722 minus804Mrk 817 0032 SY1 4499 minus796201 760 29622 875 0036 minus798 minus1811 minus893NGC 3227 0004 SY1 4386 minus800201 764 30022 996 0064 minus621 minus1634 minus716NGC 3516 0009 SY1 4429 minus772201 736 27222 897 0021 minus643 minus1656 minus738NGC 3783 0010 SY1 4441 minus730201 694 23022 801 0013 minus579 minus1592 minus674NGC 4051 0002 SY1 4356 minus649201 613 14922 724 0006 minus296 minus1310 minus391NGC 4151 0003 SY1 4373 minus749201 713 24922 907 0028 minus507 minus1520 minus602NGC 4593 0009 SY1 4409 minus727201 691 22722 827 0016 minus564 minus1577 minus659NGC 5548 0017 SY1 4483 minus839201 803 33922 977 0033 minus816 minus1830 minus911NGC 7469 0016 SY1 4528 minus720201 684 22022 694 0016 minus603 minus1616 minus698PG 0026 + 129 0142 RQQ 4539 minus794201 758 29422 831 0144 minus935 minus1948 minus1030PG 0052 + 251 0155 RQQ 4593 minus877201 841 37722 943 0158 minus1089 minus2102 minus1184PG 0804 + 761 0100 RQQ 4593 minus860201 824 36022 909 0102 minus1016 minus2029 minus1111PG 0844 + 349 0064 RQQ 4536 minus774201 738 27422 794 0065 minus823 minus1836 minus918PG 0953 + 414 0239 RQQ 4616 minus860201 824 36022 886 0240 minus1104 minus2117 minus1199PG 1211 + 143 0085 RQQ 4581 minus785201 749 28522 771 0085 minus869 minus1882 minus964PG 1229 + 204 0064 RQQ 4501 minus892201 856 39222 1065 0099 minus1029 minus2042 minus1124PG 1307 + 085 0155 RQQ 4583 minus826201 790 32622 851 0156 minus999 minus2012 minus1094PG 1351 + 640 0087 RQQ 4550 minus884201 848 38422 1000 0097 minus1044 minus2057 minus1139PG 1411 + 442 0089 RQQ 4558 minus793201 757 29322 810 0090 minus887 minus1900 minus982PG 1426 + 015 0086 RQQ 4519 minus828201 792 32822 919 0091 minus945 minus1958 minus1040PG 1613 + 658 0129 RQQ 4566 minus898201 862 39822 1012 0138 minus1107 minus2120 minus1202PG 1617 + 175 0114 RQQ 4552 minus824201 788 32422 878 0116 minus965 minus1978 minus1060PG 1700 + 518 0292 RQQ 4656 minus867201 831 36722 860 0293 minus1138 minus2151 minus1233PG 2130 + 099 0061 RQQ 4547 minus810201 774 31022 855 0063 minus881 minus1894 minus976PG 1226 + 023 0158 RLQ 4735 minus758201 722 25822 563 0158 minus882 minus1895 minus977PG 1704 + 608 0371 RLQ 4633 minus859201 823 35922 867 0372 minus1150 minus2164 minus1245


Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm



119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587

Mrk 841 0036 SY1 4584 minus846201 810 34622 890 0038 minus896 minus1909 minus991NGC 4253 0013 SY1 4440 minus690201 654 19022 722 0014 minus532 minus1545 minus627NGC 6814 0005 SY1 4392 minus764201 728 26422 918 0028 minus578 minus1591 minus6730054 + 144 0171 RQQ 4547 minus926201 890 42622 1087 0198 minus1184 minus2197 minus12790157 + 001 0164 RQQ 4562 minus806201 770 30622 832 0165 minus970 minus1983 minus10650204 + 292 0109 RQQ 4505 minus703201 667 20322 683 0109 minus749 minus1762 minus8440205 + 024 0155 RQQ 4545 minus822201 786 32222 881 0158 minus992 minus2005 minus10870244 + 194 0176 RQQ 4551 minus839201 803 33922 909 0179 minus1035 minus2048 minus11300923 + 201 0190 RQQ 4622 minus930201 894 43022 1020 0195 minus1202 minus2215 minus12971012 + 008 0185 RQQ 4551 minus815201 779 31522 861 0187 minus998 minus2011 minus10931029 minus 140 0086 RQQ 4603 minus944201 908 44422 1067 0097 minus1148 minus2161 minus12431116 + 215 0177 RQQ 4602 minus857201 821 35722 894 0179 minus1067 minus2080 minus11621202 + 281 0165 RQQ 4539 minus865201 829 36522 973 0173 minus1074 minus2087 minus11691309 + 355 0184 RQQ 4539 minus836201 800 33622 891 0186 minus1034 minus2047 minus11291402 + 261 0164 RQQ 4513 minus765201 729 26522 799 0165 minus898 minus1911 minus9931444 + 407 0267 RQQ 4593 minus842201 806 34222 873 0268 minus1084 minus2097 minus11791635 + 119 0146 RQQ 4513 minus846201 810 34622 961 0155 minus1028 minus2041 minus11230022 minus 297 0406 RLQ 4498 minus827201 791 32722 938 0414 minus1105 minus2118 minus12000024 + 348 0333 RLQ 4531 minus673201 637 17322 597 0333 minus813 minus1826 minus9080056 minus 001 0717 RLQ 4654 minus907201 871 40722 942 0718 minus1313 minus2326 minus14080110 + 495 0395 RLQ 4578 minus870201 834 37022 944 0399 minus1177 minus2190 minus12720114 + 074 0343 RLQ 4402 minus716201 680 21622 812 0349 minus891 minus1904 minus9860119 + 041 0637 RLQ 4557 minus874201 838 37422 973 0643 minus1241 minus2254 minus13360133 + 207 0425 RLQ 4583 minus988201 952 48822 1175 0474 minus1392 minus2405 minus14870133 + 476 0859 RLQ 4669 minus909201 873 40922 931 0860 minus1340 minus2353 minus14350134 + 329 0367 RLQ 4644 minus910201 874 41022 958 0369 minus1238 minus2252 minus13330135 minus 247 0831 RLQ 4664 minus949201 913 44922 1016 0834 minus1406 minus2419 minus15010137 + 012 0258 RLQ 4522 minus893201 857 39322 1046 0280 minus1170 minus2183 minus12650153 minus 410 0226 RLQ 4474 minus792201 756 29222 892 0233 minus979 minus1992 minus10740159 minus 117 0669 RLQ 4684 minus963201 927 46322 1024 0672 minus1403 minus2416 minus14980210 + 860 0186 RLQ 4492 minus690201 654 19022 670 0186 minus780 minus1793 minus8750221 + 067 0510 RLQ 4494 minus765201 729 26522 818 0512 minus1023 minus2036 minus11180237 minus 233 2224 RLQ 4772 minus888201 852 38822 786 2224 minus1439 minus2452 minus15340327 minus 241 0888 RLQ 4601 minus896201 860 39622 973 0892 minus1322 minus2335 minus14170336 minus 019 0852 RLQ 4632 minus934201 898 43422 1018 0857 minus1383 minus2396 minus14780403 minus 132 0571 RLQ 4647 minus943201 907 44322 1021 0575 minus1348 minus2361 minus14430405 minus 123 0574 RLQ 4740 minus983201 947 48322 1008 0575 minus1419 minus2432 minus15140420 minus 014 0915 RLQ 4700 minus939201 903 43922 960 0916 minus1401 minus2414 minus14960437 + 785 0454 RLQ 4615 minus915201 879 41522 997 0458 minus1272 minus2285 minus13670444 + 634 0781 RLQ 4612 minus889201 853 38922 948 0784 minus1293 minus2306 minus13880454 minus 810 0444 RLQ 4532 minus849201 813 34922 948 0450 minus1154 minus2167 minus12490454 + 066 0405 RLQ 4512 minus778201 742 27822 826 0407 minus1019 minus2032 minus11140502 + 049 0954 RLQ 4636 minus924201 888 42422 994 0957 minus1380 minus2393 minus14750514 minus 459 0194 RLQ 4536 minus791201 755 29122 828 0196 minus961 minus1974 minus10560518 + 165 0759 RLQ 4634 minus889201 853 38922 926 0761 minus1289 minus2302 minus13840538 + 498 0545 RLQ 4643 minus994201 958 49422 1127 0559 minus1432 minus2445 minus15270602 minus 319 0452 RLQ 4569 minus938201 902 43822 1089 0473 minus1311 minus2325 minus1407


Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587

0607 minus 157 0324 RLQ 4630 minus904201 868 40422 960 0326 minus1214 minus2227 minus13090637 minus 752 0654 RLQ 4716 minus977201 941 47722 1020 0656 minus1424 minus2437 minus15190646 + 600 0455 RLQ 4558 minus910201 874 41022 1044 0469 minus1263 minus2276 minus13580723 + 679 0846 RLQ 4641 minus903201 867 40322 947 0848 minus1327 minus2341 minus14230736 + 017 0191 RLQ 4641 minus836201 800 33622 857 0192 minus1038 minus2051 minus11330738 + 313 0631 RLQ 4694 minus976201 940 47622 1040 0634 minus1418 minus2431 minus15130809 + 483 0871 RLQ 4654 minus832201 796 33222 792 0871 minus1207 minus2220 minus13020838 + 133 0684 RLQ 4623 minus888201 852 38822 935 0686 minus1274 minus2287 minus13690906 + 430 0668 RLQ 4599 minus826201 790 32622 835 0669 minus1163 minus2176 minus12580912 + 029 0427 RLQ 4526 minus808201 772 30822 872 0430 minus1077 minus2090 minus11720921 minus 213 0052 RLQ 4463 minus850201 814 35022 1019 0084 minus936 minus1950 minus10320923 + 392 0698 RLQ 4626 minus964201 928 46422 1084 0708 minus1410 minus2423 minus15050925 minus 203 0348 RLQ 4635 minus882201 846 38222 911 0349 minus1183 minus2197 minus12780953 + 254 0712 RLQ 4659 minus936201 900 43622 995 0715 minus1363 minus2376 minus14580954 + 556 0901 RLQ 4654 minus843201 807 34322 814 0901 minus1231 minus2244 minus13261004 + 130 0240 RLQ 4621 minus946201 910 44622 1053 0248 minus1255 minus2268 minus13501007 + 417 0612 RLQ 4671 minus915201 879 41522 941 0613 minus1308 minus2321 minus14031016 minus 311 0794 RLQ 4663 minus925201 889 42522 969 0796 minus1358 minus2371 minus14531020 minus 103 0197 RLQ 4487 minus872201 836 37222 1039 0228 minus1104 minus2118 minus11991034 minus 293 0312 RLQ 4620 minus911201 875 41122 984 0316 minus1222 minus2235 minus13171036 minus 154 0525 RLQ 4455 minus816201 780 31622 959 0543 minus1116 minus2129 minus12111045 minus 188 0595 RLQ 4580 minus719201 683 21922 640 0595 minus961 minus1974 minus10561100 + 772 0311 RLQ 4649 minus967201 931 46722 1067 0318 minus1320 minus2333 minus14151101 minus 325 0355 RLQ 4633 minus897201 861 39722 943 0357 minus1212 minus2225 minus13071106 + 023 0157 RLQ 4497 minus786201 750 28622 857 0160 minus931 minus1944 minus10261107 minus 187 0497 RLQ 4425 minus726201 690 22622 809 0501 minus952 minus1965 minus10471111 + 408 0734 RLQ 4626 minus1018201 982 51822 1192 0770 minus1511 minus2524 minus16061128 minus 047 0266 RLQ 4408 minus708201 672 20822 790 0270 minus849 minus1862 minus9441136 minus 135 0554 RLQ 4678 minus914201 878 41422 932 0555 minus1294 minus2307 minus13891137 + 660 0656 RLQ 4685 minus972201 936 47222 1041 0659 minus1416 minus2429 minus15111150 + 497 0334 RLQ 4598 minus909201 873 40922 1002 0340 minus1226 minus2239 minus13211151 minus 348 0258 RLQ 4556 minus938201 902 43822 1102 0287 minus1226 minus2239 minus13211200 minus 051 0381 RLQ 4641 minus877201 841 37722 895 0382 minus1226 minus2239 minus13211202 minus 262 0789 RLQ 4581 minus936201 900 43622 1073 0804 minus1376 minus2389 minus14711217 + 023 0240 RLQ 4583 minus877201 841 37722 953 0244 minus1134 minus2147 minus12291237 minus 101 0751 RLQ 4663 minus964201 928 46422 1047 0755 minus1419 minus2432 minus15141244 minus 255 0633 RLQ 4648 minus940201 904 44022 1014 0637 minus1355 minus2369 minus14511250 + 568 0321 RLQ 4561 minus878201 842 37822 977 0327 minus1167 minus2181 minus12621253 minus 055 0536 RLQ 4610 minus879201 843 37922 930 0538 minus1228 minus2242 minus13241254 minus 333 0190 RLQ 4552 minus919201 883 41922 1068 0210 minus1183 minus2196 minus12781302 minus 102 0286 RLQ 4586 minus866201 830 36622 928 0289 minus1134 minus2147 minus12291352 minus 104 0332 RLQ 4581 minus851201 815 35122 903 0334 minus1124 minus2137 minus12191354 + 195 0720 RLQ 4711 minus980201 944 48022 1031 0722 minus1442 minus2455 minus15371355 minus 416 0313 RLQ 4648 minus1009201 973 50922 1152 0331 minus1394 minus2407 minus14891359 minus 281 0803 RLQ 4619 minus843201 807 34322 849 0804 minus1216 minus2229 minus13111450 minus 338 0368 RLQ 4394 minus682201 646 18222 752 0371 minus840 minus1853 minus9351451 minus 375 0314 RLQ 4616 minus918201 882 41822 1002 0319 minus1235 minus2248 minus13301458 + 718 0905 RLQ 4693 minus934201 898 43422 957 0906 minus1391 minus2404 minus14861509 + 022 0219 RLQ 4454 minus835201 799 33522 998 0247 minus1051 minus2064 minus11461510 minus 089 0361 RLQ 4638 minus901201 865 40122 946 0363 minus1221 minus2234 minus1316


Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587

1545 + 210 0266 RLQ 4586 minus929201 893 42922 1054 0278 minus1236 minus2249 minus13311546 + 027 0412 RLQ 4600 minus908201 872 40822 998 0417 minus1248 minus2261 minus13431555 minus 140 0097 RLQ 4494 minus761201 725 26122 810 0099 minus839 minus1853 minus9341611 + 343 1401 RLQ 4699 minus993201 957 49322 1069 1405 minus1554 minus2567 minus16491634 + 628 0988 RLQ 4547 minus764201 728 26422 763 0989 minus1105 minus2118 minus12001637 + 574 0750 RLQ 4668 minus954201 918 45422 1022 0753 minus1401 minus2414 minus14961641 + 399 0594 RLQ 4689 minus978201 942 47822 1049 0597 minus1414 minus2427 minus15091642 + 690 0751 RLQ 4578 minus812201 776 31222 828 0752 minus1153 minus2166 minus12481656 + 053 0879 RLQ 4721 minus998201 962 49822 1057 0882 minus1499 minus2512 minus15941706 + 006 0449 RLQ 4401 minus699210 663 29922 779 0453 minus892 minus1906 minus9871721 + 343 0206 RLQ 4563 minus840201 804 34022 899 0209 minus1053 minus2066 minus11481725 + 044 0293 RLQ 4607 minus843201 807 34322 861 0294 minus1096 minus2109 minus1191

spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory

A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872

4rarr M

4 written

in terms of the world-deformation tensor (Ω) the general(M

4) and flat (119872

4) smooth differential 4D-manifolds The

following notational conventions will be used throughoutthe appendices All magnitudes related to the space M

