Electrostatics in the Surroundings of a Topologically Charged Black Hole in the Brane

Research ArticleElectrostatics in the Surroundings of a Topologically ChargedBlack Hole in the Brane

Alexis Larrantildeaga1 Natalia Herrera2 and Sara Ramirez2

1 National Astronomical Observatory National University of Colombia Bogota 11001000 Colombia2Department of Physics National University of Colombia Bogota 11001000 Colombia

Correspondence should be addressed to Alexis Larranaga ealarranagaunaleduco

Received 27 December 2013 Revised 21 January 2014 Accepted 23 January 2014 Published 27 February 2014

Academic Editor Christian Corda

Copyright copy 2014 Alexis Larranaga et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

We determine the expression for the electrostatic potential generated by a point charge held stationary in the topologically chargedblack hole spacetime arising from the Randall-Sundrum II braneworld model We treat the static electric point charge as a linearperturbation on the black hole background and an expression for the electrostatic multipole solution is given PACS 0470-s0450Gh 1125-w 4120-q 4190+e

1 Introduction

The idea of our universe as a brane embedded in a higherdimensional spacetime has recently attracted attentionAccording to the braneworld scenario the physical fields(electromagnetic Yang-Mills etc) in our 4-dimensionaluniverse are confined to the three-brane and only gravitypropagates in the bulk spacetime One of themost interestingscenarios is Randall-Sundrum II model in which it is consid-ered aZ

2-symmetric 5-dimensional asymptotically anti-de-

Sitter bulk [1] and our brane is identified as a domain wallThe 5-dimensional metric can be written in the general form1198891199042= 119890minus119865(119910)

120578120583]119889119909120583119889119909

]+ 1198891199102 and due to the appearance of

the warp factor it reproduces a large hierarchy between thescale of particle physics and gravity Moreover even if thefifth dimension is uncompactified standard 4-dimensionalgravity on the brane is reproduced However due to the cor-rection terms coming from the extradimensions significantdeviations from general relativity may occur at high energies[2ndash4]

As is well known in general relativity the exteriorspacetime of a spherical compact object is described bySchwarzschild solution In the braneworld scenario thehigh energy corrections to the energy density together withWeyl stresses from bulk gravitons imply that the exterior

metric of a spherical compact object on the brane is nolonger described by Schwarzschild metric In fact black holesolutions in the braneworldmodel are particularly interestingbecause they have considerably richer physical aspects thanblack holes in general relativity [5ndash11] The first solutionsdescribing static and spherically symmetric exterior vacuumsolutions of the braneworldmodelwere proposed byDadhichet al [12] and Germani andMaartens [13] and later they weregeneralised by Chamblin et al [14] and revisited by Sheykhiand Wang [15] This kind of solutions carries a topologicalcharge arising from the bulkWeyl tensor and the line elementresembles Reissner-Nordstrom solution with the tidal Weylparameter playing the role of the electric charge In orderto obtain this solution there was the null energy conditionimposed on the three-brane for a bulk having nonzero Weylcurvature

On the other hand the generation of an electromagneticfield by static sources in black hole backgrounds has beenconsidered in several papers beginning with the studies ofCopson [16] Cohen and Wald [17] and Hanni and Ruffini[18] where they discussed the electric field of a point chargein Schwarzschild background Afterwards Petterson [19]and later Linet [20] studied the magnetic field of a currentloop surrounding a Schwarzschild black hole More recentlysimilar studies have been performed in other background

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 146094 6 pageshttpdxdoiorg1011552014146094

2 Advances in High Energy Physics

geometries including charged black holes of general relativityand black holes with Brans-Dicke modifications or withconical defects [21ndash25]

In this paper we are interested in obtaining an expressionfor the electrostatic potential generated by a point chargeheld stationary in the region outside the event horizon of abraneworld black holeThere are several vacuum solutions ofthe spherically symmetric static gravitational field equationson the brane with arbitrary parameters which depend onproperties of the bulk or that are simply put in by using gen-eral physical considerations At the present it is theoreticallynot known whether these parameters should be universalover all braneworld black holes or whether each separateblack hole may have different values of them Similarlythere is not a single complete solution in the sense that themetric in the bulk is uniquely known Since this situationis unsatisfactory from a theoretical point of view it may beuseful to investigate more closely the observational effectsof the black hole properties Specifically we consider theeffects due to the projections of theWeyl tensor and how theyspecify the corrections on the electrostatic potential Sincethe generic form of the Weyl tensor in the full 5-dimensionalspacetime is yet unknown the effects of known solutionsmust be studied on a case-by-case basis Although one canin principle constrain the projections this only yields verymild constraints on the 5-dimensional Weyl tensor [26 27]Therefore we decided to derive the electrostatic potentialgenerated by a point charge held stationary in the outsideregion of the event horizon of the particular solution in theRandall-Sundrum braneworld model obtained by Dadhichet al [12] and revisited by Sheykhi and Wang [15] In thiscase tidal charge is arising via gravitational effects fromthe fifth dimension that is it is arising from the projectiononto the brane of free gravitational field effects in the bulkand is this term the one that will modify the electrostaticpotential produced by the particleWe obtain the correctionsdue to the topological charge arising from the bulk Weyltensor to the electrostatic potential and deduce the necessarycorrection to incorporate Gaussrsquos law Finally we also presentthe solution of Maxwell equations in the form of series ofmultipoles

