BlackholestowhiteholesI:Acompletequasi-classicalmodel

JamesM.Bardeen

PhysicsDepartment,Box1560,UniversityofWashingtonSeattle,Washington98195-1560.USA

bardeen@uw.edu

Abstract

Thisisthefirstoftwopaperspresentingdifferentversionsofquasi-classicaltoymodelsforthenon-singularevolutionofthegeometryandtheassociatedeffectivestress-energytensorforasphericallysymmetricblackholethatevolvesintoawhiteholeandeventuallydisappearsafterevaporatingdowntothePlanckscale.Theansatzforthegeometryisinspiredbycalculationsofthesemi-classicalstress-energytensorintheSchwarzschildbackgroundandideasfromloopquantumgravityforanonsingulartransitiontothewhitehole.Inthispaperthemainemphasisisontheevolutionoftheblackhole,andtheevolutionofthewhiteholeisassumedtobeessentiallythetimereverseofthatoftheblackhole.ThenegativeenergyoftheHawking"partners"flowsoutofthewhiteholetofuturenullinfinity.Thewhiteholedisappearswhenthematterandradiationthatcollapsedtoformtheblackholeemerges.Idiscussthecompatibilityofthemodelwithsomeofthequantumenergyconditionsproposedintheliterature,and,briefly,theimplicationsfortheinterpretationofblackholeentropy.Thesecondpaperconsidershowtheevolutionofthewhiteholecanbemodifiedtoavoidprolongedemissionofnegativeenergy.

I.INTRODUCTION

ThediscoveryofHawkingradiationfromblackholes1over40yearsagoled

totheassertion2ofafundamentalbreakdownofpredictabilityintheevolutionofquantumfieldsfollowinggravitationalcollapsetoformablackhole.TheargumentwasthattheHawkingradiationisinamixedstateentangledwithnegativeenergyHawking"partners"insidetheblackholethatdecreasethemassoftheblackholetocompensateforthepositiveenergyHawkingradiationgoingtofuturenullinfinity.Iftheblackholehasaneventhorizonandevaporatescompletely,theresultisapparentlyalossofquantuminformationandabreakdownofunitarityforexternalobservers.This"informationparadox"isnowwidelyconsideredanunacceptableconflictwithfundamentalquantumfieldtheory,requiringdrasticdeparturesfromtheoriginalsemi-classicalanalysisofHawking,thoughUnruhandWald3havearguedtothecontrary.Completeevaporationoftheblackholewithoutreleaseofthetrappedquantuminformationdoesraiseseriousissues,particularlyinthelightoftheAdS/CFTconjecture4,inwhichgravityinthebulkissupposedtobedualtoamanifestlyunitaryconformalfieldtheoryontheAdSboundary.

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Thethermodynamicentropyofablackholeinteractingwithitssurroundings5isidentifiedwiththeBekenstein-Hawkingentropyproportionaltotheareaoftheeventhorizon.Inunitswith G = c = 1 , SBH = A / 4!( ) .Normallythethermodynamicentropyofaquantumsystemisidentifiedwithitstotalnumberofquantumdegreesoffreedom,whichinturnisthemaximumpossiblevalueoftheentanglement(vonNeumann)entropy SvN .IftheHawkingradiationisentangledwithdegreesoffreedominsidetheblackhole,asinthestandardsemi-classicaltheoryofHawkingradiation,Page6hasshownthat SvN becomesequalto SBH atthePagetime,whentheblackholehaslostonlyaboutonehalfofitinitialmass.IftheblackholecontinuesemittingHawkingradiationafterthePagetime,asonewouldexpectforanyblackholewithamassmuchgreaterthanthePlanckmass

mp ,either

SvN > SBH orthelateHawkingradiationmustbeentangledwiththeearlyHawkingradiation.Ifthelatter,bythemonogamyofentanglementthelateHawkingradiationcannotbeentangledwithHawking"partners"insidetheblackholehorizon,resultingina"firewall"ofhighlyexcitedquantapropagatingonorjustinsidetheblackholehorizon7.

Controversyovertheseissueshasragedrightuptothepresenttime.SeereviewsbyMarolf8andPolchinski9.Abigpartoftheproblemisthelackofawidelyacceptedtheoryofquantumgravity.Naively,forverylargeblackholesthesemi-classicaltheoryofquantumfluctuationspropagatingonaclassicalgeometryshouldbeanexcellentapproximation.TidalaccelerationsatthehorizonofaverylargeastrophysicalblackholearenolargerthanthoseinlaboratoriesontheEarth,wherequantumfieldtheoryhasbeentestedwithexquisiteprecision.Ihavearguedatlengthelsewhere10thatthesemi-classicalphysicsinthevicinityofthehorizonofalargeblackholeprecludesanysubstantialstorageofquantuminformationonornearthehorizon,andthatalmostallofthequantuminformationentangledwiththeHawkingradiationendsupinthedeepinterioroftheblackhole.

However,thatdoesnotmeanthequantuminformationisirretrievablyswallowedupbyasingularity.Theclassicalsingularitytheoremsrelyonenergyconditionsthatareviolatedinquantumfieldtheory.Variousmoreorlessadhocnonsingularblackholemodels,someinspiredbyloopquantumgravity(LQG)11,havebeenproposed.Onepossibilityisthatquantumbackreactionsimplystopscollapseshortofasingularity,whichrequiresaninnertrappinghorizon.Iftheinnerandoutertrappinghorizonseventuallymergeanddisappear,thequantuminformationcanescape,assuggestedbyHayward12.MoreorlesssimilarmodelshavebeenproposedbyHossenfelder,etal13,RovelliandVidotto14,Frolov15,DeLorenzo,etal16,andBardeen17.ReleaseofquantuminformationbythePagetimerequiresalargequantumbackreactioninregionsoflowcurvature.Thenegativesurfacegravityoftheinnertrappinghorizonraisesseriousquestionsaboutitsstabilityandtheviabilityofthesemodels.

Aninterestingalternativeistheconversionoftheblackholeintoawhitehole,asdiscussedingeneraltermsbyModesto18andbyAshtekarandBojowald19.Moreexplicitmodelsareinreferences[20,,21,22,23,24],amongothers.InsomeofthesethereisaCauchyhorizontothefutureoftheblackholeinterior,whichleavesthe

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unitarityissuesunresolved.WhatisrequiredisanonsingularquantumtransitionfromtheblackholetoawhiteholejnaspacetimewiththecausalstructureofMinkowskispacetime.Thetrappedquantuminformationescapesfromthewhiteholeandpropagatesouttofuturenullinfinity.Modelsthatinvokequantumtunnelingfromalargeblackholedirectlytoawhitehole,suchasthatofHaggardandRovelli22,Ifindlessconvincingthanthosewithasmoothtransitionofthegeometry,asinRef.[23].Ashtekar,Olmedo,andSingh24(AOS)adaptedproposalstoresolvecosmologicalsingularitiesinLQGtosuggestaparticulareffectivegeometryinwhichthe2-sphereareahasanonzerominimumonaspacelikehypersurfaceseparatingtheinterioroftheblackholefromtheinteriorofthewhitehole.However,theAOSmodelassumesafixedblackholemass,andhasnoprovisionfortheHawkingradiationthatshoulddominatequantumcorrectionsatlargeradii.Itisinconsistentwithsemi-classicalquantumtheoryatlargeradiiwherequantumcorrectionstothegeometryaresmall,asdiscussedinPartIIofthispaper.

InPartIIIIproposeablackholetoawhiteholemodelwithasmootheffectivegeometrythroughthetransition,somewhatsimilartothatoftheAOSmodelwhiletheblackholeislarge,butthatallowstheevaporationandeventualdisappearanceoftheblackholeatthePlanckscale.TherearenoCauchyhorizons,consistentwithunitaryevolutionforobserversatlargeradii.Themodelassumesaneffectivequasi-classicalmetricevenwherequantumfluctuationsinthegeometryareexpectedtobeverylarge,andthereforeshouldonlybeconsideredasuggestionofwhatmightbepossibleinquantumgravity.Theeffectivestress-energytensorderivedfromtheeffectivemetric,unlikethatoftheAOSmodel,isbroadlyconsistentwiththeformofthesemi-classicalstress-energytensor(SCSET)outsidetheblackholehorizon.APlanck-scalewhiteholeiscreated(foranexternalobserver)astheblackholedisappears,andgrowsbyemittingnegativeenergy.