4

will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M

4and the second half of Latin alphabet

(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872

4The tensorΩ can be written in the formΩ = (Ω119898

119897=

119898

120583120583

119897) where the DC-members are the invertible distortion

matrix (119898

120583) and the tensor (120583

119897equiv 120597

119897120583 and 120597

119897= 120597120597119909

119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890

119898at

given point (119901 isin 1198724) undergo the distortion transformations

by means of and (2) the diffeomorphism 120583(119909) 1198724rarr

4is constructed by seeking new holonomic coordinates

120583(119909) as the solutions of the first-order partial differential

equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)

where the conditions of integrability 120597119896120595120583

119897= 120597

119897120595120583

119896 and

nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal

displacement 119889120583 on the M4in terms of the spacetime

structures of1198724

119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)

A deformation Ω 1198724rarr M

4comprises the following

two 4D deformations∘

Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881

4is the semi-Riemannian space and

∘

Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩

A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader


Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm



119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587

NGC 1566 0005 SY1 4445 minus728201 692 22822 793 0008 minus515 minus1528 minus610NGC 2841 0002 SY1 4367 minus857201 821 35722 1129 0347 minus660 minus1674 minus755NGC 3982 0004 SY1 4354 minus645201 609 145220 718 0008 minus350 minus1363 minus445NGC 3998 0003 SY1 4354 minus931201 895 43122 1290 2561 minus825 minus1838 minus920Mrk 10 0029 SY1 4461 minus783291 747 47908 887 0036 minus766 minus1779 minus861UGC 3223 0016 SY1 4427 minus738201 702 23822 831 0022 minus634 minus1647 minus729NGC 513 0002 SY2 4252 minus801201 765 30122 1132 1345 minus562 minus1576 minus657NGC 788 0014 SY2 4433 minus787201 751 28722 923 0029 minus708 minus1721 minus803NGC 1052 0005 SY2 4384 minus855201 819 35522 1108 0228 minus737 minus1750 minus832NGC 1275 0018 SY2 4504 minus887201 851 38722 1052 0047 minus905 minus1919 minus1001NGC 1320 0009 SY2 4402 minus754201 718 25422 888 0023 minus612 minus1625 minus707NGC 1358 0013 SY2 4437 minus824201 788 32422 993 0045 minus766 minus1780 minus862NGC 1386 0003 SY2 4338 minus760201 724 26022 964 0075 minus0020 minus1539 minus621NGC 1667 0015 SY2 4469 minus824201 788 32422 961 0030 minus779 minus1792 minus874NGC 2110 0008 SY2 4410 minus866201 830 36622 1104 0166 minus797 minus1810 minus892NGC 2273 0006 SY2 4405 minus766201 730 26622 909 0024 minus597 minus1610 minus692NGC 2992 0008 SY2 4392 minus808201 772 30822 1006 0071 minus696 minus1709 minus791NGC 3185 0004 SY2 4308 minus642201 606 14222 758 0014 minus345 minus1358 minus440NGC 3362 0028 SY2 4427 minus713201 677 21322 781 0031 minus640 minus1654 minus736NGC 3786 0009 SY2 4347 minus789201 753 28922 1013 0123 minus673 minus1686 minus768NGC 4117 0003 SY2 4364 minus719201 683 21922 856 0018 minus454 minus1467 minus549NGC 4339 0004 SY2 4338 minus776201 740 27622 996 0108 minus579 minus1592 minus674NGC 5194 0002 SY2 4379 minus731201 695 23122 865 0016 minus440 minus1453 minus535NGC 5252 0023 SY2 4539 minus840201 804 34022 923 0027 minus845 minus1858 minus940NGC 5273 0004 SY2 4303 minus687201 651 18722 853 0034 minus423 minus1436 minus518NGC 5347 0008 SY2 4381 minus715201 679 21522 831 0018 minus533 minus1546 minus628NGC 5427 0009 SY2 4412 minus675201 639 17522 720 0011 minus473 minus1486 minus568NGC 5929 0008 SY2 4304 minus761201 725 26122 1000 0169 minus613 minus1627 minus709NGC 5953 0007 SY2 4405 minus730201 694 23022 837 0015 minus548 minus1561 minus643NGC 6104 0028 SY2 4360 minus796201 760 29622 1014 0128 minus786 minus1799 minus881NGC 7213 0006 SY2 4430 minus835201 799 33522 1022 0055 minus718 minus1731 minus813NGC 7319 0023 SY2 4419 minus774201 738 27422 911 0038 minus730 minus1743 minus825NGC 7603 0030 SY2 4466 minus844201 808 34422 1004 0056 minus876 minus1889 minus971NGC 7672 0013 SY2 4386 minus724201 688 22422 844 0023 minus591 minus1605 minus687NGC 7682 0017 SY2 4393 minus764201 728 26422 917 0039 minus685 minus1698 minus780NGC 7743 0006 SY2 4360 minus695201 659 19522 812 0016 minus473 minus1486 minus568Mrk 1 0016 SY2 4420 minus752201 716 25222 866 0025 minus659 minus1672 minus754Mrk 3 0014 SY2 4454 minus901201 865 40122 1130 0142 minus908 minus1921 minus1003Mrk 78 0037 SY2 4459 minus823201 787 32322 969 0056 minus858 minus1871 minus953Mrk 270 0010 SY2 4337 minus796201 760 29622 1037 0179 minus694 minus1707 minus789Mrk 348 0015 SY2 4427 minus757201 721 25722 869 0024 minus662 minus1675 minus757Mrk 533 0029 SY2 4515 minus792201 756 29222 851 0032 minus782 minus1795 minus877Mrk 573 0017 SY2 4444 minus764201 728 26422 866 0024 minus685 minus1698 minus780Mrk 622 0023 SY2 4452 minus728201 692 22822 786 0026 minus649 minus1662 minus744Mrk 686 0014 SY2 4411 minus792201 756 29222 955 0042 minus717 minus1730 minus812Mrk 917 0024 SY2 4475 minus798201 762 29822 903 0031 minus775 minus1789 minus870Mrk 1018 0042 SY2 4439 minus845201 809 34522 1033 0092 minus908 minus1921 minus1003


Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587

Mrk 1040 0017 SY2 4453 minus800201 764 30022 929 0030 minus748 minus1761 minus843Mrk 1066 0012 SY2 4455 minus737201 701 23722 801 0015 minus607 minus1620 minus702Mrk 1157 0015 SY2 4427 minus719201 683 21922 793 0019 minus595 minus1608 minus690Akn 79 0018 SY2 4524 minus790201 754 29022 838 0020 minus736 minus1749 minus831Akn 347 0023 SY2 4484 minus836201 800 33622 970 0037 minus838 minus1851 minus933IC 5063 0011 SY2 4453 minus810201 774 31022 949 0027 minus727 minus1740 minus822II ZW55 0025 SY2 4454 minus859201 823 35922 1046 0074 minus886 minus1899 minus981F 341 0016 SY2 4413 minus751201 715 25122 871 0026 minus657 minus1670 minus752UGC 3995 0016 SY2 4439 minus805201 769 30522 953 0036 minus752 minus1765 minus847UGC 6100 0029 SY2 4448 minus806201 770 30622 946 0046 minus806 minus1820 minus9011ES 1959 + 65 0048 BLL mdash minus1039501 809 34522 779 0052 minus920 minus1934 minus1015Mrk 180 0045 BLL mdash minus1051501 821 35722 791 0051 minus935 minus1949 minus1031Mrk 421 0031 BLL mdash minus1059501 829 36522 799 0038 minus916 minus1929 minus1011Mrk 501 0034 BLL mdash minus1151501 921 45722 891 0092 minus1085 minus2098 minus1180I Zw 187 0055 BLL mdash minus1016501 786 32222 756 0058 minus893 minus1906 minus9883C 371 0051 BLL mdash minus1081501 851 38722 821 0063 minus999 minus2013 minus10951514 minus 241 0049 BLL mdash minus1040501 810 34622 780 0054 minus924 minus1937 minus10190521 minus 365 0055 BLL mdash minus1095501 865 40122 835 0071 minus1031 minus2044 minus11260548 minus 322 0069 BLL mdash minus1045501 815 35122 785 0074 minus965 minus1978 minus10600706 + 591 0125 BLL mdash minus1056501 826 36222 796 0132 minus1041 minus2054 minus11362201 + 044 0027 BLL mdash minus1040501 810 34622 780 0032 minus870 minus1883 minus9652344 + 514 0044 BLL mdash minus1110501 880 41622 850 0067 minus1037 minus2050 minus11323C 29 0045 RG mdash minus1050501 820 35622 790 0051 minus934 minus1947 minus10293C 31 0017 RG mdash minus1080501 850 38622 820 0028 minus899 minus1912 minus9943C 33 0059 RG mdash minus1068501 838 37422 808 0068 minus990 minus2003 minus10853C 40 0018 RG mdash minus1016501 786 32222 756 0021 minus792 minus1805 minus8873C 62 0148 RG mdash minus1097501 867 40322 837 0165 minus1129 minus2143 minus12253C 761 0032 RG mdash minus1043501 813 34922 783 0037 minus890 minus1904 minus9863C 78 0029 RG mdash minus1090501 860 39622 830 0043 minus964 minus1977 minus10593C 84 0017 RG mdash minus1079501 849 38522 819 0028 minus897 minus1910 minus9923C 88 0030 RG mdash minus1033501 803 33922 773 0034 minus867 minus1880 minus9623C 89 0139 RG mdash minus1082501 852 38822 822 0151 minus1097 minus2110 minus11923C 98 0031 RG mdash minus1018501 788 32422 758 0034 minus844 minus1857 minus9393C 120 0033 RG mdash minus1043501 813 34922 783 0038 minus893 minus1906 minus9883C 192 0060 RG mdash minus1036501 806 34222 776 0064 minus936 minus1949 minus10313C 1961 0198 RG mdash minus1051501 821 35722 791 0204 minus1079 minus2092 minus11743C 223 0137 RG mdash minus1045501 815 35122 785 0142 minus1031 minus2044 minus11263C 293 0045 RG mdash minus1029501 799 33522 769 0048 minus897 minus1910 minus9923C 305 0041 RG mdash minus1022501 792 32822 762 0044 minus876 minus1889 minus9713C 338 0030 RG mdash minus1108501 878 41422 848 0052 minus998 minus2012 minus10933C 388 0091 RG mdash minus1148501 918 45422 888 0145 minus1171 minus2184 minus12663C 444 0153 RG mdash minus998501 768 30422 738 0155 minus960 minus1973 minus10553C 449 0017 RG mdash minus1063501 833 36922 803 0025 minus869 minus1882 minus964gin 116 0033 RG mdash minus1105501 875 41122 845 0053 minus1002 minus2015 minus1097NGC 315 0017 RG mdash minus1120501 890 42622 860 0045 minus969 minus1982 minus1064NGC 507 0017 RG mdash minus1130501 900 43622 870 0053 minus986 minus1999 minus1081NGC 708 0016 RG mdash minus1076501 846 38222 816 0026 minus886 minus1899 minus981NGC 741 0018 RG mdash minus1102501 872 40822 842 0037 minus942 minus1955 minus1037NGC 4839 0023 RG mdash minus1078501 848 38422 818 0034 minus922 minus1935 minus1017NGC 4869 0023 RG mdash minus1042501 812 34822 782 0028 minus859 minus1872 minus954


Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587

NGC 4874 0024 RG mdash minus1093501 863 39922 833 0039 minus952 minus1965 minus1047NGC 6086 0032 RG mdash minus1126501 896 43222 866 0065 minus1036 minus2049 minus1131NGC 6137 0031 RG mdash minus1111501 881 41722 851 0054 minus1007 minus2020 minus1102NGC 7626 0025 RG mdash minus1127501 897 43322 867 0058 minus1015 minus2028 minus11100039 minus 095 0000 RG mdash minus1102501 872 40822 842 0019 minus289 minus1302 minus3840053 minus 015 0038 RG mdash minus1112501 882 41822 852 0062 minus1027 minus2040 minus11220053 minus 016 0043 RG mdash minus1081501 851 38722 821 0055 minus984 minus1997 minus10790055 minus 016 0045 RG mdash minus1115501 885 42122 855 0070 minus1047 minus2061 minus11430110 + 152 0044 RG mdash minus1039501 809 34522 779 0048 minus912 minus1926 minus10070112 minus 000 0045 RG mdash minus1083501 853 38922 823 0057 minus991 minus2005 minus10870112 + 084 0000 RG mdash minus1148501 918 45422 888 0054 minus370 minus1383 minus4650147 + 360 0018 RG mdash minus1076501 846 38222 816 0028 minus370 minus1383 minus4650131 minus 360 0030 RG mdash minus1083501 853 38922 823 0042 minus955 minus1968 minus10500257 minus 398 0066 RG mdash minus1059501 829 36522 799 0073 minus985 minus1998 minus10800306 + 237 0000 RG mdash minus1081501 851 38722 821 0012 minus252 minus1266 minus3480312 minus 343 0067 RG mdash minus1087501 857 39322 827 0080 minus1035 minus2048 minus11300325 + 024 0030 RG mdash minus1059501 829 36522 799 0037 minus913 minus1926 minus10080431 minus 133 0033 RG mdash minus1095501 865 40122 835 0049 minus984 minus1997 minus10790431 minus 134 0035 RG mdash minus1061501 831 36722 801 0042 minus930 minus1943 minus10250449 minus 175 0031 RG mdash minus1002501 772 30822 742 0033 minus816 minus1829 minus9110546 minus 329 0037 RG mdash minus1159501 929 46522 899 0107 minus1107 minus2120 minus12020548 minus 317 0034 RG mdash minus958501 728 26422 698 0035 minus747 minus1760 minus8420634 minus 206 0056 RG mdash minus1039501 809 34522 779 0060 minus935 minus1948 minus10300718 minus 340 0029 RG mdash minus1131501 901 43722 871 0066 minus1036 minus2049 minus11310915 minus 118 0054 RG mdash minus1099501 869 40522 839 0072 minus1036 minus2049 minus11310940 minus 304 0038 RG mdash minus1159501 929 46522 899 0108 minus1109 minus2122 minus12041043 minus 290 0060 RG mdash minus1067501 837 37322 807 0068 minus990 minus2003 minus10851107 minus 372 0010 RG mdash minus1111501 881 41722 851 0033 minus906 minus1919 minus10011123 minus 351 0032 RG mdash minus1183501 953 48922 923 0153 minus1135 minus2149 minus12311258 minus 321 0015 RG mdash minus1091501 861 39722 831 0030 minus907 minus1920 minus10021333 minus 337 0013 RG mdash minus1107501 877 41322 847 0034 minus922 minus1935 minus10171400 minus 337 0014 RG mdash minus1119501 889 42522 859 0042 minus950 minus1963 minus10451404 minus 267 0022 RG mdash minus1111505 881 48798 851 0045 minus976 minus1989 minus10711510 + 076 0053 RG mdash minus1133501 903 43922 873 0091 minus1094 minus2107 minus11891514 + 072 0035 RG mdash minus1095501 865 40122 835 0051 minus990 minus2003 minus10851520 + 087 0034 RG mdash minus1059501 829 36522 799 0041 minus924 minus1937 minus10191521 minus 300 0020 RG mdash minus1010501 780 31622 750 0022 minus791 minus1804 minus8861602 + 178 0041 RG mdash minus1054501 824 36022 794 0047 minus932 minus1945 minus10271610 + 296 0032 RG mdash minus1126501 896 43222 866 0065 minus1036 minus2049 minus11312236 minus 176 0070 RG mdash minus1079501 849 38522 819 0081 minus1025 minus2039 minus11202322 + 143 0045 RG mdash minus1047501 817 35322 787 0050 minus928 minus1942 minus10242322 minus 122 0082 RG mdash minus1063501 833 36922 803 0090 minus1012 minus2025 minus11072333 minus 327 0052 RG mdash minus1095501 865 40122 835 0068 minus1026 minus2039 minus11212335 + 267 0030 RG mdash minus1138501 908 44422 878 0073 minus1051 minus2064 minus1146