2 The Topologically Charged Black Hole inthe Braneworld

The gravitational field on the brane is described by the Gaussand Codazzi equations of 5-dimensional gravity [2]

119866120583] = minusΛ119892120583] + 8120587119866119879120583] + 120581

4

5Π120583] minus 119864120583] (1)

where 119866120583] = 119877

120583] minus (12)119892120583]119877 is the 4-dimensional Einsteintensor and 120581

5is the 5-dimensional gravity coupling constant

1205814

5= (8120587119866

5)2

=48120587119866

120591 (2)

with 1198665the gravitational constant in five dimensions and Λ

is the 4-dimensional cosmological constant that is given in

terms of the 5-dimensional cosmological constantΛ5and the

brane tension 120591 by

Λ =1205812

5

2(Λ5+1205812

5

61205912) (3)

119879120583] is the stress-energy tensor of matter confined on the

brane Π120583] is a quadratic tensor in the stress-energy tensor

given by

Π120583]

=1

12119879119879120583] minus

1

4119879120583120590119879120590

] +1

8119892120583] (119879120572120573119879

120572120573minus1

31198792)

(4)

with 119879 = 119879120590

120590 and 119864

120583] is the projection of the 5-dimensionalbulk Weyl tensor 119862

119860119861119862119863on the brane (119864

120583] = 120575119860

120583120575119861

]119862119860119861119862119863

119899119860119899119861 with 119899119860 the unit normal to the brane) 119864

120583] encompassesthe nonlocal bulk effect and it is traceless 119864120590

120590= 0

Considering the Randall-Sundrum scenario with

Λ5= minus

1205812

5

61205912 (5)

which implies

Λ = 0 (6)

the four-dimensional Gauss and Codazzi equations for anarbitrary static spherically symmetric object have been com-pletely solved on the brane obtaining a black hole typesolution of the field equations (1) with 119879

120583] = 0 given by theline element [12 14 15 28]

1198891199042= ℎ (119903) 119889119905

2minus1198891199032

ℎ (119903)minus 1199032119889Ω2 (7)

where 119889Ω2 = 1198891205792 + sin21205791198891205932 and

ℎ (119903) = 1 minus2119866119872

119903+120573

1199032 (8)

For this model the expression of the projected Weyltensor transmitting the tidal charge stresses from the bulk tothe brane is

119864119905

119905= 119864119903

119903= minus119864120579

120579= minus119864120593

120593=120573

1199034 (9)

This result shows that the parameter 120573 can be interpretedas a tidal charge associated with the bulk Weyl tensor andtherefore there is no restriction on it to take positive aswell as negative values (other interpretations consider 120573 asa five-dimensional mass parameter as discussed in [14]) Theinduced metric in the domain wall presents horizons at theradii (taking 119866 = 1) Consider the following

119903plusmn= 119872 plusmn radic1198722 minus 120573 (10)

Advances in High Energy Physics 3

For 120573 ge 0 there is a direct analogy to the Reissner-Nordstrom solution showing two horizons that as in generalrelativity both lie inside the Schwarzschild radius 2119872 that is

0 le 119903minusle 119903+le 119903119904 (11)

Clearly there is an upper limit on 120573 namely

0 le 120573 le 1198722 (12)

However there is nothing to stop us choosing 120573 to benegative This intriguing new possibility is impossible inReissner-Nordstrom case and leads to only one horizon 119903

lowast

lying outside the corresponding Schwarzschild radius

119903lowast= 119872 + radic1198722 +

10038161003816100381610038161205731003816100381610038161003816 gt 2119872 (13)

In this case the single horizon has a greater area thatits Schwarzschild counterpart Thus one concludes that theeffect of the bulk producing a negative 120573 is to strengthen thegravitational field outside the black hole (obviously it alsoincreases the entropy anddecreases theHawking temperaturebut these facts are not important in this paper)