Howtheeffectivestress-energytensorofthemodelrelatestocertainquantumenergyconditionsisdiscussedinPartIV.Itdoesseemtosatisfytheaveragednullenergycondition(ANEC)andrelatedquantumnullenergycondition(QNEC).However,therearecontentiousissuesrelatingtotheevolutionofthewhitehole.IpointoutreasonstodoubtthattheclaimofDeLorenzoandPerez25thatinstabilityassociatedwithexponentiallyincreasingblueshiftsalongthewhiteholehorizonimpliesaveryshortlifetimeforthewhitehole.Ontheotherhand,Ref.[23]arguedforaPlanck-scalewhiteholewithalifetimemuchlongerthanthatoftheblackhole.InthemodelIdescribeheretheevolutionofthewhiteholeisroughlythetime-reverseoftheevolutionoftheblackhole,withthenegativeenergyHawking"partners"flowingoutofthewhiteholetofuturenullinfinity.TheprolongedemissionofnegativeenergywouldseemtoviolatetheFord-Roman26theoremsonminimumaverageenergydensitiesforquantumfieldsinMinkowskispacetime.Alternativesfortheevolutionofthewhiteholewillbeconsideredinacompanionpaper.

PartVhasasummaryandfurtherdiscussionofsomekeyissues,suchaswhytheMinkowskiminimumenergydensitytheoremsmaynotapply,andwhytheentanglemententropyofablackholewithaPlanck-scaleareacangreatlyexceeditsBekenstein-Hawkingentropy.

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II.EFFECTIVEMETRICSFROMLQG

TworecentdiscussionsofquantummodificationstothegeometryofSchwarzschildblackholes,basedonslightlydifferentquantizationschemesinLQG,arethoseofAshtekarandOlmedo27(AO),extendingtheresultsofAOStotheblackholeexterior,andofGambini,Olmedo,andPullin28(GOP).InAOthesquareofpropercircumferentialradius R isexpressedintermsofacoordinate r as

R2 = r 2 + a2

4r 2 , (2.1)

whichhasaminimumvalueof a2 at r2 = a2 / 2 ,wherethereisasmoothtransition

fromtrappedsurfacesintheblackhole r 2 > a2 / 2( ) toanti-trappedsurfacesinthe

whitehole r 2 < a2 / 2( ) .AOSarguefromLQGthat

a2 = γ L0δ c M = 12

γ( )4/3Δ2/3 M 2/3

4π 2( )1/3 , (2.2)

Here γ = 0.2375 istheBarbero-ImmirziparameterofLQGandintermsofthefundamentalareagapparameter Δ = 5.17! ,

L0δ c =

12

γ Δ2

4π 2 M⎛⎝⎜

⎞⎠⎟

1/3

. (2.3)

Themassparameter M ≫ a isdefinedsuchthat r = 2M attheblackholehorizon.Insidetheblackholehorizon,where r istimelikeandtheKillingvector ∂/ ∂t

isspaceliketheAOquantum-modifiedmetriccanbewrittenas

ds2 = − R

r⎛⎝⎜

⎞⎠⎟

2 γ 2δ b2

sin2 δ bb( ) dr 2 + 2MR

⎛⎝⎜

⎞⎠⎟

2 sin2 δ bb( )γ 2δ b

2 1+sin2 δ bb( )γ 2δ b

2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−2

dt2 (2.4)

plustheangularpart R2dΩ2 ,with

cos δ bb( ) = b0

b0 +1( ) r / 2M( )b0 − b0 −1( )b0 +1( ) r / 2M( )b0 + b0 −1( )

, b0 ≡ 1+ γ 2δ b2 . (2.5)

Thecoordinate t issingularat r = 2M ,where cos δ bb( ) = 1 and sin δ bb( ) = 0 ,butthecontinuationto r > 2M istrivial,with

sin2δ bbγ 2δ b

2 →−sinh2δ bbγ 2δ b

2 =2M / r( )b0 −1⎡

⎣⎢⎤⎦⎥ b0 +1( )2

− b0 −1( )22M / r( )b0⎡

⎣⎢⎤⎦⎥

b0 +1+ b0 −1( ) 2M / r( )b0⎡⎣⎢

⎤⎦⎥

2 . (2.6)

TheAOSvalueof γδ b is γδ b = 0.5995 ! / M 2( )1/6,and

b0 −1≡ ε = 0.1800 ! / M 2( )1/3

.ChangingtoEddington-Finkelstein(EF)coordinates,withanadvancedtime

coordinate v ,constantoningoingradialnullgeodesics,resolvesthecoordinatesingularityonthefuturehorizonoftheblackhole,andthemetricbecomes

ds2 = −e2ψ grrdv2 + 2eψ dvdr + R2dΩ2 , (2.7)

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with

eψ = 1

4r

2M⎛⎝⎜

⎞⎠⎟

ε

2+ ε + ε 2Mr

⎛⎝⎜

⎞⎠⎟

1+ε⎡

⎣⎢⎢

⎤

⎦⎥⎥

2

≅ r2M

⎛⎝⎜

⎞⎠⎟

ε

(2.8)

and

grr = 1− 2Mr

⎛⎝⎜

⎞⎠⎟

1+ε⎡

⎣⎢⎢

⎤

⎦⎥⎥

r 2

R2

2+ ε( )2− ε 2 2M / r( )1+ε⎡

⎣⎢⎤⎦⎥

2+ ε + ε 2M / r( )1+ε⎡⎣⎢

⎤⎦⎥

2 . (2.9)

Thisgeometryhassomeverypeculiarproperties.Itisnotasymptoticallyflatintheconventionalsense,since eψ → ∞ as r →∞ .AOarguethatbychangingthecoordinate t to !t = eψ t andtaking r →∞ atconstant !t the(nownon-static)metricdoesbecomeinaweaksenseasymptoticallyflat,withawell-definedADMmass.However,theMisner-Sharpquasi-localmassiscoordinate-invariantandgoestozeroasymptotically.AspointedoutbyFaraoniandGiusti29,noinitiallyoutgoingtimelikegeodesicscanreachinfiniteradius.

ThequitedifferenteffectivemetricofGOPisbasedonaLQGspinnetworkwithevenspacingincircumferentialradius δ ∼ " ,whichischosenfor"simplicity".TheeffectivemetricinthecoordinatesofEq.(5.3)is

ds2 = − 1− 2Mr

+ Δ4π

2M( )4

r 4 r + 2M( )2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

dv2 + 2 1+ δ2r

⎛⎝⎜

⎞⎠⎟

dvdr + r 2dΩ2. (2.10)

Theydonottrytomodelthetransitiontothewhitehole,andonlyconsidertheeffectivemetricat r > r0 ∼ "M( )1/3

.As r →∞ theenergydensityfallsoffas

δ 2M + 3δ / 4( ) / r 4 andtheradialandtransversestressesas δ / r3 .TheasymptoticMisner-Sharpmassisequalto M +δ .WhilemorereasonablethantheAOmetric,quantumcorrectionsinthesemi-classicalregimearestilllargerelativetothesemi-classicalfieldtheoryexpectationofquantumcorrectionsproportionalto ! .

BothAOS/AOandGOPcompletelyignoretheevolutionoftheblackholeduetotheemissionofHawkingradiation.Theanalysesarebasedindifferentwaysonsymmetry-reducedHamiltonians,whichIexpectareinherentlyincapableofproperlyaccountingforallquantumcorrectionstotheeffectivemetricandstress-energytensor.

III.MODELINGANEVAPORATINGBLACKHOLE

Inconstructingamodelfortheevolutionofthegeometryofanevaporating

blackholeandthetransitiontoawhitehole,assumingsphericalsymmetry,itishighlyadvantageoustoworkinEFcoordinates.Theadvancedversionwithadvancetime v isregularontheblackholetrappinghorizon.Theretardedversion,withretardedtime u constantonoutgoingradialnullgeodesics,isregularonthewhiteholeanti-trappinghorizon.Furthermore,aspointedoutbyBardeen30,theEinsteinequationsforageneralsphericallysymmetricmetricinthesecoordinatesare

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remarkablysimple.Theadvancedversionwiththecircumferentialradiusasacoordinate,fromnowonjustdenotedby r is

ds2 = −e2ψ grrdv2 + 2eψ dvdr + r 2dΩ2 , (3.1)Theretardedversiondiffersonlyinthesignof gur .TheMisner-Sharpmassfunction

m hasthecoordinate-independentdefinition .TheEinstein

equationsthatdeterminetheeffectivestress-energytensorfrom and

ψ v,r( ) are

4πTv

v = − 1r 2

∂m∂r

⎛⎝⎜

⎞⎠⎟ v

, 4πTvr = 1

r 2

∂m∂v

⎛⎝⎜

⎞⎠⎟ r

, 4πTrv = 1

r∂e−ψ

∂r⎛⎝⎜

⎞⎠⎟ v

, (3.2)

with Tθ

θ = Tϕϕ from

Tr ;α

α ≡ 0 .TheretardedversionofEqs.(3.2),with v → u ,isexactly

thesame.Thephysicalstress-energytensorcomponents(e.g., e−ψTv

r and eψTr

v )areinvariantunder r -independentrescalingsof v (or u ).