is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872

4rarr 119881

4(Ω

120583

] equiv

120575120583

] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc

We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This

field takes values in the Lie algebra of the abelian group119880

loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the


Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm


Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587

0122 + 090 0339 BLL minus1112501 882 41822 852 0363 minus1243 minus2257 minus13390145 + 138 0124 BLL minus1072501 842 37822 812 0133 minus1068 minus2081 minus11630158 + 001 0229 BLL minus1038501 808 34422 778 0233 minus1071 minus2084 minus11660229 + 200 0139 BLL minus1154501 924 46022 894 0201 minus1223 minus2236 minus13180257 + 342 0247 BLL minus1096501 866 40222 836 0263 minus1181 minus2194 minus12760317 + 183 0190 BLL minus1025501 795 33122 765 0193 minus1029 minus2042 minus11240331 minus 362 0308 BLL minus1105501 875 41122 845 0328 minus1221 minus2234 minus13160347 minus 121 0188 BLL minus1095501 865 40122 835 0204 minus1150 minus2163 minus12450350 minus 371 0165 BLL minus1112501 882 41822 852 0189 minus1167 minus2180 minus12620414 + 009 0287 BLL minus1086501 856 39222 826 0300 minus1180 minus2193 minus12750419 + 194 0512 BLL minus1091501 861 39722 831 0527 minus1254 minus2268 minus13500506 minus 039 0304 BLL minus1105501 875 41122 845 0324 minus1219 minus2232 minus13140525 + 713 0249 BLL minus1133501 903 43922 873 0287 minus1246 minus2260 minus13410607 + 710 0267 BLL minus1095501 865 40122 835 0283 minus1246 minus2260 minus13410737 + 744 0315 BLL minus1124501 894 43022 864 0346 minus1256 minus2269 minus13510922 + 749 0638 BLL minus1191501 961 49722 931 0784 minus1456 minus2469 minus15510927 + 500 0188 BLL minus1064501 834 37022 804 0196 minus1096 minus2109 minus11910958 + 210 0344 BLL minus1133501 903 43922 873 0382 minus1282 minus2295 minus13771104 + 384 0031 BLL minus1169501 939 47522 909 0119 minus1108 minus2121 minus12031133 + 161 0460 BLL minus1062501 832 36822 802 0467 minus1191 minus2204 minus12861136 + 704 0045 BLL minus1125501 895 43122 865 0077 minus1065 minus2078 minus11601207 + 394 0615 BLL minus1140501 910 44622 880 0660 minus1362 minus2376 minus14581212 + 078 0136 BLL minus1129501 899 43522 869 0171 minus1177 minus2190 minus12721215 + 303 0130 BLL minus1042501 812 34822 782 0135 minus1020 minus2033 minus11151218 + 304 0182 BLL minus1088501 858 39422 828 0196 minus1135 minus2148 minus12301221 + 245 0218 BLL minus1027501 797 33322 767 0221 minus1047 minus2060 minus11421229 + 643 0164 BLL minus1171501 941 47722 911 0256 minus1269 minus2282 minus13641248 minus 296 0370 BLL minus1131501 901 43722 871 0407 minus1287 minus2300 minus13821255 + 244 0141 BLL minus1088501 858 39422 828 0155 minus1109 minus2122 minus12041407 + 595 0495 BLL minus1160501 930 46622 900 0566 minus1371 minus2384 minus14661418 + 546 0152 BLL minus1133501 903 43922 873 0190 minus1195 minus2208 minus12901426 + 428 0129 BLL minus1143501 913 44922 883 0177 minus1196 minus2209 minus12911440 + 122 0162 BLL minus1074501 844 38022 814 0172 minus1098 minus2111 minus11931534 + 014 0312 BLL minus1110501 880 41622 850 0335 minus1231 minus2244 minus13261704 + 604 0280 BLL minus1107501 877 41322 847 0301 minus1214 minus2227 minus13091728 + 502 0055 BLL minus1043501 813 34922 783 0060 minus940 minus1953 minus10351757 + 703 0407 BLL minus1105501 875 41122 845 0427 minus1252 minus2265 minus13471807 + 698 0051 BLL minus1240501 1010 54622 980 0502 minus1278 minus2291 minus13731853 + 671 0212 BLL minus1053501 823 35922 793 0218 minus1089 minus2102 minus11842005 minus 489 0071 BLL minus1133501 903 43922 873 0109 minus1121 minus2134 minus12162143 + 070 0237 BLL minus1076501 846 38222 816 0247 minus1141 minus2154 minus12362200 + 420 0069 BLL minus1053501 823 35922 793 0075 minus979 minus1992 minus10742254 + 074 0190 BLL minus1092501 862 39822 832 0205 minus1146 minus2159 minus12412326 + 174 0213 BLL minus1104501 874 41022 844 0233 minus1179 minus2192 minus12742356 minus 309 0165 BLL minus1090501 860 39622 830 0179 minus1128 minus2141 minus12230230 minus 027 0239 RG minus1027501 797 33322 767 0242 minus1056 minus2070 minus11520307 + 169 0256 RG minus1096501 866 40222 836 0272 minus1185 minus2198 minus1280


Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587

0345 + 337 0244 RG minus942501 712 24822 682 0244 minus910 minus1923 minus10050917 + 459 0174 RG minus1051501 821 35722 791 0180 minus1065 minus2078 minus11600958 + 291 0185 RG minus1023501 793 32922 763 0188 minus1023 minus2036 minus11181215 minus 033 0184 RG minus1023501 793 32922 763 0187 minus1022 minus2035 minus11171215 + 013 0118 RG minus1050501 820 35622 790 0124 minus1025 minus2038 minus11201330 + 022 0215 RG minus1012501 782 31822 752 0217 minus1019 minus2032 minus11141342 minus 016 0167 RG minus1071501 841 37722 811 0176 minus1096 minus2109 minus11912141 + 279 0215 RG minus1012501 782 31822 752 0217 minus1019 minus2032 minus11140257 + 024 0115 RQQ minus1105501 875 41122 845 0135 minus1118 minus2132 minus12131549 + 203 0250 RQQ minus922501 692 22822 662 0250 minus878 minus1891 minus9732215 minus 037 0241 RQQ minus1050501 820 35622 790 0247 minus1098 minus2111 minus11932344 + 184 0138 RQQ minus937501 707 24322 677 0138 minus842 minus1856 minus9380958 + 291 0185 RG minus1023501 793 32922 763 0188 minus1023 minus2036 minus11181215 minus 033 0184 RG minus1023501 793 32922 763 0187 minus1022 minus2035 minus11171215 + 013 0118 RG minus1050501 820 35622 790 0124 minus1025 minus2038 minus11201330 + 022 0215 RG minus1012501 782 31822 752 0217 minus1019 minus2032 minus11141342 minus 016 0167 RG minus1071501 841 37722 811 0176 minus1096 minus2109 minus11912141 + 279 0215 RG minus1012501 782 31822 752 0217 minus1019 minus2032 minus11140257 + 024 0115 RQQ minus1105501 875 41122 845 0135 minus1118 minus2132 minus12131549 + 203 0250 RQQ minus922501 692 22822 662 0250 minus878 minus1891 minus9732215 minus 037 0241 RQQ minus1050501 820 35622 790 0247 minus1098 minus2111 minus11932344 + 184 0138 RQQ minus937501 707 24322 677 0138 minus842 minus1856 minus938

differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599

119898 Weconnect the structure group 119866

119881 further to the nonlinear

realization of the Lie group119866119863of distortion of extended space

1198726(rarr

6) (E1) underlying the119872

4This extension appears

to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866

119863and its stationary subgroup 119867 =

119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields

A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]

119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)

on the space 1198726(spanned by the coordinates 120578) with

the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)

119884and

exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable

theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and

hypercharge 119884 implying 119876119889= 119879

3+ 1198842 where 119876119889 is

the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)

119884 or that coupled to the hypercharge 119884 Spontaneous

symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)

119889subgroup intact that is leaving one Goldstone

boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872

119886= 0)

which is responsible for gravitational interactions and itsmassive component (119886) (119872

119886= 0) which is responsible for

the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the


cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space

A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886

0(119903) in presence of one-dimensional space-

like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881

4rarr

2 has the fiber1198782= 119901

minus1() isin 119881

4 with a trivial connection on it where 2

is the quotient-space 1198814119878119874(3) Considering the equilibrium

configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881

4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)

where () and () ( isin 3

) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886

0) and ID (119886) fields can be given in Feynman

gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()

(Δ minus 120582minus2

119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)

where is the concentration of particles and 120582119886= ℎ119898

119886119888 ≃

04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872

4rarr 119881

4is given as 119903 = minus 119877

1198924 A distortion of the

basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909

0= 0 119909 = 0) of the space-time continuum

tan 3= minus119909

1=

1= 0 The Poincare generators 119875

120583of

translations are transformed as follows [71]

= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)

where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881

4in holonomic coordinate basis takes the form

00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)

As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872

⊙

This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted

B A Hard Look at Spacetime Deformation

The holonomic metric on M4can be recast in the form

= 120583]

120583

otimes ]= (

120583 ])

120583

otimes ] with components

120583] = (120583 ]) in dual holonomic base

120583

equiv 119889120583 In order

to relate local Lorentz symmetry to more general deformed


log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star

Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10

36[erg cmminus3

] and 120588OV = 7194 times 1015[g cmminus3

]respectively

spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M

4

has at each point a tangent space

4 spanned by the

anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (

0

3) where

119886=

119886

120583120583

We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential

forms = (0

3

) as a shorthand for the collection of the

119887

= 119887

120583119889

120583 whose values at every point form the dual

basis such that 119886rfloor

119887

= 120575119887

119886 where rfloor denotes the interior

product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω

0

withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM

4 In components we have

119886

120583119887

120583= 120575

119887

119886 On the manifold

M4 the tautological tensor field 119894 of type (1 1) can be

definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M

4and any vector

isin M

4 one has 119894() = In terms of the frame field the

119886

give the expression for 119894 as 119894 = = 0otimes

0

+ sdot sdot sdot 3otimes

3

in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation

119886

119887

will be usedbelow for general linear frames Let us introduce so-called

first deformation matrices (120587(119909)119898119896and 119886

119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896

119897 yields local tetrad deformations

119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896

isin M4The first deformationmatrices

120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field

= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)

where the second deformation matrix 120574119897119898 reads 120574

119897119898=

119900119886119887119886

119897119887

119898 The deformed metric splits as

120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)

The anholonomic orthonormal frame field relates tothe tangent space metric 119900

119886119887= diag(+ minus minusminus) by 119900

119886119887=

(119886

119887) =

120583]119886120583119887

] which has the converse 120583] = 119900119886119887

119886

120583119887

]because

119886

120583119886

] = 120575120583

] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872

4 The 120574

119897119898can be decomposed in terms of symmetric

(119886119897)

and antisymmetric [119886119897]

parts of the matrix 119886119897= 119900

119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ

119886119897is the traceless symmetric part and

119886119897is

the skew symmetric part of the first deformation matrix


The anholonomy objects defined on the tangent space

4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)

where the anholonomy coefficients 119886119887119888 which represent the

curls of the base members are

119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)

In particular case of constant metric in the tetradic space thedeformed connection can be written as

Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)

All magnitudes related to the 1198814will be denoted by an over

ldquo ∘rdquo According to (A1) we have∘

Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)

In analogy with (B1) the following relations hold∘

119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘

119889 of the displacement 119889 ∘119909120583 on 1198814can be

written in terms of the spacetime structures of1198724as

119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)

The holonomic metric can be recast in the form∘

119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)

The anholonomy objects defined on the tangent space∘

119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)

where the anholonomy coefficients∘

119862119886

119887119888 which represent the

curls of the base members are∘

119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)

The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as

∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘

120599119888is understood as the down indexed 1-form

∘

120599119888=

119900119888119887

∘

120599

119887

The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881

4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)

Under a local tetrad deformation (B20) a general spinconnection transforms according to

119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)

We have then two choices to recast metric as follows

= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)

In the first case the contribution of the Christoffel symbolsconstructed by the metric

119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)

As before the second deformation matrix 119886119887 can be

decomposed in terms of symmetric (119886119887)

and antisymmetric[119886119887]


119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887

is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as

120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)

The inverse deformed metric reads

120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886

119887 The (anholonomic) Levi-

Civita (or Christoffel) connection is

Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)

where 119888is understood as the down indexed 1-form

119888=

119900119888119887119887

Hence the usual Levi-Civita connection is related tothe original connection by the relation

Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla

120588]120590 + nabla120590120588] minus nabla]120588120590)

(B30)

where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of

120583] suchthat

120583]]120588= 120575

120588

120583 Hence the connection deformation Π120583

120588120590

acts like a force that deviates the test particles from thegeodesic motion in the space 119881

4 A metric-affine space

(4 Γ) is defined to have a metric and a linear connection

that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form

119886119887and the affine torsion 2-form

119886

representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form

119886

119887 symbolically can be presented as [96 97]

(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)

where D is the covariant exterior derivative If the nonmetric-ity tensor

120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish

the general formula for the affine connection written in thespacetime components is

Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588

120583] is the Riemann part and 120588

120583] = 2(120583])120588

+ 120588

120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588

120583] = (12)120588

120583] = Γ120588

[120583]] given with respectto a holonomic frame 119889

120588

= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the

coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be

120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887

denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880

4 is a particular case of

the general metric-affine manifold M4 restricted by the

metricity condition 120582120583] = 0 when a nonsymmetric linear

connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886

120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)