3 The Electrostatic Field of a Point Particle

Copson [16] and Linet [20] found the electrostatic potentialin a closed formof a point charge at rest outside the horizon ofa Schwarzschild black hole and that the multipole expansionof this potential coincides with the one given by Cohen andWald [17] and by Hanni and Ruffini [18] In this sectionwe will investigate this problem in the background of thetopologically charged black hole (7)

According to the braneworld scenario we will considerthat the physical fields are confined to the three brane andif the electromagnetic field of the point particle is assumedto be sufficiently weak so its gravitational effect is negligibleand the Einstein-Maxwell equations reduce to Maxwellrsquosequations confined in the curved backgroundof the brane (7)These are written as

nabla120588119865120588120583= 4120587119869

120583 (14)

where

119865120583] = 120597120583119860] minus 120597]119860120583 (15)

with 119860120583 the electromagnetic vector potential and 119869

120583 thecurrent density Considering that the test charge 119902 is heldstationary at the point (119903

0 1205790 1205930) the associated current

density 119869119894 vanishes while the charge density 1198690 is given by

1198690=

119902

11990320sin 120579

120575 (119903 minus 1199030) 120575 (120579 minus 120579

0) 120575 (120593 minus 120593

0) (16)

From now on we will assume that the charge 119902 is heldoutside the black hole that is 119903

0gt 119903+in the case 120573 ge 0

and 1199030

gt 119903lowastin the case 120573 lt 0 We will not consider

here the possibility of having the charge inside the eventhorizon because as well known this construction leads tothe description of a charged black hole of the Reissner-Nordstrom type [15]

The spatial components of the potential vanish 119860119894 = 0while the temporal component 1198600 = 120601 is determined by the120583 = 0 component of (14) giving the differential equation

120597

120597119903(1199032 120597120601

120597119903)

+ (1 minus2119872

119903+120573

1199032)

minus1

times [1

sin 120579120597

120597120579(sin 120579

120597120601

120597120579) +

1

sin21205791205972120601

1205971205932]

= minus119903241205871198690

(17)

In order to solve this equation we will perform thesubstitutions

119903 = 119903 + 119903minus

120601 (119903 120579 120593) =119903 minus 119903minus

119903120601 (119903 minus 119903

minus 120579 120593)

(18)

which turn the differential equation into

120597

120597119903(1199032 120597120601

120597119903)

+ (1 minus2119898

119903)

minus1

[1

sin 120579120597

120597120579(sin 120579

120597120601

120597120579) +

1

sin21205791205972120601

1205971205932]

= minus4120587119902

sin 120579120575 (119903 minus 119903

0) 120575 (120579 minus 120579

0) 120575 (120593 minus 120593

0)

(19)

where119898 = radic1198722 minus 120573 and 119902 = 1199021199030(1199030+ 119903minus) Note that in the

extremal case 119898 = 0 or equivalently1198722 = 120573 (19) becomesLaplacersquos equation in Minkowski spacetime On the otherhand when119898 = 0 (including values with 0 le 120573 lt 119872

2 as wellas 120573 lt 0) (19) is formally identical to the partial differentialequation for the electrostatic potential in the Schwarzschildspacetime Hence assuming 120579

0= 0 120593

0= 0 and proceeding

in analogy with Copson [16] we find that the solution of (17)after using the substitutions (18) is

120601119862(119903 120579) =

119902

1199030119903

(119903 minus 119872) (1199030minus119872) minus (119872

2minus 120573) cos 120579

radic(119903 minus119872)2+ (1199030minus119872)2

minus 2 (119903 minus119872) (1199030minus119872) cos 120579 minus (1198722 minus 120573) sin2120579

(20)


This solution describes as stated before the potential for acharge 119902 situated at the point (119903

0 0 0) and it is regular for any

value of 119903 outside the event horizon except at the position ofthe point charge However as shown by Linet [20] and Leauteand Linet [21] it is easy to see that this solution includesanother source Note that for 119903 rarr infin the potential 120601

119862takes

the asymptotic form

120601119862(119903 120579) 997888rarr

119902

119903(1 minus

119872

1199030

) (21)

and consequently by virtue of Gaussrsquos theorem there is asecond charge with value minus119902119872119903

0 Solution (20) has only the

source 119902 outside the horizon and thus the second chargemustlie inside the horizon Moreover since the only electric fieldwhich is regular outside the horizon is spherical symmetric[29] the electrostatic potential for our physical system willbe of the form

120601 (119903 120579) = 120601119862(119903 120579) +

119902

119903

119872

1199030

(22)

The electrostatic solution can be analysed for all values 119903Considering this equation it is clear that when the charge 119902is approaching the outer horizon 119903