Somemodelsforblackholeinteriorshavesuggestedamassfunction m similartothatofHayward12,

m = Mr3

r3 + 2Ma2 . (3.3)

IntheHaywardnonsingularmodelofanevaporatingblackholethemassparameter

M = M v( ) intheblackholeinterior, eψ = 1 inadvancedEFcoordinates,and a isaconstant.Thestress-energytensorcurvatureinvariantsareregularat r = 0 ,whichisjusttheoriginofasphericalcoordinatesysteminalocallyflatgeometry.Howeveranon-singulartransitiontoawhiteholerequiresaminimumvalueof r > 0 .

Iassumeatransitiontothewhiteholeataminimumradius r = a similartothatofAOSandAO.WhilethemetricofEq.(3.1)issingularthere,thecoordinatesingularitycanberesolvedbychangingtheradialcoordinatefrom r to z suchthat

r2 = z2 + a2. (3.4)

ThisisequivalenttoEq.(2.1),buthasasimplerform.Thecoordinate z ,definedtobenegativeintheblackholeandpositiveinthewhitehole,increasestothefutureinsideboththeblackholeandwhiteholehorizonsandiszeroatthetransition.However,foranevaporatingblackhole,an a2 proportionalto M 2/3 ,asinAOSandAO,wouldbetime-dependent.Instead,Iwillmakethemuchsimplerassumptionthat a2 isaPlanck-scaleconstant,perhapsrelatedtotheareagapparameterofLQG.

With z insteadof r asacoordinateand a2 aconstant,theadvancedEFmetricgiveninEq.(3.1)becomes

ds2 = −e2ψ v g zzdv2 − 2eψ v dvdz + r 2dΩ2 , g zz = r 2

z2 grr , eψ v = − zr

eψ v ,r( ). (3.5)

Theretardedformis

ds2 = −e2ψ u g zzdu2 − 2eψ u dudz + r 2dΩ2 , g zz = r 2

z2 grr , eψ u = + zr

eψ u,r( ). (3.6)

∇αr∇αr = 1− 2m / r

m v,r( )

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Inasmoothtransitionfromtheblackholetothewhitehole, gzz , ,and vary

smoothly,implying grr = 1− 2m / r = 0 and eψ → ∞ at z = 0 .Eqs.(3.2)become

4πTv

v = − 1zr

∂m∂z

⎛⎝⎜

⎞⎠⎟ v

, 4πTvz = 1

r 2

∂m∂v

⎛⎝⎜

⎞⎠⎟

, 4πeψ vTzv = a2

r 2 −zr 2

∂ψ v

∂z⎛⎝⎜

⎞⎠⎟ z

, (3.7)

with

2m = r 1− z2

r 2 g zz⎛⎝⎜

⎞⎠⎟

. (3.8)

Justreplace v by u togettheexpressionsinretardedEFcoordinates.ThecausalrelationshipsinmymodelareillustratedinthePenrosediagram

ofFig.1.Theblackholeisformedbyaninfluxofmatter/radiationalongradialnullgeodesicsina"thick"nullshellofmass betweenadvancedtimes and .AninfinitesimallythinshellisnotphysicallyrealisticwhenconsideringgeometryatclosetothePlanckscale.TheblackholeevaporatesslowlybyemittingHawkingradiationfor 0 > v > v2 ,witha"horizon"(notaneventhorizon)definedasthe"outgoing"nullhypersurface,bydefinitionat u = 0 ,whoseradiusfor v > v2 slowlydecreasesuntiltrappedsurfacesdisappearandtheblackholeendsatthe2-surfacewhere g

zz = 0 at r = a .The"ingoing"nullhypersufaceatthis2-surface,bydefinitionat v = 0 ,becomesthewhiteholehorizonfor z > 0 .Theblackholeapparent(trapping)horizonisthetimelikehypersurfaceonwhich g

zz = 0 justoutsidetheblackholehorizon.Thewhiteholeapparent(anti-trapping)horizonisthehypersurfaceonwhich, g

zz = 0 ,justoutsidethewhiteholehorizonandtimelikeifthewhiteholemassisincreasingorjustinsidethewhiteholehorizonifthewhiteholemassisdecreasing..Intheinteriorofthecollapsingshellthereisaspacelikeoutertrappinghorizonindicatedbythelowerblueline.Thewhiteholeendsinthereboundingshellbetween u2 and u1 .

InthemodelconsideredinthispapertheHawking"partners"areassumedtopropagatealongingoingradialnullgeodesicsintheblackholeandalongoutgoingnullgeodesicsinthewhiteholeandouttofuturenullinfinity,asindicatedbytheblackarrows.Analternatepictureofpartnerpropagationinsidetheblackholeispropagationalong"outward"radialnullgeodesics(stillingoingincircumferentialradius),butthiswouldmakelittledifferenceinhowtheblackholetransitionstothewhitehole.Ifanything,itwouldevenmorestronglysupporttheassumptionthattheHawking"partners"shouldpropagatealongoutgoingradialnullgeodesicsinthewhitehole.

Imakenoattempttoexplicitlymodelthedynamicsoftheradiationandevolutionofthegeometryintheinterioroftheshell,excepttonotethatinsidetheshellthegeometryshouldbeMinkowski(regionM1).However,whentheinneredgereaches ,quantumbackreactionmustgenerateaspacelikeinnertrappinghorizon,indicatedbytheupperblueline,thatconnectswiththeinneredgeofthe

transitionhypersurfaceattheouteredgeoftheshell,asindicatedbytheupperblueline.Potentialinstabilityduetonegativesurfacegravityoftheinner

ψ v ψ u

M0 v1 v2

r = 0

r = a

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trappinghorizonshouldnotbeaproblem,sinceitdoesn'tneedtolastlong.ThereisanotherMinkowskiregionM2tothefutureofthereboundingshell.

Figure1.APenrosediagramshowingathicknullshellcominginfrompast

nullinfinitytoformablackholethattransitionsintoawhitehole.Seethetextfordetails.

APenrosediagramcanbeverymisleadingastowhateventsarecloseto

otherevents.Theadvancedtime overwhichtheblackholeevaporatesis,enormouslygreaterthantherangeofadvancedtime overwhichthe

blackholeforms,andthebounceofthecollapsingshellpresumablytakesplace.Also,theHawkingradiationreachesfuturenullinfinityoverwhatappearsasaninfinitesimalrangeofretardedtimeinthediagram,butwhichisactuallycomparableto asmeasuredbyadistantobserversousidetheblackhole.

Myansatzesforthemetricfunctions and eψ v intheregionoutsidethe

shellareinthespiritofEq.(3.3),butwithaddedflexibilitytobettermatchtheform

v = 0

u = 0

r = a

v0

u2

M1

M2

BH

WH

v1

v2

r =

0

u1

−v2

∼M03 / " ∼ M0

−v2

gzz

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oftheSCSETintheexterioroftheblackholesuggestedbynumericalcalculationsforspin0andspin1fields10.Unfortunately,theunknownspin2(graviton)contributiontotheSCSETpresumablydominates,sincethespin2traceanomalyismorethan10timesthespin1traceanomalyinmagnitude.Bothmetricfunctionsshouldberegularfunctionsof z2 (i.e., r )at ,implying there.

Myexpressionfor is

g zz = 1− 2Mr 2 +αa2r

r3 + βa2r + γ 2M( )a2 . (3.9)

Foralarge( M ≫ a )slowlyevaporatingblackhole,themetricisSchwarzschildin

thelimit r ≫ Ma2( )1/3

and M istheblackholemass.CalculationsoftheSCSETshow10thatclosetotheblackholehorizonthereisaninflowofnegativeenergy,balancingtheoutwardflowofpositiveenergyHawkingradiationatlargeradii.Justhowthisnegativeenergypropagatesinsidetheblackholeissomewhatuncertain.Iwillassumethat M = M ′t( ) ,where ′t = v forall r < 2M insidetheblackholeandshouldbecomearetardedtimeatlargeradiioutsidetheblackhole.