C Determination of and in StandardTheory of Gravitation

Let 119886119887 = 119886119887

120583and 119889

120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form

119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘

119877radicminus119889Ω

(C2)

is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866

119873119888

4 119878119876is

the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω

119901rarr Ω

119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(

1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899

119886119901+1 sdotsdotsdot119886119899 Here we used

the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901

= 1198861

and 1198862

and sdot sdot sdot and 119886119901 defined on the 119880

4space the

119886119894(119894 = 119901 + 1 119899) are understood as the down indexed

1-forms 119886119894= 119900

119886119894119887119887

and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields

120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)

The variation of the action describing themacroscopicmattersources

119898with respect to the coframe 120599119886 and connection 1-

form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)

where⋆119886is the dual 3-form relating to the canonical energy-

momentum tensor 120583

119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ

119887119886is the dual 3-form corresponding to the

canonical spin tensor which is identical with the dynamicalspin tensor

119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)

The variation of the total action = 119892+

119898 with respect to

the 119886 119886119887 and Φ gives the following field equations

(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)

In the sequel the DC-members and can readily bedetermined as follows

119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)

D The GGP in More Detail

Note that an invariance of the Lagrangian 119871Φ

under theinfinite-parameter group of general covariance (A5) in 119881

4

implies an invariance of 119871Φunder the 119866

119881group and vice

versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla

120583Φ()

are introduced by finite local119880119881isin 119866

119881gauge transformations

Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)

Here nabla120583denotes the covariant derivative agreeing with the

metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()

120583119887]() is the connection and 119869

119886119887are

the generators of Lorentz group Λ The tetrad components119886

120583() associate with the chosen representation 119863(Λ) by

which the Φ() is transformed as [119863(Λ)]1198971015840sdotsdotsdot1198961015840

119897sdotsdotsdot119896Φ() where

119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus

119887119886are the parameters of the

Lorentz group One has for example to set 120583() rarr 120583()

for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897

119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]

for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881

4 119866

119881 ) with the

structure group 119866119881 the local coordinates isin are =

( 119880119881) where isin 119881

4and 119880

119881isin 119866

119881 the total bundle space

is a smooth manifold and the surjection is a smoothmap rarr 119881

4 A set of open coverings U

119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881

4 The collection of matter

fields of arbitrary spins Φ() take values in standard fiber over

minus1(U

119894) = U

119894times

The fibration is given as ⋃

minus1() =

The local gauge will be the diffeomorphism map 119894

U119894times

1198814119866119881rarr

minus1(U

119894) isin since minus1

119894maps minus1(U

119894) onto

the direct (Cartesian) product U119894times

1198814119866119881 Here times

1198814represents

the fiber product of elements defined over space 1198814such that

(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881

for all isin U119894 where (119894119889)

119866119881is the identity element of the

group 119866119881 The fiber minus1 at isin 119881

4is diffeomorphic to

where is the fiber space such that minus1() equiv asymp The

action of the structure group119866119881on defines an isomorphism

of the Lie algebra g of 119866119881onto the Lie algebra of vertical

vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872

4 119880

loc 119904) with the base space

1198724 the structure group119880loc and the surjection 119904Thematter

fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]

The GGP accounts for the gravitation gauge group 119866119881

generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881

4must be

invariant under the finite local gauge transformations119880119881(D1)

of the Lie group of gravitation119866119881(see Scheme 1) where 119877

120595(119886)

is the matrix of unitary map

119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP

Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)

minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886

119896 In an illustration

of the point at issue the (D2) explicitly may read

Φ120583sdotsdotsdot120575

() = 120583

119897sdot sdot sdot

120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)

equiv (119877120595)120583sdotsdotsdot120575

119897sdotsdotsdot119898Φ

119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ

120583sdotsdotsdot120575

()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)

In case of zero curvature one has 120595120583119897= 119863

120583

119897= 119890

120583

119897=

(120597119909120583120597119883

119897) 119863 = 0 where 119883119897 are the inertial coordinates

In this the conventional gauge theory given on the 1198724

is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886

119860) is a vector field

only the gravitational attraction is presented in the proposedtheory of gravitation

E A Lie Group of Distortion

The extended space1198726reads 0

1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3

sgn (1198773) = (+ + +) sgn (1198793) = (minus minus minus) (E1)

The 119890(120582120572)

= 119874120582times 120590

120572(120582 = plusmn 120572 = 1 2 3) are linearly

independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773

120582 The unit vectors 119874

120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573

is the Kronecker symbol and lowast

120575120582120591= 1 minus 120575

120582120591

Three spatial 119890120572= 120585 times 120590

120572and three temporal 119890

0120572= 120585

0times 120590

120572

components are the basis vectors respectively in spaces 1198773

and 1198793 where119874plusmn= (1radic2)(120585

0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0

The 3D space 1198773plusmnis spanned by the coordinates 120578

(plusmn120572) In using

this language it is important to consider a reduction to thespace119872

4which can be achieved in the following way

(1) In case of free flat space 1198726 the subspace 1198793 is

isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872

6rarr 119872

4= 119877

3oplus 119879

1 can bereadily achieved if we use 119905 = |x0| for conventionaltime

(2) In case of curved space the reduction 1198816rarr 119881

4can

be achieved if we use the projection ( 1198900) of the tem-

poral component ( 1198900120572) of basis six-vector 119890( 119890

120572 119890

0120572) on

the given universal direction ( 1198900120572rarr 119890

0) By this we

choose the time coordinate Actually the Lagrangianof physical fields defined on 119877

6is a function of scalars

such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon

the reduction of temporal components of six-vectors119860

01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610

we may fulfill a reduction to 1198814

A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874

120591under the distortion gauge field

(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591

(120582120572)(119886)) is an element of the group 119876 This

induces the distortion transformations of the ingredient unitvectors 120590

120573 which in turn undergo the rotations

(120582120572)(120579) =

R120573

(120582120572)(120579)120590

120573 whereR(120579) isin 119878119874(3) is the element of the group

of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773

120582 In fact

distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read

120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)

provided by the generators 119883120591

120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894

are the Paulimatrices119872120572120573= 120576

120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The

transformation matrix 119863(119886 120579) = Q(119886) timesR(120579) is an elementof the distortion group 119866

119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868

120572at given120582The generators119883

119860(E4) of the group

119876 do not complete the group119867 to the dynamical group 119866119863

and therefore they cannot be interpreted as the generatorsof the quotien space 119866

119863119867 and the distortion fields 119886

119860

cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866

119863

These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866

119863= 119876 times

119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866

119863= 119878119874(3) times 119878119874(3) which is isomorphic to the

chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation

tan 120579119860= minusaelig119886

119860 (E6)


Given the distortion field 119886119860 the relation (E6) uniquely

determines six angles 120579119860of rotations around each of six (119860)

axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold

11987212= 119872

6oplus119872

6 (E7)

Here the1198726is related to the spacetime continuum (E1) but

the 1198726is displayed as a space of inner degrees of freedom

The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)

are basis vectors at the point 119901(120577) of11987212

⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr

lowast1198774=lowast1198772otimeslowast1198772 120590

120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the

decomposition (E1) together with

1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793

) = (+ + +) sgn (1198753) = (minus minus minus) (E10)

holds The 12-dimensional basis (119890) transforms under thedistortion gauge field 119886(120577) (120577 isin 119872

12)

= 119863 (119886) 119890 (E11)

where the distortion matrix 119863(119886) reads 119863(119886) = 119862(119886) otimes 119877(119886)provided

= 119862 (119886)119874 = 119877 (119886) 120590 (E12)

The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors

119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)

but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of

the planes involving two arbitrary bases of the spaces 1198773120582120583

around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth

F Field Equations at Spherical Symmetry

The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872

12

and 12[74] In accordance the equation of distortion gauge

field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862

is the energy-momentum tensor and 1205770is the

gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form

minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)

Hence we may assign to Newtonrsquos gravitational constant 119866119873

the value

119866119873=aelig2

4120587 (F3)

The curvature of manifold 1198726rarr 119872

6is the familiar

distortion induced by the extended field components

119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)

The other regime of ID presents at

119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)

To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886

0(119903) in presence of one-dimensional space-like

ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)

One can then easily determine the basis vectors (119890120582120572 119890

120591120573)

where tan 120579(plusmn3)= aelig(minus119886

0plusmn 119886) Passing back from the

6to

1198814 the basis vectors read

0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02

minus (sin 120579(+3)minus sin 120579

(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886

G SPC-Configurations

The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt

454 times 1012 g cmminus3 one uses for both configurations the

simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3

le 120588 lt

120588119889and of the n-p-e protomatter in presence of ID at 120588 gt

120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10

14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588

119889 to which the distances 04 fm lt 119903

119873119873le

16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595

1of energy

1to another one 120595

2

of energy 2 the lowering of energy of system takes place

and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of

119902119902119902119902 described by the Hamiltonian and invariant action then one has

[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)

where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877

4The strings are initially

in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590

0of the minimum surface area ldquoinstantonrdquo A

computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590

0 In domain

120588119889le 120588 lt 120588

119886119904 one has the string flip-flop regime in presence

of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system

is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588

119886119904 the system is made of quarks

in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free

parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int

0(119903) matches the exterior

one 119909ext0(119903) at the surface of the configuration The central

value of the gravitational potential 1199090(0) can be found by

reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus

119861119873 on equal footing Minimizing

the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )

119904the following values Γ

1asymp 2216 for


the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated

References

[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003

[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969

[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007

[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010

[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002

[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010

[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009

[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004

[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008

[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009

[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009

[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010

[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011

[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012

[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011

[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010

[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003

[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009

[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005

[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009

[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003

[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011

[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010

[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003

[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011

[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011

[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466

[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010

[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010

[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963

[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in


the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994

[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995

[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966

[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966

[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011

[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008

[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996

[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997

[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999

[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999

[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999


[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004

[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005

[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998

[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001

[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002

[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002

[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002

[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002

[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005

[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995

[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002

[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997

[55] D Eichler ldquoHigh-energy neutrino astronomymdasha probe ofgalactic nucleirdquoThe Astrophysical Journal vol 232 pp 106ndash1121979

[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997

[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012

[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011

[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013

[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013

[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013

[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013

[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013

[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013

[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009


[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013

[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012

[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012

[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009

[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013

[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001

[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007

[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013

[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010

[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014

[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997

[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002

[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980

[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963

[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972

[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984

[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011

[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984

[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986

[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969

[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969

[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970

[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969

[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971

[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973

[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974

[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985

[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980

[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994

[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334

[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973

[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006

[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983

Submit your manuscripts athttpwwwhindawicom

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aspects of the growth of black hole seeds are summarizedin Section 2 The other important phenomenon of ultrahighenergy cosmic rays in relevance to AGNs is discussed inSection 3 The objectives of suggested approach are outlinedin Section 4 In Section 5 we review the spherical accretionon superdense protomatter nuclei in use In Section 6 wediscuss the growth of the seed black hole at accretion andderive its intermediate mass initial redshift and neutrinopreradiation time (PRT) Section 7 is devoted to the neutrinoradiation produced in superdense protomatter nuclei Thesimulation results of the seed black hole intermediate massesPRTs seed redshifts and neutrino fluxes for 377 AGN blackholes are brought in Section 8 The concluding remarks aregiven in Section 9 We will refrain from providing lengthydetails of the proposed gravitation theory at huge energiesand neutrino flux computations For these the reader isinvited to visit the original papers and appendices of thepresent paper In the latter we also complete the spacetimedeformation theory in the model context of gravitation bynew investigation of building up the complex of distortion(DC) of spacetime continuum and showing how it restoresthe world-deformation tensor which still has been put in byhand Finally note that we regard the considered black holesonly as the potential neutrino sources The obtained resultshowever may suffer if not all live black holes at present residein final stage of their growth driven by the formation ofprotomatter disk at accretion and they radiate neutrino Weoften suppress the indices without notice Unless otherwisestated we take geometrized units throughout this paper

2 A Breakthrough in Observational andComputational Aspects on Growth of BlackHole Seeds

Significant progress has been made in the last few years inunderstanding how supermassive black holes form and growGiven the currentmasses of 106minus9119872

⊙ most black hole growth

happens in the AGN phase A significant fraction of the totalblack hole growth 60 [6] happens in the most luminousAGN quasars In an AGN phase which lasts sim108 years thecentral supermassive black hole can gain up to sim107minus8119872

⊙

so even the most massive galaxies will have only a few ofthese events over their lifetime Aforesaid gathers supportespecially from a breakthroughmade in recent observationaltheoretical and computational efforts in understanding ofevolution of black holes and their host galaxies particularlythrough self-regulated growth and feedback from accretion-powered outflows see for example [4 7ndash18] Whereas themultiwavelength methods are used to trace the growth ofseed BHs the prospects for future observations are reviewedThe observations provide strong support for the existenceof a correlation between supermassive black holes and theirhosts out to the highest redshifts The observations of thequasar luminosity function show that the most supermassiveblack holes get most of their mass at high redshift whileat low redshift only low mass black holes are still growing[19] This is observed in both optical [20] and hard X-ray luminosity functions [19 21] which indicates that this

result is independent of obscuration Natarajan [13] hasreported that the initial black hole seeds form at extremelyhigh redshifts from the direct collapse of pregalactic gasdiscs Populating dark matter halos with seeds formed inthis fashion and using a Monte-Carlo merger tree approachhe has predicted the black hole mass function at highredshifts and at the present time The most aspects of themodels that describe the growth and accretion history ofsupermassive black holes and evolution of this scenario havebeen presented in detail by [9 10] In these models at earlytimes the properties of the assembling black hole seeds aremore tightly coupled to properties of the dark matter haloas their growth is driven by the merger history of halosWhile a clear picture of the history of black hole growth isemerging significant uncertainties still remain [14] and inspite of recent advances [6 13] the origin of the seed blackholes remains an unsolved problem at present The NuSTARdeep high-energy observations will enable obtaining a nearlycomplete AGN survey including heavily obscured Compton-thick sources up to 119911 sim 15 [22] A similar mission ASTRO-H [23] will be launched by Japan in 2014These observationsin combination with observations at longer wavelengths willallow for the detection and identification of most growingsupermassive black holes at 119911 sim 1 The ultradeep X-ray andnear-infrared surveys covering at least sim1 deg2 are requiredto constrain the formation of the first black hole seeds Thiswill likely require the use of the next generation of space-based observatories such as the James Webb Space Telescopeand the International X-ray Observatory The superb spatialresolution and sensitivity of the Atacama Large MillimeterArray (ALMA) [24] will revolutionize our understanding ofgalaxy evolution Combining these new data with existingmultiwavelength information will finally allow astrophysi-cists to pave the way for later efforts by pioneering some ofthe census of supermassive black hole growth in use today