+in the case 120573 ge 0 or the

horizon 119903lowastin the case120573 lt 0 the electrostatic potential120601 tends

to become spherically symmetric recovering a charged blackhole of the Reissner-Nordstrom type as stated before

When 1199030gt 2119872 there is only one charge 119902 in the case

120573 ge 0 as well as in the case 120573 lt 0 However when 119903+le 1199030lt

2119872 in the case 120573 ge 0 there are also two other charges 1199021=

119902(1minus21198721199030) and 119902

2= minus1199021located at (119903 = 2119872minus119903

0 120579 = 120587) and

119903 = 0 respectivelyThis behaviour is obviously not present for120573 lt 0 because 119903

lowastgt 2119872 as shown in Section 2 of this paper

Finally in order to write formally the electrostatic poten-tial of a charge 119902 located at the arbitrary point with coor-dinates (119903

0 1205790 1205930) we simply replace the term cos 120579 by the

function 120582(120579 120593) = cos 120579 cos 1205790+ sin 120579 sin 120579

0cos(120593 minus 120593

0) This

gives the general solution

120601 (119903 120579 120593) =119902

1199030119903


2minus 120573) 120582 (120579 120593)


minus 2 (119903 minus119872) (1199030minus119872)120582 (120579 120593) minus (1198722 minus 120573) [1 minus 1205822 (120579 120593)]

+119902

119903

119872

1199030

(23)

31 Multipole Expansion If the angular part of the potentialin (19) is expanded in terms of Legendre polynomials in cos 120579

one may obtain the well-known electric multipole solutionsUsing again the substitutions (18) the radial parts of the twolinearly independent multipole solutions of (17) are

119892119897(119903) =

1 for 119897 = 0

2119897119897 (119897 minus 1)(119872

2minus 120573)1198972

(2119897)

1199032minus 2119872119903 + 120573

119903

119889119875119897

119889119903(

119903 minus119872

radic1198722 minus 120573

) for 119897 = 1 2 (24)

119891119897(119903) = minus

(2119897 + 1)

2119897 (119897 + 1)119897(1198722 minus 120573)(119897+1)2

1199032minus 2119872119903 + 120573

119903

119889119876119897

119889119903(

119903 minus119872


) for 119897 = 0 1 2 (25)

where 119875119897and 119876

119897are the two types of Legendre functions

These functions satisfy the following

(1) for 119897 = 0 1198920(119903) = 1 and 119891

0(119903) = 1119903

(2) for all values of 119897 as 119903 rarr infin the leading term of 119892119897(119903)

is 119903119897 while the leading term of 119891119897(119903) is 119903minus(119897+1)

(3) as 119903 rarr 119903+ 119892119897(119903) rarr 0 (for 119897 = 0) while 119891

119897(119903) rarr

finite constant However 119889119891119897119889119903 blows up for 119897 = 0

when 119903 rarr 119903+

The above properties let us infer that only the set of solutions(25) has the correct behaviour at infinity and only themultipole term 119897 = 0 does not produce a divergent field at thehorizon 119903 = 119903

+ This set of solutions reproduces Israelrsquos result

[29] when 120573 = 0 (ie for the Schwarzschild black hole)

As is well known the gravitational field modifies theelectrostatic interaction of a charged particle in such a waythat the particle experiences a finite self-force [22 23 30ndash34]whose origin comes from the spacetime curvature associatedwith the gravitational field However even in the absence ofcurvature it was shown that a charged point particle [35 36]or a linear charge distribution [37] placed at rest may becomesubject to a finite repulsive electrostatic self-force (see also[24]) In these references the origin of the force is the distor-tion in the particle field caused by the lack of global flatnessof the spacetime of a cosmic string Therefore one mayconclude that the modifications of the electrostatic potentialcome from two contributions one of geometric origin andthe other of a topological one In a forthcoming paper we willshow that considering the topologically charged black hole


described by the line element (7) as the background metricboth kinds of contributions appear in the electrostatic self-force of a charged particle

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Universidad Nacional deColombia Hermes Project Code 18140

References

[1] L Randall and R Sundrum ldquoAn alternative to compactifica-tionrdquo Physical Review Letters vol 83 no 23 pp 4690ndash46931999

[2] T Shiromizu K-I Maeda and M Sasaki ldquoThe Einsteinequations on the 3-brane worldrdquo Physical Review D vol 62 no2 Article ID 024012 6 pages 2000

[3] M Sasaki T Shiromizuand and K Maeda ldquoGravity stabil-ity and energy conservation on the Randall-Sundrum braneworldrdquo Physical Review D vol 62 Article ID 024008 8 pages2000