Consideraslowlyevaporatingblackhole,withHawkingluminosity

LH = −dM / d ′t ≪1 .Slowevaporationisplausiblethroughoutthelifetimeoftheblackhole,with LH → 0 attheendoftheblackhole.Thegeometryisquasi-staticSchwarzschildfor r > 2M aslongaslongas 2M / a≫1 and r ≪ M / LH .Fortheinterpolationbetweenadvancedandretardedtime,Idefine ′t implicitlyby ′t = v − 2r +12M −16M 2 / r − 4M ln r / 2M( ). (3.10)Tofirstorderin LH ,

∂ ′t∂v

⎛⎝⎜

⎞⎠⎟ r

= 1− 4LH ln r2M

− 4+ 8Mr

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

−1

(3.11)

and

∂ ′t∂r

⎛⎝⎜

⎞⎠⎟ v

= −2− 4Mr

+ 16M 2

r 2

⎛⎝⎜

⎞⎠⎟

∂ ′t∂v

⎛⎝⎜

⎞⎠⎟ r

(3.12)

Byconstruction ′t anditsfirstderivativesarecontinuousacross r = 2M .TheansztzofEq.(3.10)doesnotaccountforHawkingradiationemittedwhen M and

LH werelargerinthepast,butthisisirrelevantforthecurrentevolutionoftheblackhole.Forsimplicity,theparameters , ,and willjustbetakentobeconstants.

Themetricfunction ψ v controlshowthecoordinateadvancedtime v isrelatedtoalocalproperdistancesandtimes.Howitvariesfromoneingoingradialnullgeodesictoanotherisagaugechoice,butfromEqs.(3.7)howitvariesalongthesegeodesicsisrelatedtothe Tz

v componentofthestress-energytensor.Theexpectationisthatgravitationaltimedilationcouldbecomeimportantduetoquantumbackreactioninthedeepinterioroftheblackhole,correspondingto e

ψ v

z = 0 1− 2m / r = 0

gzz

α β γ

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becomingsmallrelativetoitsasymptoticvalue,whichItaketobeone.Myansatzfortheinterioroftheblackhole,consistentwiththeformofthesemi-classicalstress-energytensor,hasthreeadditionalparametersδ ,ε ,andφ :

e−ψ v = 1+δ a2

2Mr+ ε a2

r 2 +φ2Ma2

r3

⎡

⎣⎢

⎤

⎦⎥. (3.13)

Intheexterior, r > 2M ,thedominantcontributiontothestress-energytensoratlarge r istheHawkingradiation,andinretardedEFcoordinatesthedominantcomponentis

Tu

r = −LH / 4πr 2( ) .TransformingtoadvancedEFcoordinates,

Tv

r = ∂u∂v

⎛⎝⎜

⎞⎠⎟ r

Tur = Tu

r , Trv = ∂v

∂r⎛⎝⎜

⎞⎠⎟ u

∂u∂r

⎛⎝⎜

⎞⎠⎟ v

Tur = −4Tu

r 1+O 2Mr

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

. (3.14)

Thenatlarge r

∂ψ v

∂r⎛⎝⎜

⎞⎠⎟ v

= 4πrTrv ⇒ψ v = 4LH ln r

2M⎛⎝⎜

⎞⎠⎟+O LH

2Mr

⎛⎝⎜

⎞⎠⎟

. (3.15)

Thiscanbeaccommodatedbymodifyingtheansatz(3.13)for r > 2M to

e−ψ v = 1− 4LH ln r

2M− 4+ 8M

r⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ 1+δ a2

2Mr+ ε a2

r 2 +φ2Ma2

r3

⎡

⎣⎢

⎤

⎦⎥ , (3.16)

consistentwithcontinuityof Trv at r = 2M .Theaddedfactorisjust

∂ ′t / ∂v( )r

−1.

Afterthe transitiontothewhiteholeIswitchtoretardedEddington-FinkelsteincoordinatesandthemetricofEq.(3.6),regularatthewhiteholeapparenthorizon,andmakethesimple,butquestionable,assumptionthatthenegativeenergyassociatedwiththeHawking"partners"afterflowingatconstant v insidetheblackholeflowsoutthroughthewhiteholeatconstantretardedtime u .Thismeans ′t = ′t u( ) for z > 0 ,with u definedinrelationtotheadvancedtimefor

z < 0 by u = −v at z = 0 .Since ′t = v intheblackhole,inthewhitehole ′t = −u .Theexpressionfor g

zz inEq.(3.9)remainsthesame,butwith M = M u( ) = M − ′t( ) .Then

∂M / ∂u( )r

= −dM / d ′t = +LH , ∂M / ∂r( )u= 0. (3.17)

Eq.(3.13)for e−ψ u alsoremainsthesame.The z = 0 hypersurfacemustbe

spacelike, ,with gzz → 0 attheendoftheblackholeandthebeginningofthe

whiteholeat u = v = 0 .The2-surfaceswith gzz < 0 onthewhiteholesideareanti-

trappedsurfaces,with r increasingtothefutureon"ingoing"aswellasoutgoingradialnullgeodesics.

Attheevaporationendpointthemassparameter Mmin is,fromEq.(3.9),

2Mmin

a= 1+ β −α

1−γ. (3.18)

z = 0

gzz < 0

11

11

Amodestrestrictionontheparametersthatconsiderablysimplifiesthecalculationsandinterpretationofthemodelistotakeα = β + γ ,giving 2Mmin / a = 1.Then g

zz becomes

g zz = 1−γ a2

r 2

⎛⎝⎜

⎞⎠⎟

1− 2Mr

⎛⎝⎜

⎞⎠⎟

/ 1+ β a2

r 2 + γ2Ma2

r3

⎛⎝⎜

⎞⎠⎟

. (3.19)

Aphysicallysensiblemodelrequires 0 < γ <1 , β + γ > −1.Withtheserestrictionsthereisonlyoneapparenthorizonfortheblackholeandoneforthewhitehole,bothat r = 2M everywhereoutsidethemattershell.TheexistenceofHawkingradiationrequirestheexistenceofatrappinghorizonfortheblackhole,soIassumethattheHawkingluminosity LH smoothlygoestozeroas M → Mmin and v → 0 .

Themassfunction m fromEq.(3.8)canbeinsertedintothefirstofEqs.(3.7),usingEq.(3.19)for g

zz ,withtheresultfor r ≤ 2M

8πTvv = a2

r 4 1−1+ a2

r 2

⎛⎝⎜

⎞⎠⎟β + γ( ) + 2 2M

r⎛⎝⎜

⎞⎠⎟

− z2

r 2 2γ − 3γ 2Mr

+ g zz 2β + 3γ 2Mr

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

1

1+ β a2

r 2 + γ2Ma2

r3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

. (3.20)

Also,

4πTvz = 4π r

z⎛⎝⎜

⎞⎠⎟

Tvr = −LH

zr

∂ ′t∂v

⎛⎝⎜

⎞⎠⎟ r

r 4 + βr 2a2 −αγ a4⎡⎣ ⎤⎦r3 + βra2 + γ 2M( )a2⎡⎣ ⎤⎦

2 . (3.21)

Thevanishingof Tvz at z = 0 isconsistentwithasmoothtransitionfrominflowof

(negative)energyintheblackholetooutflowofnegativeenergyatconstant u inthewhitehole.

The Rzv = 8πTz

v Einsteinequationgivesinsidetheblackhole

4πeψ vTzv = a2

r 4 1− z2δ / 2Mr( ) + 2ε / r 2 + 3φ 2M / r3( )⎡⎣

⎤⎦

1+δa2 / 2Mr( ) + εa2 / r 2 +φ 2Ma2 / r3( )⎡⎣

⎤⎦

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪. (3.22)

Then Tzz canbefoundfromtheidentity

Tzz = −eψ vT zv = Tv

v − g zzeψ vTzv . (3.23)

Afterthetransitiontothewhitehole, z ispositive.Iassumethattheonlychangeintheexpressionsfor g

zz and e−ψ u from e

−ψ v isthat M = M u( ) .Derivativesof M areevaluatedusingEqs.(3.17).Theexpressionsfor , ,and Tz

u arethesameasEq.(3.20),Eq.(3.21),and(3.22). Tu

z like Tvz ispositive.