3 UHE Cosmic-Ray Particles

The galactic sources like supernova remnants (SNRs) ormicroquasars are thought to accelerate particles at least upto energies of 3 times 1015 eV The ultrahigh energy cosmic-ray (UHECR) particles with even higher energies have sincebeen detected (comprehensive reviews can be found in [25ndash29]) The accelerated protons or heavier nuclei up to energiesexceeding 1020 eV are firstly observed by [30] The cosmic-ray events with the highest energies so far detected haveenergies of 2 times 1011 GeV [31] and 3 times 1011 GeV [32] Theseenergies are 107 times higher than the most energetic man-made accelerator the LHC at CERN These highest energiesare believed to be reached in extragalactic sources like AGNsor gamma-ray bursts (GRBs) During propagation of suchenergetic particles through the universe the threshold forpion photoproduction on the microwave background is sim2times 1010 GeV and at sim3 times 1011 GeV the energy-loss distance isabout 20Mpc Propagation of cosmic rays over substantiallylarger distances gives rise to a cutoff in the spectrum at sim1011 GeV as was first shown by [33 34] the GZK cutoffThe recent confirmation [35 36] of GZK suppression in the



2

119885(2119898]) ≃ 42 times 10













AGN

ADAD

EH

infin

(a)

AGN

PDPC

ADAD

EHSPC

(b)




2119871 = (1 minus 1199090)2 1199052

minus (1 + 1199090)2 119903

2

minus 2sin2 1205932 minus 2 120579

2

(1)


1015840)) such that


119890=10038161003816100381610038161 minus 1199090

1003816100381610038161003816

minus1

119890119905 119890

= (1 + 119909

0)minus1

119890119903

119890=

minus1119890120579 119890

= ( sin )

minus1

119890120579

(2)




2



(119864 gt 0) = 16120587 (119866)2



AGN

SPC

x0 = 1x0 = 2

x0 = 0

t = infin

vr lt 0 vr = 0

r = infin

EH

(a)

1minus 1 1+x0

10

15

25

ntimes10

minus40(g

cmminus

3 )

(b)







infin≪ 1 During


infin)1199090rarr1

asymp

minus(ln 00)2 V


119889()

minus13=





119904outside the


00119864

2) where 119864 = 119864

infin119898 = (

00(1 minus 119906

2))12 and

119864infin








= 212058711989811989911990411990352

119904(ln

00)1015840

119904 (4)



119899infin

asymp11990352

119904

21199032[(ln

00)1015840

119904

1 + 119903119903()]

12

(5)






rarr


Δ119877119892= 119877

BH119892minus 119877

Seed119892=2119866

1198882119872

119889=2119866

1198882120588119889119881119889


SeedBHΔ119877

119892

119877Seed119892

(6)

where 119872119889 120588

119889 and 119881





PC PD120579

1205880

120593Z

Z0Z1

EHBH EHSeed

1205881120588

d120576d = d2Rg asymp 120579

RSPCRBHg

RSeedg

Rd



(120588 119911 120593) from Figure 3

119889= int

BH119892

1205880

119889120588int

2120587

0

120588119889120601int

1199111(120588)

minus1199111(120588)

119889119911

minus int

119877119889

1205880

119889120588int

2120587

0

120588119889120601int

1199110(120588)

minus1199110(120588)

119889119911

(119877119889≪BH119892

)

≃radic2120587

3119877119889(

BH119892)

2

(7)

where 1199111(120588) ≃ 119911

0minus 119911

0(120588 minus 120588

0)(

BH119892minus 120588

0) 119911

0(120588) = radic119877

2

119889minus 1205882


BH119892

we set 1199110(120588

0) ≃ 120588

0≃ 119877

119889radic2



2

119896119877Seed119892) (8)

where 2119896 = 873 [km]119877119889120588119889119872


119896)119877Seed119892le 1 namely

119877⊙

119877119889

ge 209[km]119877⊙

120588119889

120588⊙

119877Seed119892

119877⊙

(9)



119892≃


119889ge

234 times 108(1 to 103) or [cm]119877

119889ge 034(10


≃ 11 times 106 to 13 times 1010



119872SeedBH119872

⊙

≃119872BH119872

⊙


[cm]119872BH119872

⊙

) (10)



119889≪ 119877

119892 the PRT

reads

119879BH = 120588119889119881119889

≃ 933 sdot 10

15[g cmminus3

]

1198771198891198772

119892

(11)


119879BH ≃ 26 sdot 1016 119877119889

cm10

minus24 g cmminus3

120588infin

Vinfin

10 km 119904minus1yr (12)


119879BH ≃ 88 sdot 10381198771198891198772

119892cmminus3

11989911990411990352

119904 (ln11989200)1015840

119904

(13)




119879BH ≃ 032119877119889

119903OV(119872BH119872

⊙

)

210

39119882

119871bol[yr] (14)



119904 120579

119904 and 120593

119904 at time 119905





0





1 and 120593

1 at time 119905

1 where 119905

1=







yields

119911Seed≃ 119911 + 2292 times 10

28 119877119889

119903OV(119872BH119872

⊙

)

2119882

119871bol (15)



2


119871(119911) is the



1015840

]119877(1199051)119877(1199050) = 1015840

](1 + 119911)minus1 of


119872= 120588

119872120588


119881=

120588119881120588

119888 therebyΩ

119881+Ω


120588119888= 3119867

2

0(8120587119866


1198670[77]

119863119871(119911) =

(1 + 119911) 119888

1198670radicΩ

119872

int

1+119911

1

119889119909

radicΩ119881Ω

119872+ 1199093

= 24 times 1028119868 (119911) cm

(16)


0=


119872= 03


URCA]120576 = 38 times 10

50120576119889(119872

⊙

)

175

[erg sminus1] (17)

where 120576119889= 1198892 119877



119869URCA]120576 ≃ 522 times 10

minus8

times120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(18)


119889≪ 1 119868(119911) = (1 + 119911) int1+119911

1119889119909radic23 + 1199093 As it is



120587minus+ 119899 997888rarr 119899 + 119890

minus+ ]

119890 120587

minus+ 119899 997888rarr 119899 + 120583

minus+ ]

120583(19)


120587

]120576 = 578 times 1058120576119889(119872

⊙

)

175

[erg sminus1] (20)


119869120587

]120576 ≃ 791120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(21)



119889 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119889 + ]

119890 (22)

119904 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119904 + ]

119890 (23)


119897]119897


119869119902

]120576 ≃ 7068120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(24)

8 Simulation



5E6

3E7

2E7

25E7

1E7

15E7

0


z = 001

z = 007

z = 002

z = 07

z = 05

z = 01

z = 003

z = 005

10000

20000

30000

40000

50000

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1 srminus

1 )

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1srminus

1 )

00

4 8 12 16 200 4 8 12 16


5E7

2E8

2E5

1E8

15E8

16E5

12E5

28E5

24E5

00 4 8 12 16


(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

00

4 8 12 16 20

z = 001

z = 002

z = 003

z = 005

40000

80000

z = 007

z = 07

z = 05

z = 01






Seed and119869119894


BH 119872⊙





6 7 8 9 10 11

log (MBHM⊙)

1

2

3

4

5

6

log

(MSe

edBH

M⊙)









]120576 with the average119869120587






119889= 1198892 119903

119892≪ 1



120576119889≃ 169times10


⊙(2 119877

119892= 59times10



9 Conclusions






119872BH119872⊙≃ 11 times 10


Appendices







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587





119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2

119885(2119898]) ≃ 42 times 10













AGN

ADAD

EH

infin

(a)

AGN

PDPC

ADAD

EHSPC

(b)




2119871 = (1 minus 1199090)2 1199052

minus (1 + 1199090)2 119903

2

minus 2sin2 1205932 minus 2 120579

2

(1)


1015840)) such that


119890=10038161003816100381610038161 minus 1199090

1003816100381610038161003816

minus1

119890119905 119890

= (1 + 119909

0)minus1

119890119903

119890=

minus1119890120579 119890

= ( sin )

minus1

119890120579

(2)




2



(119864 gt 0) = 16120587 (119866)2



AGN

SPC

x0 = 1x0 = 2

x0 = 0

t = infin

vr lt 0 vr = 0

r = infin

EH

(a)

1minus 1 1+x0

10

15

25

ntimes10

minus40(g

cmminus

3 )

(b)







infin≪ 1 During


infin)1199090rarr1

asymp

minus(ln 00)2 V


119889()

minus13=





119904outside the


00119864

2) where 119864 = 119864

infin119898 = (

00(1 minus 119906

2))12 and

119864infin








= 212058711989811989911990411990352

119904(ln

00)1015840

119904 (4)



119899infin

asymp11990352

119904

21199032[(ln

00)1015840

119904

1 + 119903119903()]

12

(5)






rarr


Δ119877119892= 119877

BH119892minus 119877

Seed119892=2119866

1198882119872

119889=2119866

1198882120588119889119881119889


SeedBHΔ119877

119892

119877Seed119892

(6)

where 119872119889 120588

119889 and 119881





PC PD120579

1205880

120593Z

Z0Z1

EHBH EHSeed

1205881120588

d120576d = d2Rg asymp 120579

RSPCRBHg

RSeedg

Rd



(120588 119911 120593) from Figure 3

119889= int

BH119892

1205880

119889120588int

2120587

0

120588119889120601int

1199111(120588)

minus1199111(120588)

119889119911

minus int

119877119889

1205880

119889120588int

2120587

0

120588119889120601int

1199110(120588)

minus1199110(120588)

119889119911

(119877119889≪BH119892

)

≃radic2120587

3119877119889(

BH119892)

2

(7)

where 1199111(120588) ≃ 119911

0minus 119911

0(120588 minus 120588

0)(

BH119892minus 120588

0) 119911

0(120588) = radic119877

2

119889minus 1205882


BH119892

we set 1199110(120588

0) ≃ 120588

0≃ 119877

119889radic2



2

119896119877Seed119892) (8)

where 2119896 = 873 [km]119877119889120588119889119872


119896)119877Seed119892le 1 namely

119877⊙

119877119889

ge 209[km]119877⊙

120588119889

120588⊙

119877Seed119892

119877⊙

(9)



119892≃


119889ge

234 times 108(1 to 103) or [cm]119877

119889ge 034(10


≃ 11 times 106 to 13 times 1010



119872SeedBH119872

⊙

≃119872BH119872

⊙


[cm]119872BH119872

⊙

) (10)



119889≪ 119877

119892 the PRT

reads

119879BH = 120588119889119881119889

≃ 933 sdot 10

15[g cmminus3

]

1198771198891198772

119892

(11)


119879BH ≃ 26 sdot 1016 119877119889

cm10

minus24 g cmminus3

120588infin

Vinfin

10 km 119904minus1yr (12)


119879BH ≃ 88 sdot 10381198771198891198772

119892cmminus3

11989911990411990352

119904 (ln11989200)1015840

119904

(13)




119879BH ≃ 032119877119889

119903OV(119872BH119872

⊙

)

210

39119882

119871bol[yr] (14)



119904 120579

119904 and 120593

119904 at time 119905





0





1 and 120593

1 at time 119905

1 where 119905

1=







yields

119911Seed≃ 119911 + 2292 times 10

28 119877119889

119903OV(119872BH119872

⊙

)

2119882

119871bol (15)



2


119871(119911) is the



1015840

]119877(1199051)119877(1199050) = 1015840

](1 + 119911)minus1 of


119872= 120588

119872120588


119881=

120588119881120588

119888 therebyΩ

119881+Ω


120588119888= 3119867

2

0(8120587119866


1198670[77]

119863119871(119911) =

(1 + 119911) 119888

1198670radicΩ

119872

int

1+119911

1

119889119909

radicΩ119881Ω

119872+ 1199093

= 24 times 1028119868 (119911) cm

(16)


0=


119872= 03


URCA]120576 = 38 times 10

50120576119889(119872

⊙

)

175

[erg sminus1] (17)

where 120576119889= 1198892 119877



119869URCA]120576 ≃ 522 times 10

minus8

times120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(18)


119889≪ 1 119868(119911) = (1 + 119911) int1+119911

1119889119909radic23 + 1199093 As it is



120587minus+ 119899 997888rarr 119899 + 119890

minus+ ]

119890 120587

minus+ 119899 997888rarr 119899 + 120583

minus+ ]

120583(19)


120587

]120576 = 578 times 1058120576119889(119872

⊙

)

175

[erg sminus1] (20)


119869120587

]120576 ≃ 791120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(21)



119889 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119889 + ]

119890 (22)

119904 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119904 + ]

119890 (23)


119897]119897


119869119902

]120576 ≃ 7068120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(24)

8 Simulation



5E6

3E7

2E7

25E7

1E7

15E7

0


z = 001

z = 007

z = 002

z = 07

z = 05

z = 01

z = 003

z = 005

10000

20000

30000

40000

50000

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1 srminus

1 )

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1srminus

1 )

00

4 8 12 16 200 4 8 12 16


5E7

2E8

2E5

1E8

15E8

16E5

12E5

28E5

24E5

00 4 8 12 16


(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

00

4 8 12 16 20

z = 001

z = 002

z = 003

z = 005

40000

80000

z = 007

z = 07

z = 05

z = 01






Seed and119869119894


BH 119872⊙





6 7 8 9 10 11

log (MBHM⊙)

1

2

3

4

5

6

log

(MSe

edBH

M⊙)









]120576 with the average119869120587






119889= 1198892 119903

119892≪ 1



120576119889≃ 169times10


⊙(2 119877

119892= 59times10



9 Conclusions






119872BH119872⊙≃ 11 times 10


Appendices







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587





119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









AGN

ADAD

EH

infin

(a)

AGN

PDPC

ADAD

EHSPC

(b)




2119871 = (1 minus 1199090)2 1199052

minus (1 + 1199090)2 119903

2

minus 2sin2 1205932 minus 2 120579

2

(1)


1015840)) such that


119890=10038161003816100381610038161 minus 1199090

1003816100381610038161003816

minus1

119890119905 119890

= (1 + 119909

0)minus1

119890119903

119890=

minus1119890120579 119890

= ( sin )

minus1

119890120579

(2)




2



(119864 gt 0) = 16120587 (119866)2



AGN

SPC

x0 = 1x0 = 2

x0 = 0

t = infin

vr lt 0 vr = 0

r = infin

EH

(a)

1minus 1 1+x0

10

15

25

ntimes10

minus40(g

cmminus

3 )

(b)







infin≪ 1 During


infin)1199090rarr1

asymp

minus(ln 00)2 V


119889()

minus13=





119904outside the


00119864

2) where 119864 = 119864

infin119898 = (

00(1 minus 119906

2))12 and

119864infin








= 212058711989811989911990411990352

119904(ln

00)1015840

119904 (4)



119899infin

asymp11990352

119904

21199032[(ln

00)1015840

119904

1 + 119903119903()]

12

(5)






rarr


Δ119877119892= 119877

BH119892minus 119877

Seed119892=2119866

1198882119872

119889=2119866

1198882120588119889119881119889


SeedBHΔ119877

119892

119877Seed119892

(6)

where 119872119889 120588

119889 and 119881





PC PD120579

1205880

120593Z

Z0Z1

EHBH EHSeed

1205881120588

d120576d = d2Rg asymp 120579

RSPCRBHg

RSeedg

Rd



(120588 119911 120593) from Figure 3

119889= int

BH119892

1205880

119889120588int

2120587

0

120588119889120601int

1199111(120588)

minus1199111(120588)

119889119911

minus int

119877119889

1205880

119889120588int

2120587

0

120588119889120601int

1199110(120588)

minus1199110(120588)