[4] S Nojiri S D Odintsov and S Ogushi ldquoFriedmann-Robertson-Walker brane cosmological equations from the five-dimensional bulk (A)dS black holerdquo International Journal ofModern Physics A vol 17 no 32 pp 4809ndash4870 2002

[5] S Nojiri O Obregon S D Odintsov and S Ogushi ldquoDilatonicbrane-world black holes gravity localization and Newtonrsquosconstantrdquo Physical Review D vol 62 no 6 Article ID 06401710 pages 2000

[6] P Kanti and K Tamvakis ldquoQuest for localized 4D black holesin brane worldsrdquo Physical Review D vol 65 no 8 Article ID084010 2002

[7] R Casadio A Fabbri and L Mazzacurati ldquoNew black holes inthe brane worldrdquo Physical Review D vol 65 no 8 Article ID084040 2002

[8] R Casadio and L Mazzacurati ldquoBulk shape of brane-worldblack holesrdquo Modern Physics Letters A vol 18 no 9 pp 651ndash660 2003

[9] R Casadio ldquoOn brane-world black holes and short scalephysicsrdquo Annals of Physics vol 307 no 2 pp 195ndash208 2003

[10] A Chamblin S W Hawking and H S Reall ldquoBrane-worldblack holesrdquo Physical Review D vol 61 no 6 Article ID 0650076 pages 2000

[11] R Casadio and B Harms ldquoBlack hole evaporation and largeextra dimensionsrdquo Physics Letters B vol 487 no 3-4 pp 209ndash214 2000

[12] N Dadhich R Maartens P Papadopoulos and V RezanialdquoQuarkmixings in SU(6)X SU(2)(R) and suppression ofV(ub)rdquoPhysics Letters B vol 487 no 1-2 pp 104ndash109 2000

[13] C Germani and RMaartens ldquoStars in the braneworldrdquo PhysicalReview D vol 64 no 12 Article ID 124010 2001

[14] A Chamblin H S Reall H-A Shinkai and T ShiromizuldquoCharged brane-world black holesrdquo Physical Review D vol 63no 6 Article ID 064015 2001

[15] A Sheykhi and B Wang ldquoOn topological charged braneworldblack holesrdquoModern Physics Letters A vol 24 no 31 pp 2531ndash2538 2009

[16] E Copson ldquoOn electrostatics in a gravitational fieldrdquo Proceed-ings of the Royal Society A vol 118 pp 184ndash194 1928

[17] J M Cohen and R M Wald ldquoPoint charge in the vicinity of aSchwarzschild black holerdquo Journal of Mathematical Physics vol12 no 9 pp 1845ndash1849 1971

[18] R S Hanni and R Ruffini ldquoLines of force of a point charge neara schwarzschild black holerdquo Physical Review D vol 8 no 10 pp3259ndash3265 1973

[19] J A Petterson ldquoMagnetic field of a current loop around aschwarzschild black holerdquo Physical Review D vol 10 no 10 pp3166ndash3170 1974

[20] B Linet ldquoElectrostatics and magnetostatics in theSchwarzschild metricrdquo Journal of Physics A vol 9 no 7pp 1081ndash1087 1976

[21] B Leaute and B Linet ldquoElectrostatics in a Reissner-Nordstromspace-timerdquo Physics Letters A vol 58 pp 5ndash6 1976

[22] B Leaute and B Linet ldquoPrinciple of equivalence and electro-magnetismrdquo International Journal ofTheoretical Physics vol 22no 1 pp 67ndash72 1983

[23] B Leaute and B Linet ldquoSelf-interaction of an electric ormagnetic dipole in the Schwarzschild space-timerdquo ClassicalQuantum Gravity vol 1 no 1 p 5 1985

[24] J Spinelly and V B Bezerra ldquoElectrostatic in Reissner-Nordstrom space-time with a conical defectrdquo Modern PhysicsLetters A vol 15 no 32 pp 1961ndash1966 2000

[25] M Watanabe and A W C Lun ldquoElectrostatic potential ofa point charge in a Brans-Dicke Reissner-Nordstrom fieldrdquoPhysical Review D vol 88 Article ID 045007 11 pages 2013

[26] RMaartens ldquoBrane-world gravityrdquo Living Reviews in Relativityvol 7 pp 1ndash99 2004

[27] C S J Pun Z Kovacs and T Harko ldquoThin accretion disksonto brane world black holesrdquo Physical Review D vol 78 no8 Article ID 084015 2008

[28] C G Bohmer T Harko and F S N Lobo ldquoWormholegeometries with conformal motionsrdquo Classical and QuantumGravity vol 25 no 7 Article ID 075016 2008