Tofurtherclarifytheblackholetowhiteholetransition,project ontoan

orthonormaltetradwithfuture-directed4-velocity andradialunitvector pointingawayfromtheshell.Where insidetheblackholeandwhitehole

Tuu

Tuz

Tαβ

uα nα

gzz < 0

12

12

apparenthorizons,andparticularlyinthevicinityof z = 0 ,itisnaturaltoset uv = 0 ,sothe4-velocityisorthogonaltoaspacelikedisplacementatconstant z .Since uz > 0 ,theremainingcomponentsinadvancedcoordinatesare

uv = e−ψ v / −g zz , uz = −g zz , uz = −1/ −g zz . (3.24)

Theradialbasisvectorhas nv > 0 so n

v = e−ψ v / −g zz , nz = 0, nv = eψ v −g zz , nz = −1/ −g zz . (3.25)Theenergydensity ,theenergyflux ,andtheradialstress are

E = −Tz

z − −g zz( )−1e−ψ vTv

z = −Tzz − F , Pr = Tv

v − F. (3.26)Inretardedcoordinatesinsidethewhiteholeapparenthorizon,

uu = e−ψ u / −g zz , uz = −g zz , uz = −1/ −g zz , (3.27)

nu = −e−ψ u / −g zz , nz = 0, nu = −eψ u −g zz , nz = −1/ −g zz . (3.28)

Theenergydensity,energyflux,andradialstressare

E = −Tz

z − −g zz( )−1e−ψ uTu

z = −Tzz + F , Pr = Tu

u + F. (3.29)

Since e−ψ uTu

z and e−ψ vTv

z areidenticalfunctionsof z andtheblackholeandwhite

holeframesareidenticalat z = 0 ,theenergyfluxgoessmoothly C1( ) frompositive

intheblackholetonegativeinthewhitehole.Theenergyfluxissingularat g

zz = 0 ,becausethe uv = 0 frameisinfinitelyboostedrelativetoanylocalinertialframe.Asimplechoiceofframevalidwhere

isthestaticframe,definedby .Thenoutsidetheblackhole

E = −Tv

v − g zz( )−1e−ψ vTv

z = −Tvv − F , Pr = Tz

z − F. (3.30)Outsidethewhitehole

E = −Tu

u − g zz( )−1e−ψ uTu

z = −Tuu + F , Pr = Tz

z + F. (3.31)

Takingintoaccountthechangeinframeacross gzz = 0 ,thesignsof F and E + Pr do

notchangeacrossanapparenthorizon, E ≅ Pr ≅ −F fortheblackhole,and

E ≅ Pr ≅ +F forthewhitehole.Thereisnosingularityin F inafree-fallframe.

The componentoftheEinsteintensorisrathercomplicated,and Tθ

θ = Tϕϕ

canmosteasilybefoundfromthe conservationequation.Inadvancedcoordinatesfortheblackhole,

2Tθ

θ = 1r

r 2Tzz( )

,r+ re−ψ v eψ vTr

v( ),v− rψ v ,r g

zz + r2

g zz,r

⎛⎝⎜

⎞⎠⎟

eψ vTzv , (3.32)

andsimilarlyforthewhitehole. isfiniteat z = 0 inspiteofasingulartermin

,becausethesingulartermdoesnotdependon .

E F Pr

gzz > 0 uz = 0

Gθθ

Tr ;µ

µ = 0

Tθθ

Trv = r / z( )Tz

v v

13

13

At z = 0 ,

g zz = 1−γ( ) 1− 2M / a( ) / 1+ β + γ 2M / a( )( ), (3.33)

Thestress-energytensorreducesto

a2Tv

v = −1+ 2g zz( ) / 8πa2( ), Tvz = 0, eψ vTz

v = −2 8πa2( ), (3.34)

Theenergydensityisthesameandpositive, E = −Tz

z = +1/ 8πa2( ) ,everywhereonthetransitionhypersurface.However,

E + Pr = g zz / 4πa2( ) isnegative.The

expressionfor Tθθ israthercomplicatedingeneral,butinthelimit 2M / a≫1 at

z = 0 , 8πa2Tθθ = 10−11/ γ .

At r ≫ 2Ma2( )1/3

,inthesemi-classicalregimewherequantumcorrectionstothegeometryaresmall,theSCSETisfirst-orderin ! ,i.e.,first-orderinanexpansioninpowersof a2 .Inthislimit

2m = 2M + 1+α( ) a2

r− β 2Ma2

r 2 −γ2M( )2

a2

r3 . (3.35)

ThecomponentsoftheSCSETarepolynomialsin .Hawkingradiation

termsonlypresentfor r > 2M areenclosedincurlybrackets.With LH = q a / 2M( )2

,

8πTvv = a2

2M( )4 q −4− 4x +8x2{ }x2 + 1+α( )x4 − 2 1+ β( )x5 − 3γ x6⎡⎣ ⎤⎦ , (3.36)

8πTvr = −8πTv

z = −2qa2

2M( )4 x2 , (3.37)

8πTzv = −2 a2

2M( )4 q 4− 4x{ }x2 +δ x3 + 2ε −1( )x4 + 3φx5⎡⎣ ⎤⎦ , (3.38)

8πTzz = a2

2M( )4

q 4− 20x +16x2{ }x2 + 2δ x3 + α − 2δ + 4ε −1( )x4

− 2β + 4ε − 6φ( )x5 − 3γ + 6φ( )x6

⎡

⎣⎢⎢

⎤

⎦⎥⎥, (3.39)

16πTθθ = a2

2M( )4

q 24x − 36x2{ }x2 − 2δ x3 − 2α −5δ +8ε − 2( )x4

+ 6β +14ε −1−18φ( )x5 + 12γ + 27φ( )x6

⎡

⎣⎢⎢

⎤

⎦⎥⎥. (3.40)

Thereisasmalldiscontinuityin Tθθ at x = 1 .ThetraceoftheSCSETis

8πTµµ = a2

2M( )4

−12qx4{ }+ 3δ − 4ε + 2( )x4

+ 2β +10ε − 3−12φ( )x5 + 6γ + 21φ( )x6

⎡

⎣⎢⎢

⎤

⎦⎥⎥. (3.41)

NumericalcalculationsoftheUnruhstateSCSETintheexteriorofaSchwarzschildblackholehavebeencarriedoutformassless,conformallycoupledscalarandvectorfields31andmasslessminimallycoupledscalarfields32.Thesecan

x ≡ 2M / r

14

14

befit10withintheirnumericalaccuracyby6thorderpolynomialsin .Theyallhavepositivecoefficientsforthe x6 termin Tv

v ,correspondingtoanegativecontributiontoγ ,inapparentconflictwithmymodel'srequirementthat γ > 0 .However,theasyetunknowncontributiontotheSCSETfromspin2gravitonsshoulddominate.Additionaltypesifquantumfieldswillcontributeforblackholeswithmassessmallcomparedwiththoseofknownastrophysicalblackholes.

Figure2.TheenergydensityandradialstressfortheSetAparameterswhen

inthecoreoftheblackholeforthelocalframedefinedbyEqs.(3.24)-(3.26).

AllthatisknownaboutthecontributionstotheSCSETfromquantumfluctuationsinthegravitationalfieldistheHawkingluminosityandthespin2traceanomaly.AstheHawkingtemperatureincreasesmorequantumfieldswillcontributetotheSCSET.Ifallthequantumfieldsareconformallycoupled,onlythe x6 terminthetraceisnonzero.Intheblackholeinterior,thecorrespondingconstraintsfromEq.(3.41)onthecoefficientsinmymodelare

ε = 1/ 2+ 3δ / 4, β +1+ 2δ − 6φ = 0. (3.42)Asetofmodelparameters(SetA)consistentwiththeseis

(3.43)

Intheinterioroftheblackhole,with and negligible,theenergydensityandradialstressintheorthonormalframewithradialbasisvectoratconstant are and (seeEq.(3.26)).TheseareplottedinFig.2fortheSetAparametersattheadvancedtimeforwhich .Quantummodificationstothegeometrystartbecomingunimportantat z / a ∼ −20 .

2M / r

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

a2 (E + Pr)

- a2 Pr

z/a

2M / a = 8000

α = 0.4, β = 0.2, γ = 0.2, δ = 0, ε = 0.5, φ = 0.2.

gzz < 0 F

z Pr = Tvv

E + Pr = g zzeψ vTzv

2M / a = 8000

15

15

E + Pr isnegative,dueto becomingpositiveclosetothetransitiontothewhitehole,for −0.6 < z / a < 0.6 .

