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Massinversioninacriticalneutronstar:Analternativetotheblackholemodel.

J.P.Petit

FIRSTPART

_______________________________________________________________________________________Keywords:Blackhole,Einsteinfieldequations,Schwarzschildmetric._______________________________________________________________________________________Abstract:Inthisfirstpart,wepresentadocumentedhistoricalbackgroundabouttheSchwarzschildsolutiontothefieldequationsofgeneralrelativity,asoriginallyconsideredbyKarlSchwarzschild,aswellasJohannesDroste,HermannWeylandAlbertEinstein,initslinearizedform.WealsodetailhowDavidHilbertassumedthisquestion,withapureimaginarytime.Finally,wediscussextensionsof the solution throughanalytic continuationand theirgeometryimplications.

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Introduction

Theblackholemodelremainsabeliefwithinthescientificcommunity.

Itistruethatintherecenthistoryofscience,theexistenceofmanyobjectsandphenomenahasbeenconjecturedlongbeforetheyhavebeeneffectivelyobserved.Yetmanyofthemhadalottostirupskepticism.Anexamplewasantimatter,imaginedbySirArthurSchusterin1898,theorizedbyPaulDiracin 1928, before the first observation of positrons in 1932. Later, thephenomenondescribedin1935byEinstein,PodolskyandRosen,namedtheEPRparadox,wasdemonstrateddecadesafteritsauthorspresenteditasadenunciationoftheinfallibilityofquantummechanics.

Swiss-American astrophysicist Fritz Zwicky described howmassive starscouldviolentlyendassupernovæ,inamemorableCaltechlecturecoursein1931.Accordingtohim,suchastarrunningoutoffusionfuelwouldendinaviolent gravitational collapse at an incredible high speed, converting itspotentialenergyintokineticenergy.Thisoutburstofaverylargequantityofmass would compress the internal iron core into a totally new object: aneutronstar.

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Thismodelisatfirstgreetedwithgeneralskepticism,andveryfewscientistsagree with such a singular idea. But after tremendous efforts, Zwickymanagedtograduallydemonstratethephenomenon.

Nowadays,noonewouldventuretoquestiontheexistenceofantimatter,orquantumentanglementinvolvedintheEPRparadox,orsupernovae.Asforastronomicalphenomena,thesituationisverysimple.Thecosmosisvast.Ifthesupernovaphenomenonstatisticallyoccursinourgalaxyatarateofonlyonepercentury,theextensiontoallobservablegalaxiesbringstheirnumbertotensofthousands.

Relatedtosupernovæwasthequestionoftheexistenceofneutronstars.Thefirstoneswerediscovered in1967aspulsars(highlymagnetizedrotatingneutron stars acting as pulsed radiation sources). The relation withsupernovæwassoonestablished,oneofthesepulsarsbeinglocatedinthemiddleoftheCrabNebula,remnantofasupernovawhosevisibleexplosionin 1054 is testified by Chinese records. Today, their number has reachedhundredsofsamples,ineverycornerofthegalaxy,includingneardistances.Noreasonablescientistwouldnowdoubtoftheirveryexistence.

Butthis isnotthesameatallaboutstellarblackholes,whoseexistenceispracticallyinferredbyjustoneobservation:theCygnusX-1binarysystem,whosedistancefromusisevaluatedat6,000light-years.Thisrarityisveryabnormal. At such a large distance, its estimatedmass (about eight solarmasses)couldresultfromcumulativeobservationalerrors.

Thisveryanomaloussituationgivesthishypotheticalobjectthenatureofabelief,inconflictwiththescientificmethod.

Asforsupermassiveblackholes,allthatcanbesaidforthemomentisthattheseareveryimportantconcentrationsofmatter(hundredsofthousandstobillionsofsolarmasses)locatedinthecenterofgalaxies.Theyhaveaverylargemass,butnotaveryhighdensity;theonelocatedinthecenterofMilkyWaycanberepresentedbyaspherehalfthesolarsystemindiameter,filledwithmatterhaving the samedensityaswater. Inanyevent theyarenewobjects, which could be the remainder of quasar phenomena, but theirsignaturedoesnotallow,initscurrentstate,tonamethemblackholes.