119889119911

(119877119889≪BH119892

)

≃radic2120587

3119877119889(

BH119892)

2

(7)

where 1199111(120588) ≃ 119911

0minus 119911

0(120588 minus 120588

0)(

BH119892minus 120588

0) 119911

0(120588) = radic119877

2

119889minus 1205882


BH119892

we set 1199110(120588

0) ≃ 120588

0≃ 119877

119889radic2



2

119896119877Seed119892) (8)

where 2119896 = 873 [km]119877119889120588119889119872


119896)119877Seed119892le 1 namely

119877⊙

119877119889

ge 209[km]119877⊙

120588119889

120588⊙

119877Seed119892

119877⊙

(9)



119892≃


119889ge

234 times 108(1 to 103) or [cm]119877

119889ge 034(10


≃ 11 times 106 to 13 times 1010



119872SeedBH119872

⊙

≃119872BH119872

⊙


[cm]119872BH119872

⊙

) (10)



119889≪ 119877

119892 the PRT

reads

119879BH = 120588119889119881119889

≃ 933 sdot 10

15[g cmminus3

]

1198771198891198772

119892

(11)


119879BH ≃ 26 sdot 1016 119877119889

cm10

minus24 g cmminus3

120588infin

Vinfin

10 km 119904minus1yr (12)


119879BH ≃ 88 sdot 10381198771198891198772

119892cmminus3

11989911990411990352

119904 (ln11989200)1015840

119904

(13)




119879BH ≃ 032119877119889

119903OV(119872BH119872

⊙

)

210

39119882

119871bol[yr] (14)



119904 120579

119904 and 120593

119904 at time 119905





0





1 and 120593

1 at time 119905

1 where 119905

1=







yields

119911Seed≃ 119911 + 2292 times 10

28 119877119889

119903OV(119872BH119872

⊙

)

2119882

119871bol (15)



2


119871(119911) is the



1015840

]119877(1199051)119877(1199050) = 1015840

](1 + 119911)minus1 of


119872= 120588

119872120588


119881=

120588119881120588

119888 therebyΩ

119881+Ω


120588119888= 3119867

2

0(8120587119866


1198670[77]

119863119871(119911) =

(1 + 119911) 119888

1198670radicΩ

119872

int

1+119911

1

119889119909

radicΩ119881Ω

119872+ 1199093

= 24 times 1028119868 (119911) cm

(16)


0=


119872= 03


URCA]120576 = 38 times 10

50120576119889(119872

⊙

)

175

[erg sminus1] (17)

where 120576119889= 1198892 119877



119869URCA]120576 ≃ 522 times 10

minus8

times120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(18)


119889≪ 1 119868(119911) = (1 + 119911) int1+119911

1119889119909radic23 + 1199093 As it is



120587minus+ 119899 997888rarr 119899 + 119890

minus+ ]

119890 120587

minus+ 119899 997888rarr 119899 + 120583

minus+ ]

120583(19)


120587

]120576 = 578 times 1058120576119889(119872

⊙

)

175

[erg sminus1] (20)


119869120587

]120576 ≃ 791120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(21)



119889 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119889 + ]

119890 (22)

119904 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119904 + ]

119890 (23)


119897]119897


119869119902

]120576 ≃ 7068120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(24)

8 Simulation



5E6

3E7

2E7

25E7

1E7

15E7

0


z = 001

z = 007

z = 002

z = 07

z = 05

z = 01

z = 003

z = 005

10000

20000

30000

40000

50000

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1 srminus

1 )

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1srminus

1 )

00

4 8 12 16 200 4 8 12 16


5E7

2E8

2E5

1E8

15E8

16E5

12E5

28E5

24E5

00 4 8 12 16


(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

00

4 8 12 16 20

z = 001

z = 002

z = 003

z = 005

40000

80000

z = 007

z = 07

z = 05

z = 01






Seed and119869119894


BH 119872⊙





6 7 8 9 10 11

log (MBHM⊙)

1

2

3

4

5

6

log

(MSe

edBH

M⊙)









]120576 with the average119869120587






119889= 1198892 119903

119892≪ 1



120576119889≃ 169times10


⊙(2 119877

119892= 59times10



9 Conclusions






119872BH119872⊙≃ 11 times 10


Appendices







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587





119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




AGN

ADAD

EH

infin

(a)

AGN

PDPC

ADAD

EHSPC

(b)




2119871 = (1 minus 1199090)2 1199052

minus (1 + 1199090)2 119903

2

minus 2sin2 1205932 minus 2 120579

2

(1)


1015840)) such that


119890=10038161003816100381610038161 minus 1199090

1003816100381610038161003816

minus1

119890119905 119890

= (1 + 119909

0)minus1

119890119903

119890=

minus1119890120579 119890

= ( sin )

minus1

119890120579

(2)




2



(119864 gt 0) = 16120587 (119866)2



AGN

SPC

x0 = 1x0 = 2

x0 = 0

t = infin

vr lt 0 vr = 0

r = infin

EH

(a)

1minus 1 1+x0

10

15

25

ntimes10

minus40(g

cmminus

3 )

(b)







infin≪ 1 During


infin)1199090rarr1

asymp

minus(ln 00)2 V


119889()

minus13=





119904outside the


00119864

2) where 119864 = 119864

infin119898 = (

00(1 minus 119906

2))12 and

119864infin








= 212058711989811989911990411990352

119904(ln

00)1015840

119904 (4)



119899infin

asymp11990352

119904

21199032[(ln

00)1015840

119904

1 + 119903119903()]

12

(5)






rarr


Δ119877119892= 119877

BH119892minus 119877

Seed119892=2119866

1198882119872

119889=2119866

1198882120588119889119881119889


SeedBHΔ119877

119892

119877Seed119892

(6)

where 119872119889 120588

119889 and 119881





PC PD120579

1205880

120593Z

Z0Z1

EHBH EHSeed

1205881120588

d120576d = d2Rg asymp 120579

RSPCRBHg

RSeedg

Rd



(120588 119911 120593) from Figure 3

119889= int

BH119892

1205880

119889120588int

2120587

0

120588119889120601int

1199111(120588)

minus1199111(120588)

119889119911

minus int

119877119889

1205880

119889120588int

2120587

0

120588119889120601int

1199110(120588)

minus1199110(120588)

119889119911

(119877119889≪BH119892

)

≃radic2120587

3119877119889(

BH119892)

2

(7)

where 1199111(120588) ≃ 119911

0minus 119911

0(120588 minus 120588

0)(

BH119892minus 120588

0) 119911

0(120588) = radic119877

2

119889minus 1205882


BH119892

we set 1199110(120588

0) ≃ 120588

0≃ 119877

119889radic2



2

119896119877Seed119892) (8)

where 2119896 = 873 [km]119877119889120588119889119872


119896)119877Seed119892le 1 namely

119877⊙

119877119889

ge 209[km]119877⊙

120588119889

120588⊙

119877Seed119892

119877⊙

(9)



119892≃


119889ge

234 times 108(1 to 103) or [cm]119877

119889ge 034(10


≃ 11 times 106 to 13 times 1010



119872SeedBH119872

⊙

≃119872BH119872

⊙


[cm]119872BH119872

⊙

) (10)



119889≪ 119877

119892 the PRT

reads

119879BH = 120588119889119881119889

≃ 933 sdot 10

15[g cmminus3

]

1198771198891198772

119892

(11)


119879BH ≃ 26 sdot 1016 119877119889

cm10

minus24 g cmminus3

120588infin

Vinfin

10 km 119904minus1yr (12)


119879BH ≃ 88 sdot 10381198771198891198772

119892cmminus3

11989911990411990352

119904 (ln11989200)1015840

119904

(13)




119879BH ≃ 032119877119889

119903OV(119872BH119872

⊙

)

210

39119882

119871bol[yr] (14)



119904 120579

119904 and 120593

119904 at time 119905





0





1 and 120593

1 at time 119905

1 where 119905

1=







yields

119911Seed≃ 119911 + 2292 times 10

28 119877119889

119903OV(119872BH119872

⊙

)

2119882

119871bol (15)



2


119871(119911) is the



1015840

]119877(1199051)119877(1199050) = 1015840

](1 + 119911)minus1 of


119872= 120588

119872120588


119881=

120588119881120588

119888 therebyΩ

119881+Ω


120588119888= 3119867

2

0(8120587119866


1198670[77]

119863119871(119911) =

(1 + 119911) 119888

1198670radicΩ

119872

int

1+119911

1

119889119909

radicΩ119881Ω

119872+ 1199093

= 24 times 1028119868 (119911) cm

(16)


0=


119872= 03


URCA]120576 = 38 times 10

50120576119889(119872

⊙

)

175

[erg sminus1] (17)

where 120576119889= 1198892 119877



119869URCA]120576 ≃ 522 times 10

minus8

times120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(18)


119889≪ 1 119868(119911) = (1 + 119911) int1+119911

1119889119909radic23 + 1199093 As it is



120587minus+ 119899 997888rarr 119899 + 119890

minus+ ]

119890 120587

minus+ 119899 997888rarr 119899 + 120583

minus+ ]

120583(19)


120587

]120576 = 578 times 1058120576119889(119872

⊙

)

175

[erg sminus1] (20)


119869120587

]120576 ≃ 791120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(21)



119889 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119889 + ]

119890 (22)

119904 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119904 + ]

119890 (23)


119897]119897


119869119902

]120576 ≃ 7068120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(24)

8 Simulation



5E6

3E7

2E7

25E7

1E7

15E7

0


z = 001

z = 007

z = 002

z = 07

z = 05

z = 01

z = 003

z = 005

10000

20000

30000

40000

50000

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1 srminus

1 )

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1srminus

1 )

00

4 8 12 16 200 4 8 12 16


5E7

2E8

2E5

1E8

15E8

16E5

12E5

28E5

24E5

00 4 8 12 16


(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

00

4 8 12 16 20

z = 001

z = 002

z = 003

z = 005

40000

80000

z = 007

z = 07

z = 05

z = 01






Seed and119869119894


BH 119872⊙





6 7 8 9 10 11

log (MBHM⊙)

1

2

3

4

5

6

log

(MSe

edBH

M⊙)









]120576 with the average119869120587






119889= 1198892 119903

119892≪ 1



120576119889≃ 169times10


⊙(2 119877

119892= 59times10



9 Conclusions






119872BH119872⊙≃ 11 times 10


Appendices







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587





119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




AGN

SPC

x0 = 1x0 = 2

x0 = 0

t = infin

vr lt 0 vr = 0

r = infin

EH

(a)

1minus 1 1+x0

10

15

25

ntimes10

minus40(g

cmminus

3 )

(b)







infin≪ 1 During


infin)1199090rarr1

asymp

minus(ln 00)2 V


119889()

minus13=





119904outside the


00119864

2) where 119864 = 119864

infin119898 = (

00(1 minus 119906

2))12 and

119864infin








= 212058711989811989911990411990352

119904(ln

00)1015840

119904 (4)



119899infin

asymp11990352

119904

21199032[(ln

00)1015840

119904

1 + 119903119903()]

12

(5)






rarr


Δ119877119892= 119877

BH119892minus 119877

Seed119892=2119866

1198882119872

119889=2119866

1198882120588119889119881119889


SeedBHΔ119877

119892

119877Seed119892

(6)

where 119872119889 120588

119889 and 119881





PC PD120579

1205880

120593Z

Z0Z1

EHBH EHSeed

1205881120588

d120576d = d2Rg asymp 120579

RSPCRBHg

RSeedg

Rd



(120588 119911 120593) from Figure 3

119889= int

BH119892

1205880

119889120588int

2120587

0

120588119889120601int

1199111(120588)

minus1199111(120588)

119889119911

minus int

119877119889

1205880

119889120588int

2120587

0

120588119889120601int

1199110(120588)

minus1199110(120588)

119889119911

(119877119889≪BH119892

)

≃radic2120587

3119877119889(

BH119892)

2

(7)

where 1199111(120588) ≃ 119911

0minus 119911

0(120588 minus 120588

0)(

BH119892minus 120588

0) 119911

0(120588) = radic119877

2

119889minus 1205882


BH119892

we set 1199110(120588

0) ≃ 120588

0≃ 119877

119889radic2



2

119896119877Seed119892) (8)

where 2119896 = 873 [km]119877119889120588119889119872


119896)119877Seed119892le 1 namely

119877⊙

119877119889

ge 209[km]119877⊙

120588119889

120588⊙

119877Seed119892

119877⊙

(9)



119892≃


119889ge

234 times 108(1 to 103) or [cm]119877

119889ge 034(10


≃ 11 times 106 to 13 times 1010



119872SeedBH119872

⊙

≃119872BH119872

⊙


[cm]119872BH119872

⊙

) (10)



119889≪ 119877

119892 the PRT

reads

119879BH = 120588119889119881119889

≃ 933 sdot 10

15[g cmminus3

]

1198771198891198772

119892

(11)


119879BH ≃ 26 sdot 1016 119877119889

cm10

minus24 g cmminus3

120588infin

Vinfin

10 km 119904minus1yr (12)


119879BH ≃ 88 sdot 10381198771198891198772

119892cmminus3

11989911990411990352

119904 (ln11989200)1015840

119904

(13)




119879BH ≃ 032119877119889

119903OV(119872BH119872

⊙

)

210

39119882

119871bol[yr] (14)



119904 120579

119904 and 120593

119904 at time 119905





0





1 and 120593

1 at time 119905

1 where 119905

1=







yields

119911Seed≃ 119911 + 2292 times 10

28 119877119889

119903OV(119872BH119872

⊙

)

2119882

119871bol (15)



2


119871(119911) is the



1015840

]119877(1199051)119877(1199050) = 1015840

](1 + 119911)minus1 of


119872= 120588

119872120588


119881=

120588119881120588

119888 therebyΩ

119881+Ω


120588119888= 3119867

2

0(8120587119866


1198670[77]

119863119871(119911) =

(1 + 119911) 119888

1198670radicΩ

119872

int

1+119911

1

119889119909

radicΩ119881Ω

119872+ 1199093

= 24 times 1028119868 (119911) cm

(16)


0=


119872= 03


URCA]120576 = 38 times 10

50120576119889(119872

⊙

)

175

[erg sminus1] (17)

where 120576119889= 1198892 119877



119869URCA]120576 ≃ 522 times 10

minus8

times120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(18)


119889≪ 1 119868(119911) = (1 + 119911) int1+119911

1119889119909radic23 + 1199093 As it is



120587minus+ 119899 997888rarr 119899 + 119890

minus+ ]

119890 120587

minus+ 119899 997888rarr 119899 + 120583

minus+ ]

120583(19)


120587

]120576 = 578 times 1058120576119889(119872

⊙

)

175

[erg sminus1] (20)


119869120587

]120576 ≃ 791120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(21)



119889 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119889 + ]

119890 (22)

119904 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119904 + ]

119890 (23)


119897]119897


119869119902

]120576 ≃ 7068120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(24)

8 Simulation



5E6

3E7

2E7

25E7

1E7

15E7

0


z = 001

z = 007

z = 002

z = 07

z = 05

z = 01

z = 003

z = 005

10000

20000

30000

40000

50000

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1 srminus

1 )