Arbitrarilysetting ,about70timesthevalueforphotonsplusgravitons33if ,theenergyfluxinthecoreoftheblackholeforsameSetAparametersasinFig.2isplottedinFig.3.Evenwith q = 1 theenergyfluxissmaller

thanthedominanttermsinthestress-energytensorbyafactoroforder .OncetheblackholehasevaporateddowntoclosetothePlanckscale,thereis

nosemi-classicalregimeinsidethehorizonandtheverynotionofaquasi-classicalevolutionishardtojustify.Still,themodeldoesdemonstratethepossibilityofanevolutioninwhichtheblackholeendsandthewhiteholebeginswithoutanysingularityandwithoutanyneedforquantumtunneling.

Figure3.Theenergyflux inthecoreofablackholefortheSetA

parameterswhen ,with .ComparewithFig.2,

notingthat issmallerthan and byafactortheorderof .

Forwhatitisworth,IplotinFig.4 E + Pr and Pr fortheSetAstress-energytensorfortheSetAparameterswhen 2M / a = 3 .Quantumcorrectionsaresignificantthroughthewholeblackholeinterior,andthereisnodistinct"core"inwhichthemagnitudeoftheeffectivestress-energytensorisslowlyvarying.Theblackholetrappinghorizonisat , .Thesurfacegravityofthetrappinghorizonatthispointisjustabitsmallerthantheclassicalvalueof .

Tzv

q = 0.001

a2 = !

a / 2M( )4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

1018

a2 F

z/a

F

2M / a = 8000 LH = 0.001 a / 2M( )2

F E Pr 1018

r / a = 3 z / a ≅ −2.83 1/ 4M

16

16

Figure4.Thedominantcomponentsofthestress-energytensorforSetA

parameterswhen .Theblackholeapparenthorizonisat ..

IV.QUANTUMENERGYCONDITIONSAninterestingquestiontoaskofthemodeliswhetheritisconsistentwith

quantumenergyconditionsthathavebeenproveninsomegeneralityinasemi-classicalcontest.Onesuchconditionistheachronalaveragednullenergycondition34(ANEC).Thisstatesthat

(4.1)

wheretheintegralisoveracompleteachronal(notwopointsconnectedbyatimelikecurve)nullgeodesicwithaffineparameter andtangentvector

.Ifirstconsiderradialnullgeodesicscrossingtheblackholeandwhiteholehorizonsandthenthenullgeneratorsoftheblackholeandwhiteholehorizons.

Intheblackholeregionan"ingoing"radialnullgeodesichas

and kz = −e−ψ v kv > 0 ,so

Tαβkαk β = −e−ψ vTz

v kv( )2. (4.2)

FromEq.(3.22)withtheSetAparameters, isnegativeandslowlyvaryinginmostofthecoreoftheblackhole,butitmustbecomespositiveclosetothetransitionwhere z / a <1 .Theevaporationtimescaleismuchlongerthana

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

a2 (E + Pr)

- a2 Pr

a2 (E +

Pr)

z/a

2M / a = 3 z / a = −2.83

Tαβkαk β dλ

−∞

∞

∫ ≥ 0,

λ

kα = dxα / dλ

kv = 0⇒ kz = 0

eψ vTz

v

17

17

dynamicaltimescale,andtoagoodapproximation kv < 0 isconstantalongthetrajectory.Theintegralasthegeodesicgoesfrom to is

Tαβkαk β k z( )−1dz = +

−∞

0

∫ e−ψ v eψ vTzv( )kv dz

−∞

0

∫ . (4.3)

Because e−ψ v fallsoffquiterapidlyfor r > a when 2M / a≫1 ,thenetresultforthe

integralofEq.(4.3)istypicallynegative.Aslongas ,asassumed,thecontinuationofthe"ingoing"null

geodesicintothewhiteholeregionstaysinsidetheanti-trappinghorizonwhere

gzz = 0 .Thegeodesicequationfortheingoingtangentvectorintheretarded

Eddington-Finkelsteincoordinatesgives

d eψ u ku( ) / du = eψ u g zz( )

,zeψ u ku( ) / 2. (4.4)

Once r / a≫1 eψ u ku growsexponentially,andinthesemi-classicalregimewitha

timeconstant ≅ 4M .Theintegralofthenullenergycanbewrittenas

− dz / du( )2Tz

u −Tuz⎡

⎣⎢⎤⎦⎥eψ u ku∫ du. (4.5)

Duringthegrowthofthewhitehole,correspondingtotheevaporationoftheblackhole, and .Thefirsttermispositiveexceptcloseto r = a ,butissuppressedasthegeodesicapproachesthewhiteholehorizonand dz / du becomesverysmall.ThesecondterminEq.(4.5)isnegative,andwhileinitiallysmallcomparedtothefirstterm,itquicklybecomesdominant.Thesecondtermdoesbecomepositivewhenthematterandradiationthatcollapsedtoformtheblackholestartsescapingfromthewhiteholeand .Theexponentialgrowthoff e

ψ u ku meansthatonlythelaste-foldingofthenegativecontributionsissignificant,andthisissmallcomparedwiththepositivecontributionasthegeodesiccrossesthemattershell.Thesamereasoningappliesinreversesequenceappliestoradially"outgoing"nullgeodesicspassingthroughthecollapsingmatter,theinterioroftheblackhole,andexitingacrossthewhiteholehorizon.

TheANECisalsosatisfiedforthenullgeneratorsoftheblackholeandwhiteholehorizons,sincetheintegralsarealsodominatedbythepositivecontributionsastheypassthroughthecollapsingmatter/radiationshellastheblackholeformsandtheexpandingshellasthewhiteholedisappears.

Thequantumnullenergycondition35(QNEC)isaquasi-locallowerlimitonthenullenergybasedonthevonNeumannentropy SvN oftheregionoutsideazero-expansionnullhypersurface,

Tαβkαk β ≥ !2π A

d 2SvN

dλ 2 , (4.6)

where A istheareaofacross-section.Inthepresentcontext,thiscanbeappliedattheblackholeandwhiteholehorizons,withthecross-sectionatwo-surfaceofconstant r and,respectively,ofconstant v or u .Whileneitherhorizonisexactly

z = −∞ z = 0

!M u( ) ≥ 0

!M u( ) > 0 Tu

z > 0

!M u( ) < 0

18

18

zero-expansion,theyarecloseenough,atleastwhile 2M / a≫1 .Then SvN v( ) isslowlyincreasingontheblackholehorizon,andslowlydecreasingonthewhiteholehorizon.Toagoodapproximationwhen M ≫ a ,andwithsurfacegravity

κ = 1/ 4M v( )⎡⎣ ⎤⎦ ontheblackholehorizon,

d 2SvN

dλ 2 = ddλ

dSvN

dvk v⎛

⎝⎜⎞⎠⎟≅ −κ

dSvN

dvk v( )2

, (4.7)

since k v ∝ e−κ v and

d 2SvN / dv2( ) / dSvN / dv( ) ≪ − dk v / dv( ) / k v ≅κ . (4.8)

Onthewhiteholehorizonat u ≅ −v , ku ≅ eκu and

d 2SvN

dλ 2 ≅ ddλ

dSvN

duku⎛

⎝⎜⎞⎠⎟≅ +κ

dSvN

duku( )2

, κdSvN

du≅ −κ

dSvN

dv. (4.9)

Onbothhorizonsaslongas 2M / a≫1 ,

Tαβkαk β = −LH

4πr 2 . (4.10)

UsingtheHawkingluminosityand dSvN / dv ascalculatedinasemi-classicalapproximationbyPage36forphotonsandgravitons,onecanconfirmthattheQNECissatisfiedwhilethesemi-classicalapproximationisvalid,consistentwiththerecentclaimofaquitegeneralproofoftheQNECinasemi-classicalcontextbyCeyhanandFaulkner37.

Acontroversialaspectofthemodelasformulatedinthispaperisthenegativeenergypropagatingouttofuturenullinfinityfromthewhitehole.TheasymptoticgeometryisMinkowski,andformasslessquantumfieldsinMinkowskispacetimeFordandRoman27haveestablishedthatalowerboundtoenergydensitymeasuredbyaninertialobserveraveragedoverapropertime is .Withamass ,ataradius thetimeoverwhichtidalaccelerationscanbeneglectedmeans canbeaslargeas ,correspondingtoaminimum

averagedenergydensity .Thenegativeenergydensityassociatedwiththenegativeenergyfluxfromthewhiteholeinmymodel,fallsoffroughlyas

! / M 2r 2( ) ,stronglyviolatingtheFord-Romanboundonce r ≫ M .AnyleakageofnegativeenergyfromthewhiteholelastingmuchlongerthanseveralPlancktimeshasthisproblem.