It is thereforeperfectly licit todaytoquestiontheblackholemodel,whileproposinganalternativescenariotothefateofacriticalneutronstar.Indeed,inX-raybinariesforexample,astableneutronstar(theaccretor)cancapturematerialfromacompanionstar(thedonor)andthusbecomesatsomepointdestabilized. To refuse such a study would be unscientific and would betantamounttothedefenseofadogma.

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Preludetothebirthoftheblackholemodel

Beforeliterallygettingtotheheartofthematter,itisnecessarytosituatethecontextinwhichSchwarzschildfirstproducedanonlinear,exactsolutionofEinstein's field equation in 1916. In public's imagination, Einstein got asuperhero origin story by the media. Actually, thanks to his remarkableability to assimilate both the mathematics of his time and the palette ofphysical phenomena, hewas able to produce an impressive successionofresults,includingin1905thediscoveryofthelawofthephotoelectriceffect.ButatthattimeanumberofGermanandDutchscientistsfollowedthispathcarefully, and were immediately able to understand and interpret anyprogressmade.Schwarzschildwasoneofthem.

In 1915, we find first important texts resulting from the collaboration-competition between Einstein and the great mathematician Hilbert. InNovember 1915, Einstein presented a finalized version of his theory ofgeneralrelativityinaseriesoffourpapersattheRoyalPrussianAcademyofSciencesinBerlin[1][2][3][4].Atfirst,hisfieldequationisnotdivergence-free, but he has the intuition that for distances small enough and formoderatevelocities(whichcorrespondstotheNewtonianapproximation)his equation should give the corresponding equations of fluidmechanics,namelytheEulerconservationequations.Hemodifieshisfieldequationtomakeitdivergence-freein[4],whoseoriginaltitleis:

Fig.1–Einstein'soriginalpaper,November25,1915

"TheFieldEquationsofGravitation."

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Andhereisthecorrespondingfieldequation:

Fig.2–ThefieldequationfinalizedbyEinstein,25November1915.

Inthelefthand-sideofhisequation(6):theRiccitensor.ThefollowingistheEnglishtranslation[4]from1997:

Fig.3–TranslationofEinstein'sfoundingpaper.

Thefieldequationwithzero-divergenceafterthemodificationofNovember25,1915.

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EinsteinisobviouslyinterestedinphenomenaNewtonianmechanicscannotexplain.OnNovember18,1915,hepublishesalinearizedsolutionofthefieldequationwithoutasecondmember,resultingofanullRiccitensor.[3]

Fig.4–EinsteinexplainstheperihelionmotionofMercuryonNovember18,1915.

Inthissolution,Einsteinchoosestogivethevalue-1tothedeterminantofhismetricsolution,ahypothesisSchwarzschildwilltakeagain.

Fig.5–Fieldequationwithnorighthand-side,anddeterminant-1.

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It isworthmentioning, and thiswill beof great importance lateron, thatEinsteinoptsforametricsignaturewithatimevariable(aswellasapropertime)havingrealvalues:

Fig.6–Einstein'schoiceofametricsignature(+–––).

The solutionofhis fieldequation is a four-dimensionalhypersurface. It isclearthathischoiceofvariableiswithin !4 andtheelementoflengthdsisreal.

Inthefollowingexcerpt,hegivesanapproximateexpressionofthemetricpotential g44 , introducing a quantityα , a Greek letter that SchwarzschildalsochosestodesignatetheconstantofintegrationofhisnonlinearsolutioninJanuary1916.

Fig.7–Einstein'sapproximation,linearizedsolutionofhisfieldequation.

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Letters x1 , x2 , x3 denoterealspacecoordinatesfromwhichhedesignatesaspolarcoordinate x1

2 + x22 + x3

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Usingpolarcoordinates,hefindstheNewtonianarealaw:

Fig.8–Einstein:arealaw(Kepler'ssecondlawofplanetarymotion).