[29] W Israel ldquoEvent horizons in static electrovac space-timesrdquoCommunications in Mathematical Physics vol 8 no 3 pp 245ndash260 1968

[30] A Vilenkin ldquoSelf-interaction of charged particles in the grav-itational fieldrdquo Physical Review D vol 20 no 2 pp 373ndash3761979

[31] F Piazzese and G Rizzi ldquoOn the electrostatic self-force in aReissner-Nordstrom spacetimerdquo Physics Letters A vol 119 no1 pp 7ndash9 1986

[32] B Boisseau C Charmousis and B Linet ldquoElectrostatic self-force in a static weak gravitational field with cylindrical symme-tryrdquoClassical and QuantumGravity vol 13 no 7 pp 1797ndash18031996

[33] A G Smith and C M Will ldquoForce on a static charge outside aSchwarzschild black holerdquo Physical Review D vol 22 no 6 pp1276ndash1284 1980

[34] A I Zelrsquonikov and V P Frolov ldquoInfluence of gravitation on theself-energy of charged particlesrdquo Journal of Experimental andTheoretical Physics vol 55 no 2 p 191 1982

[35] B Linet ldquoForce on a charge in the space-time of a cosmic stringrdquoPhysical Review D vol 33 no 6 pp 1833ndash1834 1986


[36] A G Smith ldquoGravitational effects of an infinite straight cosmicstring on classical and quantum fields self-forces and vacuumFluctuationsrdquo in Proceedings of the Symposium of the Formationand Evolution of Cosmic Strings G W Gibbons S W Hawkingand T Vaschapati Eds p 263 Cambridge University Press1990

[37] E R Bezerra deMello V B Bezerra C Furtado and F MoraesldquoSelf-forces on electric andmagnetic linear sources in the space-time of a cosmic stringrdquo Physical Review D vol 51 no 12 pp7140ndash7143 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Journal of



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Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


geometries including charged black holes of general relativityand black holes with Brans-Dicke modifications or withconical defects [21ndash25]

In this paper we are interested in obtaining an expressionfor the electrostatic potential generated by a point chargeheld stationary in the region outside the event horizon of abraneworld black holeThere are several vacuum solutions ofthe spherically symmetric static gravitational field equationson the brane with arbitrary parameters which depend onproperties of the bulk or that are simply put in by using gen-eral physical considerations At the present it is theoreticallynot known whether these parameters should be universalover all braneworld black holes or whether each separateblack hole may have different values of them Similarlythere is not a single complete solution in the sense that themetric in the bulk is uniquely known Since this situationis unsatisfactory from a theoretical point of view it may beuseful to investigate more closely the observational effectsof the black hole properties Specifically we consider theeffects due to the projections of theWeyl tensor and how theyspecify the corrections on the electrostatic potential Sincethe generic form of the Weyl tensor in the full 5-dimensionalspacetime is yet unknown the effects of known solutionsmust be studied on a case-by-case basis Although one canin principle constrain the projections this only yields verymild constraints on the 5-dimensional Weyl tensor [26 27]Therefore we decided to derive the electrostatic potentialgenerated by a point charge held stationary in the outsideregion of the event horizon of the particular solution in theRandall-Sundrum braneworld model obtained by Dadhichet al [12] and revisited by Sheykhi and Wang [15] In thiscase tidal charge is arising via gravitational effects fromthe fifth dimension that is it is arising from the projectiononto the brane of free gravitational field effects in the bulkand is this term the one that will modify the electrostaticpotential produced by the particleWe obtain the correctionsdue to the topological charge arising from the bulk Weyltensor to the electrostatic potential and deduce the necessarycorrection to incorporate Gaussrsquos law Finally we also presentthe solution of Maxwell equations in the form of series ofmultipoles

2 The Topologically Charged Black Hole inthe Braneworld

The gravitational field on the brane is described by the Gaussand Codazzi equations of 5-dimensional gravity [2]

119866120583] = minusΛ119892120583] + 8120587119866119879120583] + 120581

4

5Π120583] minus 119864120583] (1)

where 119866120583] = 119877

120583] minus (12)119892120583]119877 is the 4-dimensional Einsteintensor and 120581

5is the 5-dimensional gravity coupling constant

1205814

5= (8120587119866

5)2

=48120587119866

120591 (2)

with 1198665the gravitational constant in five dimensions and Λ

is the 4-dimensional cosmological constant that is given in

terms of the 5-dimensional cosmological constantΛ5and the

brane tension 120591 by

Λ =1205812

5

2(Λ5+1205812

5

61205912) (3)