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1srminus

1 )

00

4 8 12 16 200 4 8 12 16


5E7

2E8

2E5

1E8

15E8

16E5

12E5

28E5

24E5

00 4 8 12 16


(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

00

4 8 12 16 20

z = 001

z = 002

z = 003

z = 005

40000

80000

z = 007

z = 07

z = 05

z = 01






Seed and119869119894


BH 119872⊙





6 7 8 9 10 11

log (MBHM⊙)

1

2

3

4

5

6

log

(MSe

edBH

M⊙)









]120576 with the average119869120587






119889= 1198892 119903

119892≪ 1



120576119889≃ 169times10


⊙(2 119877

119892= 59times10



9 Conclusions






119872BH119872⊙≃ 11 times 10


Appendices







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587





119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




PC PD120579

1205880

120593Z

Z0Z1

EHBH EHSeed

1205881120588

d120576d = d2Rg asymp 120579

RSPCRBHg

RSeedg

Rd



(120588 119911 120593) from Figure 3

119889= int

BH119892

1205880

119889120588int

2120587

0

120588119889120601int

1199111(120588)

minus1199111(120588)

119889119911

minus int

119877119889

1205880

119889120588int

2120587

0

120588119889120601int

1199110(120588)

minus1199110(120588)

119889119911

(119877119889≪BH119892

)

≃radic2120587

3119877119889(

BH119892)

2

(7)

where 1199111(120588) ≃ 119911

0minus 119911

0(120588 minus 120588

0)(

BH119892minus 120588

0) 119911

0(120588) = radic119877

2

119889minus 1205882


BH119892

we set 1199110(120588

0) ≃ 120588

0≃ 119877

119889radic2



2

119896119877Seed119892) (8)

where 2119896 = 873 [km]119877119889120588119889119872


119896)119877Seed119892le 1 namely

119877⊙

119877119889

ge 209[km]119877⊙

120588119889

120588⊙

119877Seed119892

119877⊙

(9)



119892≃


119889ge

234 times 108(1 to 103) or [cm]119877

119889ge 034(10


≃ 11 times 106 to 13 times 1010



119872SeedBH119872

⊙

≃119872BH119872

⊙


[cm]119872BH119872

⊙

) (10)



119889≪ 119877

119892 the PRT

reads

119879BH = 120588119889119881119889

≃ 933 sdot 10

15[g cmminus3

]

1198771198891198772

119892

(11)


119879BH ≃ 26 sdot 1016 119877119889

cm10

minus24 g cmminus3

120588infin

Vinfin

10 km 119904minus1yr (12)


119879BH ≃ 88 sdot 10381198771198891198772

119892cmminus3

11989911990411990352

119904 (ln11989200)1015840

119904

(13)




119879BH ≃ 032119877119889

119903OV(119872BH119872

⊙

)

210

39119882

119871bol[yr] (14)



119904 120579

119904 and 120593

119904 at time 119905





0





1 and 120593

1 at time 119905

1 where 119905

1=







yields

119911Seed≃ 119911 + 2292 times 10

28 119877119889

119903OV(119872BH119872

⊙

)

2119882

119871bol (15)



2


119871(119911) is the



1015840

]119877(1199051)119877(1199050) = 1015840

](1 + 119911)minus1 of


119872= 120588

119872120588


119881=

120588119881120588

119888 therebyΩ

119881+Ω


120588119888= 3119867

2

0(8120587119866


1198670[77]

119863119871(119911) =

(1 + 119911) 119888

1198670radicΩ

119872

int

1+119911

1

119889119909

radicΩ119881Ω

119872+ 1199093

= 24 times 1028119868 (119911) cm

(16)


0=


119872= 03


URCA]120576 = 38 times 10

50120576119889(119872

⊙

)

175

[erg sminus1] (17)

where 120576119889= 1198892 119877



119869URCA]120576 ≃ 522 times 10

minus8

times120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(18)


119889≪ 1 119868(119911) = (1 + 119911) int1+119911

1119889119909radic23 + 1199093 As it is



120587minus+ 119899 997888rarr 119899 + 119890

minus+ ]

119890 120587

minus+ 119899 997888rarr 119899 + 120583

minus+ ]

120583(19)


120587

]120576 = 578 times 1058120576119889(119872

⊙

)

175

[erg sminus1] (20)


119869120587

]120576 ≃ 791120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(21)



119889 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119889 + ]

119890 (22)

119904 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119904 + ]

119890 (23)


119897]119897


119869119902

]120576 ≃ 7068120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(24)

8 Simulation



5E6

3E7

2E7

25E7

1E7

15E7

0


z = 001

z = 007

z = 002

z = 07

z = 05

z = 01

z = 003

z = 005

10000

20000

30000

40000

50000

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1 srminus

1 )

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1srminus

1 )

00

4 8 12 16 200 4 8 12 16


5E7

2E8

2E5

1E8

15E8

16E5

12E5

28E5

24E5

00 4 8 12 16


(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

00

4 8 12 16 20

z = 001

z = 002

z = 003

z = 005

40000

80000

z = 007

z = 07

z = 05

z = 01






Seed and119869119894


BH 119872⊙





6 7 8 9 10 11

log (MBHM⊙)

1

2

3

4

5

6

log

(MSe

edBH

M⊙)









]120576 with the average119869120587






119889= 1198892 119903

119892≪ 1



120576119889≃ 169times10


⊙(2 119877

119892= 59times10



9 Conclusions






119872BH119872⊙≃ 11 times 10


Appendices







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587





119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







yields

119911Seed≃ 119911 + 2292 times 10

28 119877119889

119903OV(119872BH119872

⊙

)

2119882

119871bol (15)



2


119871(119911) is the



1015840

]119877(1199051)119877(1199050) = 1015840

](1 + 119911)minus1 of


119872= 120588

119872120588


119881=

120588119881120588

119888 therebyΩ

119881+Ω


120588119888= 3119867

2

0(8120587119866


1198670[77]

119863119871(119911) =

(1 + 119911) 119888

1198670radicΩ

119872

int

1+119911

1

119889119909

radicΩ119881Ω

119872+ 1199093

= 24 times 1028119868 (119911) cm

(16)


0=


119872= 03


URCA]120576 = 38 times 10

50120576119889(119872

⊙

)

175

[erg sminus1] (17)

where 120576119889= 1198892 119877



119869URCA]120576 ≃ 522 times 10

minus8

times120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(18)


119889≪ 1 119868(119911) = (1 + 119911) int1+119911

1119889119909radic23 + 1199093 As it is



120587minus+ 119899 997888rarr 119899 + 119890

minus+ ]

119890 120587

minus+ 119899 997888rarr 119899 + 120583

minus+ ]

120583(19)


120587

]120576 = 578 times 1058120576119889(119872

⊙

)

175

[erg sminus1] (20)


119869120587

]120576 ≃ 791120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(21)



119889 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119889 + ]

119890 (22)

119904 997888rarr 119906 + 119890minus+ ]

119890 119906 + 119890

minus997888rarr 119904 + ]

119890 (23)


119897]119897


119869119902

]120576 ≃ 7068120576119889

1198682 (119911) (1 + 119911)(119872

⊙

)

175


(24)

8 Simulation



5E6

3E7

2E7

25E7

1E7

15E7

0


z = 001

z = 007

z = 002

z = 07

z = 05

z = 01

z = 003

z = 005

10000

20000

30000

40000

50000

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1 srminus

1 )

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1srminus

1 )

00

4 8 12 16 200 4 8 12 16


5E7

2E8

2E5

1E8

15E8

16E5

12E5

28E5

24E5

00 4 8 12 16


(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

00

4 8 12 16 20

z = 001

z = 002

z = 003

z = 005

40000

80000

z = 007

z = 07

z = 05

z = 01






Seed and119869119894


BH 119872⊙





6 7 8 9 10 11

log (MBHM⊙)

1

2

3

4

5

6

log

(MSe

edBH

M⊙)









]120576 with the average119869120587






119889= 1198892 119903

119892≪ 1



120576119889≃ 169times10


⊙(2 119877

119892= 59times10



9 Conclusions






119872BH119872⊙≃ 11 times 10


Appendices







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587





119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




5E6

3E7

2E7

25E7

1E7

15E7

0


z = 001

z = 007

z = 002

z = 07

z = 05

z = 01

z = 003

z = 005

10000

20000

30000

40000

50000

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1 srminus

1 )

(dJq 120576dy2)120576 d

(041

ergc

mminus

2sminus

1srminus

1 )

00

4 8 12 16 200 4 8 12 16


5E7

2E8

2E5

1E8

15E8

16E5

12E5

28E5

24E5

00 4 8 12 16


(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

(dJq

120576dy1)120576 d

(01

ergc

mminus

2sminus

1 srminus

1 )

00

4 8 12 16 20

z = 001

z = 002

z = 003

z = 005

40000

80000

z = 007

z = 07

z = 05

z = 01






Seed and119869119894


BH 119872⊙





6 7 8 9 10 11

log (MBHM⊙)

1

2

3

4

5

6

log

(MSe

edBH

M⊙)









]120576 with the average119869120587






119889= 1198892 119903

119892≪ 1



120576119889≃ 169times10


⊙(2 119877

119892= 59times10



9 Conclusions






119872BH119872⊙≃ 11 times 10


Appendices







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587





119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




6 7 8 9 10 11

log (MBHM⊙)

1

2

3

4

5

6

log

(MSe

edBH

M⊙)









]120576 with the average119869120587






119889= 1198892 119903

119892≪ 1



120576119889≃ 169times10


⊙(2 119877

119892= 59times10



9 Conclusions






119872BH119872⊙≃ 11 times 10


Appendices







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587





119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587





119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Table 3 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587




4rarr M

4 written


4) and flat (119872



4





119897=

119898

120583120583


matrix (119898


119897equiv 120597

119897120583 and 120597

119897= 120597120597119909


119898at





equations Namely

120583=

119897

120583119890119897

120583120583

119897= Ω

119898

119897119890119898 (A1)


119897= 120597

119897120595120583

119896 and




119894 = = 120583otimes

120583

= Ω119898

119897119890119898otimes 120599

119897isin M

4 (A2)




Ω 1198724rarr 119881

4and Ω 119881

4rarr

4 where 119881


∘







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Table 4 Continued


119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



4rarr 119881

4(Ω

120583

] equiv

120575120583








Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587



Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Table 5 Continued

Name 119911 Type log(119877119889

119903ov) log(119872BH

119872⊙

) log(119872

SeedBH

119872⊙

) log119879BH 119911Seed

119869119902

119869URCA

119869120587






1198726(rarr







119880loc= 119880 (1)

119884times 119880 (1) equiv 119880 (1)

119884times diag [119878119880 (2)] (A3)



119884and




3+ 1198842 where 119876119889 is






119886= 0)









4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








4rarr

2 has the fiber1198782= 119901





4

1198791

1= 119879

2

2= 119879

3

3= minus () 119879

0

0= minus () (A4)




gauge [71] as

Δ1198860=1

2

00

12059700

1205971198860

()

minus [33

12059733

1205971198860

+ 11

12059711

1205971198860

+ 22

12059722

1205971198860

] ()


119886) 119886 =

1

2

00

12059700

120597119886 ()

minus [33

12059733

120597119886+

11

12059711

120597119886+

22

12059722

120597119886] ()

times 120579 (120582119886minus

minus13)

(A5)


119886119888 ≃


4rarr 119881





tan 3= minus119909

1=


120583of


= 119864 12= 119875

12cos

3

3= 119875

3minus tan

3119898119888

=

100381610038161003816100381610038161003816100381610038161003816

(119898 minus tan 3

1198753

119888)

2

+sin23

1198752

1+ 119875

2

2

1198882minus tan2

3

1198642

1198884

100381610038161003816100381610038161003816100381610038161003816

12

(A6)



00= (1 minus 119909

0)2

+ 1199092

120583] = 0 (120583 = ])

33= minus [(1 + 119909

0)2

+ 1199092]

11= minus

2

22= minus

2sin2

(A7)


⊙




= 120583]

120583

otimes ]= (

120583 ])

120583



120583




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




log(PP

OV)

log(120588120588OV)

30

20

10

0

minus10

0 10 20 30

AGN

branch

branch

Neutron star


36[erg cmminus3


]respectively


4


4 spanned by the


0

3) where

119886=

119886

120583120583


forms = (0

3


119887

= 119887

120583119889



119887

= 120575119887



0



119886

120583119887

120583= 120575

119887




4and any vector

isin M


119886


0


3


119886

119887



119897()) isin 119866119871(4 )

for all as follows

119898

120583=

120583

119896120587119898

119896

120583

119897=

120583

119896120587119896

119897

120583

119896120583

119898= 120575

119896

119898

119886

119898=

119886

120583

119898

120583

119886

119897=

119886

120583120583

119897

(B1)

where 120583]119896

120583119904

]= 120578

119896119904 120578

119896119904is themetric on119872

4 A deformation

tensorΩ119898

119897= 120587

119898

119896120587119896


119886=

119886

119898119890119898

119886

= 119886

119897120599119897

119890119896= 120587

119898

119896119890119898 120599

119896

= 120587119896

119897120599119897

(B2)

and 119894 = 119886otimes

119886

= 119890119896otimes120599

119896



= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119897119887

119898120599119897otimes 120599

119898

= 120574119897119898120599119897otimes 120599

119898

(B3)


119897119898=

119900119886119887119886

119897119887


120583] = Υ

2120578120583] + 120583] (B4)

provided that Υ = 119886119886= 120587

119896

119896and

120583] = (120574119886119897 minus Υ

2119900119886119897)

119886

120583119897

V

= (120574119896119904minus Υ

2120578119896119904)

119896

120583119904

V

(B5)



119886119887=

(119886

119887) =

120583]119886120583119887


119886

120583119887

]because

119886

120583119886

] = 120575120583


4 The 120574


(119886119897)



119886119888119888

119897(or

resp in terms of 120587(119896119897)

and 120587[119896119897]

where 120587119896119897= 120578

119896119904120587119904

119897) as

120574119886119897= Υ

2

119900119886119897+ 2ΥΘ

119886119897+ 119900

119888119889Θ

119888

119886Θ

119889

119897

+ 119900119888119889(Θ

119888

119886119889

119897+

119888

119886Θ

119889

119897) + 119900

119888119889119888

119886119889

119897

(B6)

where

119886119897= Υ119900

119886119897+ Θ

119886119897+

119886119897 (B7)

Υ = 119886

119886 Θ


119886119897is




4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





4

read

119886

= 119889119886

=1

2119886

119887119888119887

and 119888

(B8)



119888

119886119887= minus

119888

([119886

119887]) =

119886

120583119887

](

120583119888

] minus ]119888

120583)

= minus119888

120583[

119886(

119887

120583) minus

119887(

119886

120583)]

= 2120587119888

119897119898

120583(120587

minus1119898

[119886120583120587minus1119897

119887])

(B9)


Γ119886

119887119888=1

2(

119886

119887119888minus 119900

1198861198861015840

1199001198871198871015840

1198871015840

1198861015840119888minus 119900

1198861198861015840

1199001198881198881015840

1198881015840

1198861015840119887) (B10)