Going from the variable r to its inverse1/r, he expresses the formof thegeodesicsolution:

Fig.9–QuasiNewtoniansolution.

To stick with the chronology, we have to mention the communicationpresentedbythegreatmathematicianHilbertattheGöttingenAcademyofSciences,inthesessionofNovember20,1915.[5]

He gradually got involved about a possible mathematization of physicsthroughavariationalapproach.ThesamemonthasEinstein'spresentationofhisfinalizedversionofgeneralrelativity,Hilbertpresentedafirstpaperentitled "The Foundations of Physics", extremely ambitious. At that time,

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physicsboilsdowntotwosets:gravitationandelectromagnetism.Everyonethinksthattheonewhowouldsucceedunitingthesetwoworlds(inwhatwilllaterbecalleda"unifiedfieldtheory")willmasterallthephysicsofhistime.ThisisthemeaningofHilbert'spaper.

SuchaworkwillbepursuedbyEinstein,whowillalsofailinthisventure.Weknow today that in order to combine gravitation and electromagnetismtogether,fourdimensionsarenotenough.Tobeginwith,afifthdimensionwouldbemandatory:Kaluza'sfifthdimension.

Butveryquickly,Hilbertdidnotfeelsatisfiedwithhisarticleanddecidedtowithdrawittomakechanges.1Aswillbeseenlater,hewillpresentasecondcommunicationinDecember1916[6]anditisthisversion,verysimilartothefirstone,thatwewillcommenton.

Atthattimeofwar,KarlSchwarzschild,43yearsoldandalreadythefatherofthreechildren,joinedwiththerankoflieutenant,bypatriotism,tofightontheRussianfront.Heisalreadyanastronomerandaconfirmedmathema-tician.TakingnoteofEinstein'spaper,hepublishesinJanuaryandFebruary1916, not one article, but two, in which he presents his nonlinear exactsolutionofthefieldequation.[7][8]Hereisthetitleofthefirstpaper:

Fig.10–FirstSchwarzschild'spaperonJanuary13,1916

"OnthegravitationalfieldofamasspointaccordingtoEinstein'stheory."

1AlthoughHilbert'sfirstcommunicationwaspresentedduringthesessionofNovember20,1915,Hilbertwillpreventitspublicationformonthsinordertomakemodifications.Inparticular,hewillchangethefieldequationaposterioritousethedivergencelessversionpresentedbyEinsteinonNovember25.Thefinalversionofhispaperwillbepublished in1916,keeping thedateofNovember20,1915.Hilbertdidnot"inventthefieldequation"fivedaysbeforeEinsteinasonecansometimesnaivelyhearit.

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Hequicklysituateshissolutionbysettingcoordinates:

Fig.11–Schwarzschilddefineshiscoordinates.

"One calls t the time and x, y, z the rectangular coordinates". Thesecoordinates are real. If he had opted for coordinates that could take onimaginary values, he would have mentioned it. So he chooses arepresentation in !3 . Then he goes to a polar coordinate system,writing x= r sinθ cosϕ , y= r sinθ sinϕ , z= r cosθ with

r = x 2 + y2 + z 2

which implies r ≥ 0 , at least for the choice of a representation wherevariables(x,y,z)belongto ! .Atthispoint,itshouldbenotedthatrisnotaradialdistance,butasimplespacemarker,asimplenumber.Schwarzschildthenintroduceswhathecallsan"auxiliaryquantity"(Hilfsgröße):

Fig.12–ThesolutionexpressedwithauxiliaryquantityR.

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In so doing, he calls his constant of integration α in order to tally withEinstein's 1915 paper [3]. Using this set of variables t , R ,θ ,ϕ{ } hecalculatesthegeodesics.LikeEinstein,henotesthatthesegeodesicsarepartofplanesandhechooses, likehim, theplaneθ=π/2. It isagainwith thisauxiliaryquantityRthatheexpressesthearealaw:

Fig.13–SchwarzschildcalculatesthegeodesicsΦ(R):eq.(18)

Thesameappliestoequation(18)fortheexpressionofthesolutionintheformofanintegralbasedonthevariablex=1/r(andnotonthevariabler)as in Einstein's paper. It is clear that he orientates the expression of hissolutiontostickcloselytotheEinstein'sresult.