119879120583] is the stress-energy tensor of matter confined on the

brane Π120583] is a quadratic tensor in the stress-energy tensor

given by

Π120583]

=1

12119879119879120583] minus

1

4119879120583120590119879120590

] +1

8119892120583] (119879120572120573119879

120572120573minus1

31198792)

(4)

with 119879 = 119879120590

120590 and 119864

120583] is the projection of the 5-dimensionalbulk Weyl tensor 119862

119860119861119862119863on the brane (119864

120583] = 120575119860

120583120575119861

]119862119860119861119862119863

119899119860119899119861 with 119899119860 the unit normal to the brane) 119864

120583] encompassesthe nonlocal bulk effect and it is traceless 119864120590

120590= 0

Considering the Randall-Sundrum scenario with

Λ5= minus

1205812

5

61205912 (5)

which implies

Λ = 0 (6)

the four-dimensional Gauss and Codazzi equations for anarbitrary static spherically symmetric object have been com-pletely solved on the brane obtaining a black hole typesolution of the field equations (1) with 119879

120583] = 0 given by theline element [12 14 15 28]

1198891199042= ℎ (119903) 119889119905

2minus1198891199032

ℎ (119903)minus 1199032119889Ω2 (7)

where 119889Ω2 = 1198891205792 + sin21205791198891205932 and

ℎ (119903) = 1 minus2119866119872

119903+120573

1199032 (8)

For this model the expression of the projected Weyltensor transmitting the tidal charge stresses from the bulk tothe brane is

119864119905

119905= 119864119903

119903= minus119864120579

120579= minus119864120593

120593=120573

1199034 (9)

This result shows that the parameter 120573 can be interpretedas a tidal charge associated with the bulk Weyl tensor andtherefore there is no restriction on it to take positive aswell as negative values (other interpretations consider 120573 asa five-dimensional mass parameter as discussed in [14]) Theinduced metric in the domain wall presents horizons at theradii (taking 119866 = 1) Consider the following

119903plusmn= 119872 plusmn radic1198722 minus 120573 (10)



0 le 119903minusle 119903+le 119903119904 (11)


0 le 120573 le 1198722 (12)


lowast


119903lowast= 119872 + radic1198722 +

10038161003816100381610038161205731003816100381610038161003816 gt 2119872 (13)





nabla120588119865120588120583= 4120587119869

120583 (14)

where

119865120583] = 120597120583119860] minus 120597]119860120583 (15)





1198690=

119902

11990320sin 120579

120575 (119903 minus 1199030) 120575 (120579 minus 120579

0) 120575 (120593 minus 120593

0) (16)


0gt 119903+in the case 120573 ge 0

and 1199030




120597

120597119903(1199032 120597120601

120597119903)

+ (1 minus2119872

119903+120573

1199032)

minus1

times [1

sin 120579120597

120597120579(sin 120579

120597120601

120597120579) +

1

sin21205791205972120601

1205971205932]

= minus119903241205871198690

(17)


119903 = 119903 + 119903minus

120601 (119903 120579 120593) =119903 minus 119903minus

119903120601 (119903 minus 119903

minus 120579 120593)

(18)


120597

120597119903(1199032 120597120601

120597119903)

+ (1 minus2119898

119903)

minus1

[1

sin 120579120597

120597120579(sin 120579

120597120601

120597120579) +

1

sin21205791205972120601

1205971205932]

= minus4120587119902

sin 120579120575 (119903 minus 119903

0) 120575 (120579 minus 120579

0) 120575 (120593 minus 120593

0)

(19)




0= 0 120593

0= 0 and proceeding


120601119862(119903 120579) =

119902

1199030119903


2minus 120573) cos 120579



(20)





119862takes

the asymptotic form

120601119862(119903 120579) 997888rarr

119902

119903(1 minus

119872

1199030

) (21)




120601 (119903 120579) = 120601119862(119903 120579) +

119902

119903

119872

1199030

(22)








119902(1minus21198721199030) and 119902


0 120579 = 120587) and






0cos(120593 minus 120593

0) This


120601 (119903 120579 120593) =119902

1199030119903


2minus 120573) 120582 (120579 120593)



+119902

119903

119872

1199030

(23)



119892119897(119903) =

1 for 119897 = 0

2119897119897 (119897 minus 1)(119872

2minus 120573)1198972

(2119897)

1199032minus 2119872119903 + 120573

119903

119889119875119897

119889119903(

119903 minus119872


) for 119897 = 1 2 (24)

119891119897(119903) = minus

(2119897 + 1)