Ω

119898

119897=∘

119863

119898

120583

∘

120595120583

119897and Ω

120583

] =

120583

120588120588

] provided

∘

119890120583=∘

119863

119897

120583119890119897

∘

119890120583

∘

120595120583

119897=∘

Ω

119898

119897119890119898

120588=

120583

120588

∘

119890120583

120588120588

] = Ω120583

V∘

119890120583

(B11)


119863

119898

120583=∘

119890120583

119896 ∘

120587119898

119896

∘

120595120583

119897=∘

119890120583

119896

∘

120587119896

119897

∘

119890120583

119896 ∘

119890120583

119898= 120575

119896

119898

∘

120587119886

119898

=∘

119890119886

120583 ∘

119863

119898

120583

∘

120587119886

119897=∘

119890119886

120583

∘

120595120583

119897

(B12)

where∘

Ω

119898

119897=∘

120587119898

120588

∘

120587120588

119897and Ω

120583

] = 120583

120588120588

] We also have∘

119892120583]∘

119890119896

120583 ∘

119890119904

]= 120578

119896119904and

120583

120588= 119890]

120583]120588

120588

] = 119890120588

120583120583

]

119890]120583119890]120588= 120575

120583

120588

119886

120583= 119890

119886

120588

120583

120588

119886

] = 119890119886

120588120588

]

(B13)

The norm 119889 ∘119904 equiv 119894∘



119894

∘

119889 =∘

119890

∘

120599 =∘

Ω

119898

119897119890119898otimes 120599

119897isin 119881

4 (B14)


119892 =∘

119892120583]∘

120599

120583

otimes

∘

120599

]=∘

119892 (∘

119890120583∘

119890])∘

120599

120583

otimes

∘

120599

] (B15)


119879 ∘1199091198814

read∘

119862

119886

= 119889

∘

120599

119886

=1

2

∘

119862

119886

119887119888

∘

120599

119887

and

∘

120599

119888

(B16)


119862119886



119862

119888

119887119888= minus

∘

120599

119888

([∘

119890119886∘

119890119887])

=∘

119890119886

120583 ∘

119890119887

](

∘

120597120583

∘

119890119888

] minus∘

120597]∘

119890119888

120583)

= minus∘

119890119888

120583[∘

119890119886(∘

119890119887

120583

) minus∘

119890119887(∘

119890119886

120583

)]

(B17)


∘

Γ119886119887=∘

119890[119886rfloor119889

∘

120599119887]minus1

2(∘

119890119886rfloor∘

119890119887rfloor119889

∘

120599119888) and

∘

120599

119888

(B18)

where∘


∘

120599119888=

119900119888119887

∘

120599

119887


4and119872

4as

119894 = = 120588otimes

120588

= 119886otimes

119886

= Ω120583

]∘

119890120583otimes

∘

120599

]

= Ω119886

119887119890119886119887

= Ω119898

119897119890119898otimes 120599

119897isin M

4

(B19)

provided

Ω119886

119887=

119886

119888119888

119887= Ω

120583

]∘

119890119886

120583

∘

119890119887

]

120588=

]120588

∘

119890]

120588

= 120583

120588∘

120599

120583

119888=

119888

119886 ∘

119890119886

119888

= 119888

119887

∘

120599

119887

(B20)


119886

119887120583=

119888

119886 ∘

120596119888

119889120583119889

119887+

119888

119886120583119888

119887= 120587

119897

119886120583120587119897

119887 (B21)


= 119900119886119887119886

otimes 119887

= 119900119886119887119886

119888119887

119889

∘

120599

119888

otimes

∘

120599

119889

= 119888119889

∘

120599

119888

otimes

∘

120599

119889

(B22)


119886119887= 119900

119888119889119888

119886119889

119887reads

Γ119886

119887119888=1

2(∘

119862

119886

119887119888minus

1198861198861015840

1198871198871015840

∘

119862

1198871015840

1198861015840119888minus

1198861198861015840

1198881198881015840

∘

119862

1198881015840

1198861015840119887)

+1

21198861198861015840(∘

119890119888rfloor119889

1198871198861015840 minus∘

119890119887rfloor119889

1198881198861015840 minus∘

1198901198861015840rfloor119889

119887119888)

(B23)





119886119888119888

119887 So

119886119887= Υ119900

119886119887+ Θ

119886119887+

119886119887 (B24)

where Υ = 119886119886 Θ


119886119887


120583] () = Υ

2

()∘

119892120583] + 120583] () (B25)


where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where

120583] () = [119886119887 minus Υ

2

119900119886119887]∘

119890119886

120583

∘

119890119887

] (B26)


120583]() = 119900

119888119889minus1119886

119888minus1119887

119889

∘

119890119886

120583 ∘

119890119887

] (B27)

where minus1119886119888119888

119887=

119888

119887minus1119886

119888= 120575

119886



Γ119886119887=

[119886rfloor119889

119887]minus1

2(

119886rfloor

119887rfloor119889

119888) and

119888

(B28)


119888=

119900119888119887119887


Γ120583

120588120590= Γ

120583

120588120590+ Π

120583

120588120590 (B29)

provided

Π120583

120588120590= 2

120583]] (120588nabla120590)Υ minus

120588120590119892120583]nabla]Υ

+1

2120583](nabla


(B30)


120583] suchthat

120583]]120588= 120575

120588


120588120590






119886


119886


(119886119887

119886

119886

119887

) sim D (119886119887

119886

Γ119886

119887

) (B31)




Γ120588

120583] =∘

Γ

120588

120583] + 120588

120583] minus 120588

120583] +1

2

120588

(120583]) (B32)

where∘

Γ

120588


120583] = 2(120583])120588

+ 120588


120583] = (12)120588

120583] = Γ120588


120588



120583997888rarr D

120583=

120583minus119894

2(

119886119887

120583minus

119886119887

120583) 119869

119886119887 (B33)

with 119869119886119887






120583only or

120583]

() equiv 119886

120583119887

]120583]

119886119887

() = ( Γ)

equiv 120588]120583

120588120583] (Γ)

(B34)


Let 119886119887 = 119886119887

120583and 119889


119892=∘

119878 + 119876 (C1)

where the integral

∘

119878 = minus1

4aeligint⋆

∘

119877 = minus1

4aeligint⋆

∘

119877119888119889and

119888

and 119889

= minus1

2aeligint∘


(C2)


119873119888

4 119878119876is


119901rarr Ω


1198861 sdotsdotsdot119886119901

) = 1205761198861 sdotsdotsdot119886119899



= 1198861

and 1198862


4space the


1-forms 119886119894= 119900

119886119894119887119887


120575119876=1

aeligint⋆T

119886119887and 120575

119886119887 (C3)

where

⋆ T119886119887=1

2⋆ (

119886and

119887) =

119888

and 119889

120576119888119889119886119887

=1

2

119888

120583] and 119889

120572120576119886119887119888119889120583]120572

(C4)


and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




and also

119886

= 119886

= 119889119886

+ 119886

119887and

119887

(C5)



form 119886119887 reads

120575119898= int120575

119898

= int(minus ⋆ 119886and 120575

119886

+1

2⋆ Σ

119886119887and 120575

119886119887)

(C6)



119886 by

⋆119886=1

3120583

119886120576120583]120572120573

]120572120573(C7)

and ⋆Σ119886119887= minus ⋆ Σ



119886119887119888 namely

⋆Σ119886119887=

120583

119886119887120576120583]120572120573

]120572120573 (C8)




(1)1

2

∘

119877119888119889and

119888

= aelig119889= 0

(2) ⋆ T119886119887= minus1

2aelig ⋆ Σ

119886119887

(3)120575

119898

120575Φ

= 0120575

119898

120575Φ

= 0

(C9)


119897

119886= 120578

119897119898⟨

119886 119890

119898⟩

119886

119897= 120578

119897119898119886

(120599minus1)119898

(C10)




4




120583Φ()



Φ1015840

() = 119880119881() Φ ()

[120583() nabla

120583Φ ()]

1015840

= 119880119881() [

120583() nabla

120583Φ ()]

(D1)


metric 120583] = (12)(120583] + ]120583) nabla120583=

120583+ Γ

120583 where

Γ120583() = (12)119869

119886119887119886

]()


119886119887are





119863(Λ) = 119868+ (12)119886119887119869119886119887

119886119887= minus




119896=

120575119897

119886120578119887119896minus 120575

119897

119887120578119886119896 but 120583() =

119886

120583()120574

119886 and 119869119886119887= minus(14)[120574

119886 120574

119887]


4 119866

119881 ) with the


( 119880119881) where isin 119881

4and 119880

119881isin 119866




119894 of 119881

4with

isin U119894 sub 119881

4satisfy ⋃

120572U

120572= 119881



minus1(U

119894) = U

119894times


minus1() =


U119894times

1198814119866119881rarr

minus1(U


119894maps minus1(U

119894) onto


1198814119866119881 Here times

1198814represents


(119894( 119880

119881)) = and

119894( 119880

119881) =

119894( (119894119889)

119866119881)119880

119881=

119894()119880

119881









4 119880






4must be



120595(119886)


119877120595(119886) Φ 997888rarr Φ

119878 (119886) 119877120595(119886) (120574

119896119863

119896Φ) 997888rarr (

]() nabla]Φ)

(D2)


Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Φ998400( x) =UVΦ(x)

R998400120595(x x)

UlocΦ998400(x) =U

locΦ(x)

UV = R998400120595U

locRminus1120595

Φ(x)

Φ(x)

R120595(x x)

Scheme 1 The GGP


minus1(119865) = (14)

119897

120583

120583

119897119863

119896= 120597

119896minus 119894aelig 119886




() = 120583


120575

119898119877 (119886)Φ

119897sdotsdotsdot119898(119909)



119897sdotsdotsdot119898(119909)

(D3)

and also

]() nabla]Φ


()

= 119878 (119865) 120583


120575

119898119877 (119886) 120574

119896119863

119896Φ

119897sdotsdotsdot119898(119909)

(D4)


120583

119897= 119890

120583

119897=

(120597119909120583120597119883








1198726= 119877

3

+oplus 119877

3

minus= 119877

3oplus 119879

3


The 119890(120582120572)

= 119874120582times 120590




120582

and 120590120572imply

⟨119874120582 119874

120591⟩ =

lowast120575120582120591 ⟨120590

120572 120590

120573⟩ = 120575

120572120573 (E2)

where 120575120572120573


120575120582120591= 1 minus 120575

120582120591



0120572= 120585

0times 120590

120572



0plusmn 120585) 1205852

0= minus120585

2= 1 ⟨120585

0 120585⟩ = 0







6rarr 119872

4= 119877

3oplus 119879



4can



120572 119890

0120572) on


0) By this we



such that 119860(120582120572)119861(120582120572)= 119860

120572119861120572+ 119860

01205721198610120572 then upon


01205721198610120572= 119860

0120572⟨ 119890

0120572 119890

0120573⟩119861

0120573= 119860

0⟨ 119890

0 119890

0⟩119861

0= 119860

01198610




(119886)

(+120572)(119886) = Q

120591

(+120572)(119886) 119874

120591= 119874

++ aelig119886

(+120572)119874minus

(minus120572)(119886) = Q

120591

(minus120572)(119886) 119874

120591= 119874

minus+ aelig119886

(minus120572)119874+

(E3)

where Q (=Q120591




(120582120572)(120579) =

R120573

(120582120572)(120579)120590



120582 In fact


120575Q120591

119860(119886) = aelig120575119886

119860119883

120591

120582isin 119876

120575R (120579) = minus119894

2119872

120572120573120575120596

120572120573isin 119878119874 (3)

(E4)


120582=lowast120575120591

120582and 119868

119894= 120590

1198942 where 120590

119894


120572120573120574119868120574 and120575120596120572120573 = 120576

120572120573120574120575120579

120574The


119863= 119876 times 119878119874(3)

119863(119889119886119860119889120579119860)= 119868 + 119889119863

(119886119860120579119860)

119889119863(119886119860120579119860)= 119894 [119889119886

119860119883

119860+ 119889120579

119860119868119860]

(E5)

where 119868119860equiv 119868






119860


119863


119863= 119876 times




tan 120579119860= minusaelig119886

119860 (E6)





11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







11987212= 119872

6oplus119872

6 (E7)



The

119890(120582120583120572)

= 119874120582120583otimes 120590

120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)


⟨119874120582120583 119874

120591]⟩=lowast120575120582120591

lowast120575120583] 119874

120582120583= 119874

120582otimes 119874

120583

119874120582120583larrrarr


120572larrrarr 119877

3

(E9)

where 120577 = (120578 119906) isin 11987212(120578 isin 119872

6and 119906 isin 119872

6) So the


1198726= 119877

3

+oplus 119877

3

minus= 119879

3

oplus 1198753

sgn (1198793



12)

= 119863 (119886) 119890 (E11)


= 119862 (119886)119874 = 119877 (119886) 120590 (E12)


119862120591](120582120583120572)

(119886) = 120575120591

120582120575]120583+ aelig119886

(120582120583120572)

lowast120575120591

120582

lowast

120575]120583 (E13)






12


field 119886119860= (119886

(120582120572) 119886

(120591120573)) reads

120597119861120597119861119886119860minus (1 minus 120577

minus1

0) 120597

119860120597119861119886119861

= 119869119860= minus1

2radic119892120597119892

119861119862

120597119886119860

119879119861119862

(F1)

where 119879119861119862



minus1

2(4120587119866

119873)radic119892 (119909)

120597119892119861119862(119909)

120597119909119860

119861119862 (F2)


the value

119866119873=aelig2

4120587 (F3)


6is the familiar


119886(11120572)

= 119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= 119886(22120572)

equiv1

radic2

119886(minus120572)

(F4)


119886(11120572)

= minus119886(21120572)

equiv1

radic2

119886(+120572)

119886(12120572)

= minus119886(22120572)

equiv1

radic2

119886(minus120572)

(F5)



ID-field 119886

119886(113)

= 119886(223)

= 119886(+3)=1

2(minus119886

0+ 119886)

119886(123)

= 119886(213)

= 119886(minus3)=1

2(minus119886

0minus 119886)

119886(1205821205831)

= 119886(1205821205832)

= 0 120582 120583 = 1 2

(F6)


120591120573)



6to


0= 119890

0(1 minus 119909

0) + 119890

3119909

3= 119890

3(1 + 119909

0) minus 119890

03119909


1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

1

+ (sin 120579(+3)+ sin 120579

(minus3)) 119890

2

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

01

+ (sin 120579(+3)minus sin 120579

(minus3)) 119890

02

2=1

2(cos 120579

(+3)+ cos 120579

(minus3)) 119890

2

minus (sin 120579(+3)+ sin 120579

(minus3)) 119890

1

+ (cos 120579(+3)minus cos 120579

(minus3)) 119890

02


(minus3)) 119890

01

(F7)where 119909

0equiv aelig119886

0 119909 equiv aelig119886





le 120588 lt





119873119873le


1of energy


2










0 In domain

120588119889le 120588 lt 120588














1asymp 2216 for





Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







Acknowledgment


References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics