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But this detail is quite secondary to him, inasmuch as the conditions ofplanetaryastronomymakesthe twoquantitiespracticallyequal,whichhenotesalittlefurtheron(EsistalsopraktischRmitridentisch):

Fig.14–Schwarzschild'ssolutionmeetsEinstein's.

Translationoftheunderlinedpassage:

"Therefore r is virtually identical to R"

AfterSchwarzschild'sdeath,hisworkwasfirstpresentedtothecommunityofmathematiciansbyFrank[9]thenquicklytakenupbyDroste[10],Weyl[11]andofcourse,Hilbert[6].

It is themedia,mostly American, thatwill fashion Einstein's image as anisolatedgenius,authorofatheorythatatinynumberofpeoplewouldhavebeenabletounderstand.ThisdoesnotinanywayreduceEinstein'smerits,but hewas in fact at the forefront of physics, in which a relatively largenumber of researchers, mainly German-speaking, were already involved.Amongthem,Schwarzschild.

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ScientificarticleswerethenpublishedinGerman,andatthattimeaccessingsuchworkisdonebyreading"offprints",samplecopiesonpaper,whicharetransmitted by postal services. Subsequently, the same papers are madeavailable, grouped inbooks,but still inGerman. Itwasonly in the1970s,more thanhalf a century later, that this documentationwas translated inEnglish,andlateronmadeavailableasPDFfiles.ThemoderndistributionofthesefilesthroughtheInternetissomethingveryrecent.Onlyafewdecadesago,inordertodistributesimplecopiesofarticles,researchershadtotypetheirworksontracingpapers,thenusethemtoexposeaUV-sensitiveyellowsheet.Andfinally,embeddingthecopieswithinammoniavaporstorevealthe content. I personally experienced this technique in the 1960s. Thescanner,anessentialpartofphotocopiers,willonlybeputonthemarketinthelate1960s.

All thistotell theworksofEinsteinandSchwarzschildhavespreadtothescientificcommunityintheformofcommentaries,notintheiroriginalform.

However,oneshouldnotethatDroste[10]andWeyl[11]chosetheelementoflengthasalwayspositive,associatedwithametricsignature(+---):

Fig.15–ExcerptofWeyl's1917paper.

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StillinWeyl'spaper,wefindsuchasignatureandthementionofarealandessentiallypositivepolarcoordinater.

Fig.16–InWeyl(1917):thepolarcoordinateandthesignature.

Similar mention with Droste, in his paper communicated in English byHendrikLorentzsixteendaysafterthedeathofSchwarzschild,butpublishedonlythefollowingyear.[10]

Fig.17–DrostespecifiesonMay27,1916thesignaturehechooses.

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Itdoesnotseemtocometoanyone'smindtoextendthissolutionfor r <α whichwouldimplyamodificationofthesignatureofthemetric.

ItisthenthatHilbertpresentsthesecondcommunicationofhislongarticle"TheFoundationsofPhysics":

Fig.18–Hilbert'ssecondcommunication,December23,1916.

Inthissequeltohisfirstcommunicationpresentedtheyearbefore,hetakeshisworkoverandincludestheSchwarzschildsolutionpublishedinJanuary.Hilbertknows thathedied.Einsteindeliveredamemorial lectureonKarlSchwarzschild [12]on June29, 1916at theopeningmeetingof theRoyalPrussianAcademyof Sciences, Berlin. ButHilbert does not seem to focusmuchonthisresult,whichappearstohimasadetail in thevastscientificsagaheintendstocarryoutonbothgravitationandelectromagnetism.Let'sseehismentionofSchwarzschild'ssolution.