BianchiandSmerlak38havemadearguments,basedona2Dapproximationtoblackholeevaporation,thatanepisodeofnegativeenergyoutflowtofuturenullinfinityisrequiredinanyunitaryblackholeevaporationscenario.Theirresultisanecessaryconditionforunitaryevolutionoftheblackhole,inwhichthevonNeumannentropyoftheexteriorisinitiallyandfinallyzero,

(4.11)

t0 Emin ∼ −mp

2 / t04

M r

t0 r3/2 / M 1/2

Emin ∼ −"M2 / r6

!M u( )

−∞

∞

∫ exp 6SvN u( )⎡⎣ ⎤⎦du = 0.

19

19

Thisconditionistriviallysatisfiedformymodel,butitcanalsobesatisfiedbyabriefepisodeofemissionofnegativeenergywhentheentropyisnearitsmaximum,inthiscasejustafterformationofthewhitehole,thatwouldnotviolatetheFord-Romanbound.

Prolongedemissionofthenegativeenergyaccumulatedbytheblackholetolargeradiicanavoided,ifalmostallofitendsuppropagatingalong"ingoing"nullgeodesicsinsideoronthewhiteholehorizon,orontimelikegeodesicsthatfallbacktowardthewhiteholehorizonwithoutreachinglargeradii.Thenthenegativeenergycouldeventuallybeabsorbedbythereboundingmattershellthatcollapsedtoformtheblackhole.Otherwise,theFord-Romanboundwillbeviolated.Suchanalternativescenariofortheevolutionofthewhiteholeisdiscussedinthecompanionpaper.Notethatthechangeinsignofthelocalenergyfluxfrompositiveintheblackholetonegativeinthewhiteholeimpliedbytheenergyfluxintheblackholegoingtozeroat z = 0 ,togetherwithpropagationalong"ingoing"ratherthan"outgoing"radialnullgeodesicsimpliesapositiveenergydensityassociatedwiththeenergyflowasmeasuredbylocalobservers.However,"ingoing"radialnullgeodesicsinthewhitehole,like"outgoing"nullgeodesicsintheblackhole,havenegativeKillingenergyrelativetoinfinity,duetodominanceofnegativegravitationalpotentialenergy,sothecontributiontothemassofthewhiteholeisstillnegative.

Whileitmayseemplausiblethattheinflowalong"ingoing"radialnullgeodesicsintheblackholejustcontinuesacrossthetransitiontothewhitehole,thisisnotnecessarilythecase.TheHawking"partners"arenotpointparticlesfollowinggeodesics.TheyarewavepacketsofvacuumfluctuationswithatleastaPlanckscalesize.AstronglydynamicPlanck-scaletransitiontothewhiteholecanquiteplausiblycausealargedeviationfromgeodesicpropagation,andconvertan"ingoing"nulltrajectorytoan"outgoing"nulltrajectory.

TheadvancedEFcoordinatesintheblackholecannotbecontinuedintothewhitehole.Considertheequationforan"outgoing"radialnullgeodesicintheadvancedcoordinates,

∂z / ∂v( )u

= −eψ v g zz / 2. (4.12)

Startingfrom z justgreaterthanzero, isinitiallynegativeandbecomespositivecrossingthewhiteholeapparenthorizon.However,atthesamepoint

∂z / ∂v( )u

mustremainpositive,whichrequiresthat eψ v →∞ andchangesignattheapparent

horizon.WhathappensatthewhiteholehorizonoftheSchwarzschildgeometry,with e

ψ v ≡ 1,isthat v (ifdefinedasheretoincreasetothefuture)goesfrom+∞ to−∞ .WhileintheBHtoWHscenariothegeometryinthevicinityoftheWHhorizonwellafterthetransitionfromtheBHmaybeSchwarzschildtoagoodapproximation,globallytheSchwarzschildWHhorizonisaCauchyhorizon.ThetransformationtoKruskalcoordinates,whichremovestheSchwarzschildcoordinatesingularities,isincompatiblewithasmoothBHtoWHtransition.

IwillshowinthecompanionpaperthatasmoothtransitionfromtheBHrequiresinitialoutflowofnegativeofnegativeenergyacrosstheWHhorizon,but

gzz

20

20

thatitispossibletoconstructscenariosfortheevolutionoftheWHinwhichthisislimitedtoarelativelyshortPlanck-scaleintervalofretardedtimeanddoesnotconflictwiththeFord-Romanenergydensitybound.AlmostallofthenegativeenergyoftheHawkingpartnersthenremainsinsideasmallPlanck-scalewhiteholeuntilthematter/radiationshellemerges.

Finally,theexponentiallyincreasingblueshiftofanyexternalenergypropagatingalongthewhiteholehorizonshouldnotbeaproblem.Thereisnoreasonforasubstantialamountofsuchenergyinthecontextofmymodel,sincetheonlysourceforanisolatedwhiteholeisthebackscatteroffofthebackgroundcurvatureoftheoutgoingHawkingradiationfromtheblackholeandoftheoutgoingnegativeenergyradiationfromthewhitehole.Thestress-energytensorofanullfluidis T

αβ =σ kαk β ,where kα isanulltangentvectorobeyingthegeodesicequation.IntheretardedcoordinateswhenthegeometryisclosetoSchwarzschildthegeodesicequationgives

dku / du ≅ M / r 2( )ku ≅κ ku closetothehorizon,withthe

solution ku ≅ ku( )

0eκu .Then k

r = − 1− 2M / r( )ku / 2 ,fromwhich

r − 2M ≅ r − 2M( )0

e−κu , ku ≅ −κ r − 2M( )0

ku( )0and kr ≅ −ku .Conservationofthe

stress-energygives dσ / du +σ k ;α

α / ku = 0 .Since k;αα = 2 / r( ) dr / du( )k u

,

dσ / du = r − 2M( )σ / r 2 ∝ e−κu and σ →σ 0 ,aconstant.Thecontributiontothemassfunction m fromthestress-energytensoronthehorizonis

Δm ∼ −16π M 2σ 0 kuku dr∫ ∼ +2π Mσ 0 r − 2M( )2

ku( )2, (4.13)

whichisconstantinspiteoftheexponentialblueshift,asisrequiredbyenergyconservation,Thechangein e

−ψ u acrossthehorizonisalsounaffectedbytheblueshift.Ofcourse,theseareclassicalestimatesthatdonotprecludequantuminstabilities.Actually,theblueshiftislocallyjustanartifactofevaluatingtheenergyinframesacceleratingintheoppositedirectionfromthedirectionoftheflowofenergyalongthehorizon.TotheextentthatthequantumtheoryisinvariantunderlocalLorentztransformations,suchquantuminstabilitiesshouldnotbepresent.

Inthemodelpresentedinthispaper,theconcernexpressedinRef.[25]thatpositiveenergypropagatingalongthewhiteholehorizonwouldcauseconversionofthewhiteholeintoablackholewhenitintersectstheoutgoingshellofreboundingradiation(at u = u2 inFig1)isnotanissueforthecurrentmodel,sinceatthatpointthebackscattershouldbepredominantlyoriginatefromnegativeenergypropagatingoutofthewhitehole.

V.DISCUSSION

Atbestthetoymodelofthispaperisperhapsrepresentativeofthedominant

quasi-classicalhistoriescontributingtoaquantumpathintegralforevolutionoftheblackhole.Afullquantumgravitytreatmentisrequiredforanyfinalresolutionofthefateofablackholeandtheinformationproblem.ThemodelisnotconsistentwiththeexistingframeworkforLQGcalculationsdevelopedtoresolvecosmological

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singularities.Withmychoiceofparameters,theminimumtwo-sphereareaintheblackholeinteriorisaPlanckscaleconstantperhapsrelatedtothefundamental"areagap"parameterofLQGandisindependentofthemassoftheblackhole.WhiledirectquantumtunnelingfromtheblackholetothewhiteholeatthepointthespacetimecurvaturebecomesPlanckian,asarguedinRef.[23],mightbepossible,Iwouldexpectthequantumamplitudewouldbeverysmallcomparedtothatofnonsingularquasi-classicalevolution.