2119897 (119897 + 1)119897(1198722 minus 120573)(119897+1)2

1199032minus 2119872119903 + 120573

119903

119889119876119897

119889119903(

119903 minus119872


) for 119897 = 0 1 2 (25)

where 119875119897and 119876



(1) for 119897 = 0 1198920(119903) = 1 and 119891

0(119903) = 1119903




119897(119903) rarr


when 119903 rarr 119903+









Acknowledgment


References












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





0 le 119903minusle 119903+le 119903119904 (11)


0 le 120573 le 1198722 (12)


lowast


119903lowast= 119872 + radic1198722 +

10038161003816100381610038161205731003816100381610038161003816 gt 2119872 (13)





nabla120588119865120588120583= 4120587119869

120583 (14)

where

119865120583] = 120597120583119860] minus 120597]119860120583 (15)





1198690=

119902

11990320sin 120579

120575 (119903 minus 1199030) 120575 (120579 minus 120579

0) 120575 (120593 minus 120593

0) (16)


0gt 119903+in the case 120573 ge 0

and 1199030




120597

120597119903(1199032 120597120601

120597119903)

+ (1 minus2119872

119903+120573

1199032)

minus1

times [1

sin 120579120597

120597120579(sin 120579

120597120601

120597120579) +

1

sin21205791205972120601

1205971205932]

= minus119903241205871198690

(17)


119903 = 119903 + 119903minus

120601 (119903 120579 120593) =119903 minus 119903minus

119903120601 (119903 minus 119903

minus 120579 120593)

(18)


120597

120597119903(1199032 120597120601

120597119903)

+ (1 minus2119898

119903)

minus1

[1

sin 120579120597

120597120579(sin 120579

120597120601

120597120579) +

1

sin21205791205972120601

1205971205932]

= minus4120587119902

sin 120579120575 (119903 minus 119903

0) 120575 (120579 minus 120579

0) 120575 (120593 minus 120593

0)

(19)




0= 0 120593

0= 0 and proceeding


120601119862(119903 120579) =

119902

1199030119903


2minus 120573) cos 120579



(20)





119862takes

the asymptotic form

120601119862(119903 120579) 997888rarr

119902

119903(1 minus

119872

1199030

) (21)




120601 (119903 120579) = 120601119862(119903 120579) +

119902

119903

119872

1199030

(22)








119902(1minus21198721199030) and 119902


0 120579 = 120587) and






0cos(120593 minus 120593

0) This


120601 (119903 120579 120593) =119902

1199030119903


2minus 120573) 120582 (120579 120593)



+119902

119903

119872

1199030

(23)



119892119897(119903) =

1 for 119897 = 0

2119897119897 (119897 minus 1)(119872

2minus 120573)1198972

(2119897)

1199032minus 2119872119903 + 120573

119903

119889119875119897

119889119903(

119903 minus119872


) for 119897 = 1 2 (24)

119891119897(119903) = minus

(2119897 + 1)

2119897 (119897 + 1)119897(1198722 minus 120573)(119897+1)2

1199032minus 2119872119903 + 120573

119903

119889119876119897

119889119903(

119903 minus119872


) for 119897 = 0 1 2 (25)

where 119875119897and 119876



(1) for 119897 = 0 1198920(119903) = 1 and 119891

0(119903) = 1119903




119897(119903) rarr


when 119903 rarr 119903+









Acknowledgment


References












































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119862takes

the asymptotic form

120601119862(119903 120579) 997888rarr

119902

119903(1 minus

119872

1199030

) (21)




120601 (119903 120579) = 120601119862(119903 120579) +

119902

119903

119872

1199030

(22)








119902(1minus21198721199030) and 119902


0 120579 = 120587) and






0cos(120593 minus 120593

0) This


120601 (119903 120579 120593) =119902

1199030119903


2minus 120573) 120582 (120579 120593)



+119902

119903

119872

1199030

(23)



119892119897(119903) =

1 for 119897 = 0

2119897119897 (119897 minus 1)(119872

2minus 120573)1198972

(2119897)

1199032minus 2119872119903 + 120573

119903

119889119875119897

119889119903(

119903 minus119872


) for 119897 = 1 2 (24)

119891119897(119903) = minus

(2119897 + 1)

2119897 (119897 + 1)119897(1198722 minus 120573)(119897+1)2

1199032minus 2119872119903 + 120573

119903

119889119876119897

119889119903(

119903 minus119872


) for 119897 = 0 1 2 (25)

where 119875119897and 119876



(1) for 119897 = 0 1198920(119903) = 1 and 119891

0(119903) = 1119903




119897(119903) rarr


when 119903 rarr 119903+









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



Documents

Electrostatics in the Surroundings of a Topologically Charged Black Hole in the Brane