Inhispaper,Hilbertmentionsacentralsymmetry(zentrischsymmetrisch):

Fig.19–Hilbert'shypothesisofacentralsymmetry.

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Hethenintroducespolarcoordinatesanddesignatesafourthvariablebytheletterl:

Fig.20–Hilbertoptsforpolarcoordinates.

Thereareseveralthingstonoteinthisexcerptoffig.20.Wecanreadintheunderlinedpassageofhypothesis#3:

"The gravitation is centrally symmetric with respect to the origin of coordinates."

Wefindagain the themeofasolutionwithacentral symmetry,andnotasphericalsymmetry.

In equation (42) he presents the bilinear form of his metric, and thenintroduces the hypothesis r *= G(r) where this variable r* quicklytransformsintor.Thetranslationis:

"If we introduce r* in (42) instead of r and then eliminate the sign *, the result is the expression (43)."

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This hypothesis will then totally guide his solution. Incidentally, Hilbertindicatesthenatureofhistemporalcoordinatel:

Fig.21–Hilbert'spureimaginarytimeandhisformulationofthesolution.

ForHilbert,spacetimeisafiberbundle,builtfromarealspace(x,y,z)butthefiber,i.e.thefourthcoordinatel,ispurelyimaginary.

Regarding Schwarzschild's solution, this passage is absolutely crucial andwillguideacenturyofscientificwork.Hilbertmakesaconfusionwhenhewrites:

"then for l = i t (43) results in the desired metric in the form first found by Schwarzschild."

Ifhissolutioniscompleted(45)itshouldbewritten:

ds2 = − (1−αr

) dt2 + dr 2

1−αr

+ r 2dϑ 2 + r 2sin2ϑ dϕ 2

whereastherealSchwarzschildsolutionis:

ds2 = (1−αR

) dt2 − dR2

1−αR

− R 2dϑ 2 − R 2sin2ϑ dϕ 2 R 3= ( r 3+α 3 )1/3

Thedifferenceisobvious.HilbertconfusedhiscoordinaterwiththeauxiliaryvariableRintroducedbySchwarzschild.Healsochoosesareversesignature.

In therestof thepaper,Hilbert comesback to this choiceofvariable inafootnote,consideringitmerelypointless.Heseesitonlyasawishtorejectthesingularitybacktotheorigin.

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Fig.22–ThefootnoteinHilbert'spapermentioningSchwarzschild'swork.

Translation:

"To transform the locations to the origin, as Schwarzschild does, is not to be recommended in my opinion; Schwarzschild’s transformation is moreover not the simplest that achieves this goal."

HilbertconsidersthatwhathappensatthisSchwarzschildsphereisatruesingularity,whereas itwillbeclassically shownafterward that it isonlyacoordinatesingularity,thatcanbecancelledoutwithanappropriatechangeofvariable.

Geometrysignification

The solution to Einstein's equation is a four-dimensional hypersurface.Einstein,Schwarzschild,Frank,Droste,andWeylreliedontheideathattheelementoflengths,hencethepropertime,theonlythingindependentofthechoiceofcoordinatesonthishypersurface,isarealquantity.

Let'sgobacktoSchwarzschild's1916solution.Schwarzschilddoesnotmakeitexplicit,simplybecausehedoesnotseethepointinit.Hemerelywantstoshow the same result as Einstein's explanation of the precession of theperihelion of Mercury. No one at this time thinks astronomical objectsrequiring a nonlinear treatment of the problem could really exist. HisquantityαisnothingbutwhatwilllaterbecalledtheSchwarzschildradiusRs.He immediately notices that for the Sun, it is equivalent to one hundredthousandth of its diameter. As it happens, in his second paper publish inFebruary1916[8]whichwewillcomebackto,hebuiltageometrywithinaspherefilledwithafluidofconstantdensity,anon-singularsolution,sohe

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doesnot find relevant todeal inhis first "exterior solution"paperwith aregionofspacethatisrelatedtothe"interiorsolution".

In2011,ChristianCorda[13]makesthissolutionexplicit:

Keepinmindthatsuchametriccanbeexpressedinanyarbitrarycoordinatesystem.Thesecoordinatesarethenmerelynumbers,spacemarkers.Theonlysizehavinganintrinsicpropertyisthelengths.Then,allboilsdowntothechoiceforthislength.

Ifwedecidethatthislengthsisreal,thensuchanexpressiondesignatesa4Dhypersurface. The choice of angles θ andφ implies a solution with asphericalsymmetry(insteadofacentralone).Helicalgeodesicsexistonthishypersurface,thatcanbeexpressedgivingafixedvaluetoθ,andfixingthecoordinater.Thevariablestandφlinearlydependontheparameters.Thus,theyarebothrelatedbyalinearfunction.Spatialprojectionofsuchgeodesiccanbecalculated:theysimplybecomecircles.

It is thereforepossible to travel along this hypersurface, drawingparallelcircles.Iftheirperimetercanreach+∞itishoweverlimitedtolessthan2πα.If the value of the angleq is varied then this family of parallel circlesgeneratesafamilyofparallelspheres.Letusnotethatwedeliberatelyusetheword"parallel"andnot"concentric"becausetheSchwarzschildsolutionhasasphericalsymmetryandnot,aswrittenbyHilbert,acentralone.

Asweskimthroughthisseriesofparallelspheres,byreducingtheirarea,atsomepointaminimalvalueisreached(ifwedecidethatthehypersurfaceisreal).Thissituationismet:

• Either for the value R = α in the case of what we call "Hilbert'srepresentation" ( t , R ,θ ,ϕ ) (introducing the auxiliary quantity RaccordingtoSchwarzschild).

• Orforthevaluer=0inSchwarzschild'srepresentation ( t , R ,θ ,ϕ ) .

Thishypersurfaceisthusuncontractible.Howtointerpretit?

One can of course decide it is a manifold with boundary. A boundary,incidentally, along which the determinant of the metric is zero inSchwarzschild'srepresentation.Thiswillbediscussedinthesecondpartofthisarticle.Butinthe1960s,otherpathswereconsidered.

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Extensionsofthesolutionandtheirmeaning

DecadesaftertheemergenceoftheSchwarzschildsolution,itisknownonlythroughsubsequentcommentspublishedafterward.Iwasverysurprisedata recent international conference on black holes 2 held in Frankfurt(Schwarzschild's birthplace), specifically entitled "TheKarl SchwarzschildMeeting", that no attendee, includingGermannatives, hadnever read thefoundingpapersIhavejustmentioned,asiftheoriginofthemodelwerelostinthemistsofthepast.Inhismasterlylectureinfrontofallthedelegates,JuanMaldacenasaid:3

—TheSchwarzschildsolutionhasconfusedusoverahundredyearsandithas

forcedustosharpenourviewsonspaceandtime.Ithasledtoasharper

understandingofEinstein’stheory.Experimentally,itisexplainingseveral

astrophysical observations. Its quantum aspects have been a source of

theoreticalparadoxesthatareforcingustounderstandbettertherelation

betweenspacetime,geometryandquantummechanics.

This textsuggests that the immediateunderstandingof theSchwarzschildsolutionremainedlimitedforawhile,butthankstofurtherstudies,acenturyofworkmadeitpossibletoimprovethisreadingbyintroducinganextendedviewofspacetime.

WhydoastrophysicistsrefertotheSchwarzschildsolution?

Aswill be discussed in the second part of this article, Schwarzschild hadcompleted inFebruary1916his first"exteriorsolution"[7]withasecond"interiorsolution"[8]relatedtothegeometrywithinaspherefilledbyafluidofconstantdensity.Atthispoint,thegeometrydescribingastarimmersedinavacuumwascompletelyachieved.

Butprogressinobservationswouldrevealanewproblem.Inourgalaxy,halfofthestellarsystemsaremultiplestarsystems.Sothereshouldexistaquitelargenumberofbinarieswhereoneof the twostarsbecamea subcriticalneutronstar,thatcouldstillbedescribedwiththetwoexteriorandinteriorSchwarzschild metrics. Its companion star is then called the donor, as it

23rdKarlSchwarzschildMeetingonGravitationalPhysicsandtheGauge/GravityCorrespondence(KSM2017),24–28July2017,FIAS,FrankfurtamMain,Germany.