Iarguethatitisreasonabletoconsiderthequantumgeometryassmallfluctuationsaboutaquasi-classicalgeometryaslongas r ≫ a ,evenifthisbackgroundgeometryissubstantiallymodifiedfromaclassicalsolutionofthevacuumEinsteinequationsbyquantumbackreaction.Theeffectivestress-energytensorinthisquasi-classicalgeometryisderivedfromtheEinsteintensorcalculatedfromthemodelmetrictensorandisconsideredtoincludethemacroscopiceffectsofquantumfluctuationsinthegravitationalfieldaswellasthoseofnon-gravitationalfields.Thiscanmakesenseaslongasindividualmodesofthequantumfieldsaresmallperturbationsofabackgroundgeometry,eventhoughthecumulativeeffectofalargenumberofthesemodesmaysubstantiallymodifythegeometry.InthecontextofSchwarzschild,thesemi-classicalapproximationofquantumfieldsonafixedclassicalbackgroundgeometryshouldbevalidwherethespacetimecurvature

isverysub-Planckian, M / r3 ≪ mp

−2 ,or r ≫ Mmp

2( )1/3.

Whilemyguessattheformofthemetricinthequasi-classicalregimeisquiteadhoc,itdoesmatchthegeneralformoftheSCSETasfoundbynumericalcalculationsintheliteratureforspin0andspin1fieldsintheUnruhstate30asextrapolatedtotheblackholeinterior,butnotnecessarilytheparticularvaluesofthecoefficients.Thegeometryinthemodelvariessmoothlyinthetransitionbetweentheblackholeandthewhiteholethroughouttheblackholeevaporation,evenwhentheblackholehorizonareaisclosetothePlanckscale.Ofcourse,oneexpectslargequantumfluctuationsinthegeometrywhere r / a isoforderone.ItwouldnotbesurprisingiftheQNECwereviolatedthere,sinceitisbasicallyasemi-classicalresult.Themodelrequiresthatthequantumfocusingconjecture39isnotvalidinthevicinityofthetransitiontothewhitehole.

Thedisturbingfeatureofthismodelisthatthewhiteholeevolvesformostofitslifetimebyemittingnegativeenergy.Thisisthesamenegativeenergythatflowedintotheblackholeduringitsevaporation.Thisnegativeenergymustgosomewhere.Withoutprolongedemissionofnegativeenergy,theinitiallyPlanckscalewhiteholeremainsnearthePlanckscale,andthenegativeenergyiseventuallyabsorbedbythereboundingmatterandradiationthatformedtheblackhole..Iwillconsiderthispossibilityinacompanionpaper.

Istheresomewaytorationalizetheextendedoutflowofnegativeenergyfromthewhitehole?ThegenerationofHawkingradiationshouldbethoughtofasthetidaldisruptionofvacuumfluctuationsinthevicinityoftheblackholehorizon,partofwhichpropagatetofuturenullinfinitydirectlywithpositiveenergyandpartofwhichendupinsidetheblackholewithnegativeenergy.Thesepartsarenotindependentofeachother.Theyarestronglyentangledandcorrelated.Ifthepartinsidetheblackholelaterpropagatesoutofthewhiteholetofuturenullinfinity,it

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doesnotdosoasnormal"particles",whichmusthavepositiveenergyrelativetoasymptoticMinkowskivacuum.ThenegativeenergyemissionstogetherwiththeearlierHawkingradiationarestillpartsofvacuumfluctuations,albeitveryhighlydistortedbytheblackholegeometry.

Asomewhatsimilarsituationarisesforazero-energyvacuumfluctuationstraddlingandpropagatingalonganullhypersurfaceinMinkowskispacetime.AuniformlyacceleratingobserverforwhomthathypersurfaceisaRindlerhorizonbecomesinfinitesimallyclosetothehorizonintheoriginalinertialframeandonlypartofthefluctuationisaccessibletohim.Ifheeventuallystopsaccelerating,hewillgainaccesstothehiddenpartofthefluctuationandbeabletoverifythattheenergyoftheentirefluctuationiszero,butuntilthentheparthecanobservemayhaveasmallnon-zeroenergy.ImportantdifferencesfromtheblackholehorizonarenosystematicpreferenceinthesignoftheenergyaveragedovermanysuchfluctuationsandnoconflicttheFord-Romanbound,whichappliestoinertialobservers.TheUnruhthermalradiationmeasuredbyanacceleratingparticledetectorisnotrelevanthere,sincethisisapropertyofthedetectorinteractingwiththevacuum,andhasnothingtodowiththestress-energytensorthatisthesourceintheEinsteinequations.

Myscenarioisincomplete,sincethereisnoexplicitmodelingofhowthecollapseofthemattershellisreversed.The r = a minimumradiusoutsidetheshelldoesnotapplyinitsinterior,asinceatitscenter r = 0 isjusttheoriginofthesphericalcoordinatesinalocallyflatregion,assumingthebouncecanoccurwithoutacurvaturesingularity.WhatisdepictedinFig.1isnothingmorethanacrudeandveryschematicguess.

IftheblackholedoesevaporatedowntothePlanckscale,withnosignificantreleaseofquantuminformationacrosstheblackholehorizon,asIassume,itis

apparentthattheBekenstein-Hawkingentropy5 SBH = A / 4!( ) = 4π M / mp( )2

should

notbeinterpretedasameasureofthetotalnumberofquantumdegreesoffreedomassociatedwiththeblackhole.The"partners"oftheHawkingradiationquantasimplycrossfromtheblackholeregiontothewhiteholeregionasinFig.1andthenflowoutwardacrossthewhiteholehorizon.Neartheendoftheblackholeevaporation SBH istinycomparedwiththeentropyoftheHawkingradiationandthevonNeumannentropyoftheblackholeexterior.Itisamistaketothinkoftheblackholeinteriordegreesoffreedomasbeinginanykindofthermalequilibrium.Thedegreesoffreedomofthebouncingshellandentangledvacuummodescrossingthe z = 0 spacelikehypersurfacearecompletelyoutofcausalcontactwiththehorizondegreesoffreedomofthelatestagesoftheblackholeevaporation.While

SBH ispresumablyameasureofthemaximumnumberofquantumdegreesoffreedomassociatedwiththeblackholehorizonatanyonetime,quantumfluctuationsonthehorizondonotstayonthehorizon.TheyenduppartiallyintheHawkingradiationandpartiallyafterfallingdeepinsidetheblackholeinwhatemergesfromthewhitehole.SimilarviewshavebeenexpressedbyGarfinkle40andRovelli41.Thiscontradictsthe"centraldogma"behindmostpapersontheblack

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holeinformationproblem,asreviewedrecentlybyAlmheiri,etal42.Thecompanionpaperwillhaveamoreextensivediscussionofblackholeentropy.

Finally,theassumptionofsphericalsymmetryisunrealistic.Anysmalldeviationsfromsphericalsymmetryinthecollapsethatformstheblackholeareamplifiedasthecollapseproceeds,andclassicallythesingularitystructureofaKerrblackholewithanynonzeroangularmomentumistimelike,ratherthanthespacelikesingularityofaSchwarzschildblackhole.Sodoestheblackholetowhiteholetransitiondiscussedherehaveanyrelevancetoanevenslightlygenericblackholes?BianchiandHaggard43havemadeaninitialattempttoaddressthisquestion.Theyarguethatatleasttheinitialbreakdownofthesemi-classicalapproximationinblackholesisonaspacelikehypersurfaceforquantumgeometrieswithsmallnonzeroangularmomentumfromquantumfluctuations.AblackholetowhiteholetransitionwiththeblackholedisappearingatafiniteadvancedtimeavoidshavingtodealwithaCauchyhorizonanditsassociatedinstabilities,aspresentintheinteriorofaclassicalKerrblackhole,andwhichwouldpotentiallymakeunitarityforexternalobserversimpossible.

ACKNOLEDGEMENTS

ThisoriginalinspirationforthispapercamefromdiscussionswithHalHaggardwhilewewerebothvisitingthePerimeterInstitute.ResearchatthePerimeterInstituteissupportedbytheGovernmentofCanadathroughtheDepartmentInnovation,Science,andEconomicDevelopment,andbytheProvinceofOntariothroughtheMinistryofResearchandInnovation.IalsothankAmosOriandTommasoDeLorenzoforcommentsonanearlierversion.REFERENCES1S.W.Hawking,Nature248,30(1974);Commun.Math.Phys.43,199(1975).2S.W.Hawking,Phys.Rev.D14,2460(1976).3W.G.UnruhandR.M.Wald,Rep.Prog.Phys.80,092002(2017)

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