3https://indico.fias.uni-frankfurt.de/event/4/session/17/contribution/39

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continues toemit stellarwind that isgravitationallycapturedby its smallcompanion,namedtheaccretor.

Inhisfirstpaper,evaluatingthecriticalradius (α = Rs = 2m) andrelatingittothemass responsible for this geometry, he had concluded that beyond acertainvalue,this"Schwarzschildradius"couldexceedthestarradius.

Thisisthequestionaskedbyobserverstotheoreticians:

— Assuming that a supernova only left a subcritical neutron star after itsgravitationalcollapse,whathappenswhentheadditionalmasssuppliedby

thestellarwindofadonorcompanionstartriggersthegeometriccriticality

intheneutronstar?

Aquestion impossible toavoid.Theoreticians therefore imagined that theexteriorSchwarzschildsolution,yetreferringtoaportionofanemptyspace,coulddescribeanewstateofmatter,popularizedbythename"blackhole".

Theproposedanalyticextensions

What is the current presentationof this Schwarzschild solution?Take forexamplethebookofAdler,BazinandSchiffer.[14]Page187,equations(6.4)and(6.5)theauthorsreiterateHilbert'shypothesis,writing:

ds2 = Ac 2dt 2 − B dr 2 − C ( dθ 2 + sin 2θ dϕ 2 ) r̂ = C(r) r

Afewlineslater,thesolutionwasagainparticularized,seeequation(6.9):

Quelques lignesplus loin, lasolutionadenouveauétéparticularisée,voirl’équation(6.9):

ds2 = eν (r ) c 2dt 2 − eλ (r )dr 2 − r 2 ( dθ 2 + sin 2θ dϕ 2 )

Unless the functions ν(r) and λ(r) have thepossibility tobe imaginary, theimpressionisthiswritingmilitatesforasignature ( + − − − ) .Inthisrealisticview,themetricreferstoahypersurfacewheretheonlyrelevantvariable,independentofthechoiceofcoordinates,isthequantitys.Thisisthepropertime,livedbyanytestparticle.Foradistantobserverhowever,thispropertimeisidentifiedtothevariablet.Thus,weconcludethatthecoordinatetimetistheoneexperiencedbythedistantobserver.Thefollowingfigureisfoundin every book about general relativity. It compares the free fall timemeasured either by the clock tied to a test particle falling towards theSchwarzschildsphere,ortheoneofadistantobserver.

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Fig.23–FalltowardtheoriginofaSchwarzschildgeometry

intermsofcoordinatetimetandpropertimeonthetestparticles/c.

Thus,whateverthephenomenonwitnessedbytheobserver,itissupposedtounfoldbeforehiseyesduringaninfiniteamountoftime,whichallowsthetheoreticiantoadd:

— Idonot feelboundtodescribe theresultofaprocesswhich, tomyeyes,takesplaceinaninfinitetime.

Inpassing,theNewZealandmathematicianRoyKerr[15]extendedthistypeof solution to a "rotating black hole" thanks to a metric with a lessconstrainingsymmetry.Thesurfacethathasbeencalledthe"eventhorizon"takesthetopologyofatorus.Butthefrozentime,thisfreeze-frame,isalsopresentinthismodel.

In1960, JosephKruskal [16]andGeorgeSzekeres [17]constructeda firstanalyticextension,whichMaldacenadescribedas"extendingthesolutiontocoverthefullspacetime".

WeshallalsociteCorda,whoimplementsanotheranalyticextensionofthesolution.[13]

In1989,CanadianphysicistLeonardS.Abramspublishedapaper[18]ontheSchwarzschildsolutionentitled"BlackHoles:TheLegacyofHilbert'sError"pointingoutthesameproblemsasthosementionedabove.QuestionsalsotakenupbyItalianphysicistSalvatoreAntoci.[19][20]Thosearticleshadpractically no echo within the scientific community, except for a text

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positioned in a blog 4 by a mathematician and computer scientist,W.D.Clinger, presenting himself as a topology specialist. He resumes theargumentsofAbrams,writing:

— Thepaperiswell-written,anditsmathisalmost(butnotquite)correct.

A little further on, his criticism gets stronger: "Where Abramswent badlywrong."Hethenprovidesthecorrectwaytoproceed:

— Inreality, the"quasiregularsingularity"at thecentralpointmassof theoriginalSchwarzschildspacetimecanberemovedbyallowingtheradial

coordinatertogonegative.

Inotherwords,itisenoughtoconsidertheextensionofthesolutionforthevaluesof r = x2 + y2 + z2 < 0 i.e.inanimaginaryportionofspacetime…

For that matter he quotes Corda, who does the same and constructs adescriptionofthegravitationalcollapseoftheneutronstarwhichendsatanegativevalue r = − α = − Rs < 0 .

To the extent that these authors grant themselves the freedom to extendspacetimetoanimaginaryportion,thiscannolongerbecriticized.Ontheotherhand,thesepeoplecall"crackpots"anyonewhodoesnotfeelsatisfiedwithsuchaformula.

Conclusion

In 1916, Einstein then Schwarzschild developed solutions to the fieldequation,thefirstthroughlinearizationandthesecondnonlinearly,whichrefer to real values of the coordinates and to a real value of the length s,measuredonthefour-dimensionalhypersurfaceusingthemetric.Thisworkisthentakenupbydifferentauthors,likeFrank,Droste,Weyl,alwayswiththe same perspective. This approach implies the constancy of an explicitmetricsignature ( + − − − ) .

On this panorama comes the interpretation given by the mathematicianDavidHilbert,whostartsfromadifferentvisionwheretime,andpropertime,are presented as pure imaginary quantities, as opposed to real spacecoordinates, which goes hand in hand with a choice of metric signature ( − + + + ) andexplainsthehyperbolicpropertyofthissolution.

4http://www.internationalskeptics.com/forums/showthread.php?t=231833

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Whenhalfacenturylater,astrophysicistsareaskedtoprovideamodelabletodescribeadestabilizedneutronstar, theyopt for thesolution foundbyHilbert, deciding to extend it, via analytic continuation, to an imaginaryportionofspacetime,wheretheelementoflengthbecomespurelyimaginaryandwherethesignatureofthemetricisinverted.Thegoalisnottostopatthe surface of the Schwarzschild sphere, and to be able to carry theirinvestigationstowardsthe"interioroftheobject".

Italldependsonwhatisconsideredtofallwithinthescopeofphysicsornot.

Inthesecondpartofthispaper,wewillpresentadifferentscenario,inwhichallquantitiesremainrealand,asacorollary, themetrickeepsasignature ( + − − − ) .

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References

[1]Einstein,A. (4Nov.1915). "ZurallgemeinenRelativitätstheorie".KöniglichPreußischeAkademiederWissenschaften(Berlin).Sitzungsberichte.778–786.translatedinEnglishas:Engel,A.;Schücking,E.(1997)."OntheGeneralTheoryofRelativity".TheCollectedPapersofAlbertEinsteinVol.6:TheBerlinYears:Writings,1914-1917(Englishtranslationsupplement).Doc.21,98–107.UK:PrincetonUniversityPress.

[2]Einstein,A.(11Nov.1915)."ZurallgemeinenRelativitätstheorie(Nachtrag)".KöniglichPreußischeAkademiederWissenschaften(Berlin).Sitzungsberichte.799–801.translatedinEnglishas:Engel, A.; Schücking, E. (1997)."On the General Theory of Relativity (Addendum)".The CollectedPapersofAlbertEinsteinVol.6:TheBerlinYears:Writings,1914-1917(Englishtranslationsupplement).Doc.22,108–110.UK:PrincetonUniversityPress.

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