ResearchArticle Revisiting the Black Hole Entropy and the Information...

Research ArticleRevisiting the Black Hole Entropy and the Information Paradox

Ovidiu Cristinel Stoica

Department of Theoretical Physics National Institute of Physics and Nuclear Engineering ndash Horia Hulubei Bucharest Romania

Correspondence should be addressed to Ovidiu Cristinel Stoica holotronixgmailcom

Received 12 July 2018 Revised 13 September 2018 Accepted 20 September 2018 Published 10 October 2018

Academic Editor Theophanes Grammenos

Copyright copy 2018 Ovidiu Cristinel Stoica 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

The black hole information paradox and the black hole entropy are currently extensively researched The consensus about thesolution of the information paradox is not yet reached and it is not yet clear what can we learn about quantum gravity from theseand the related research It seems that the apparently irreducible paradoxes force us to give up on at least one well-establishedprinciple or another Since we are talking about a choice between the principle of equivalence from general relativity and someessential principles from quantum theory both being the most reliable theories we have it is recommended to proceedwith cautionand search more conservative solutions These paradoxes are revisited here as well as the black hole complementarity and thefirewall proposals with an emphasis on the less obvious assumptions Some arguments from the literature are reviewed and newcounterarguments are presented Some less considered less radical possibilities are discussed and a conservative solution which ismore consistent with both the principle of equivalence from general relativity and the unitarity from quantum theory is discussed

1 Introduction

By applying general relativity and quantum field theory oncurved spacetime Hawking arrives at the conclusion that theinformation is lost in the black holes and this breaks thepredictability [1] Apparently no matter how was formed andwhat information was contained in the matter falling in ablack hole the only degrees of freedom characterizing it areits mass angular momentum and electric charge so blackholes are ldquohairlessrdquo [2ndash5] This means that the informationdescribing the matter crossing the event horizon is lostbecause nothing outside the black hole reminds us of it Ingeneral relativity this information loss is irreversible notonly because we cannot extract it from beyond the eventhorizon but also because in a finite time the infalling matterreaches the singularity of the black hole And the occurrenceof singularities is unavoidable according to the singularitytheorems [6ndash8] This already seemed to be a problem but itwould not be so severe if we at least know that the informationis still there censored behind the horizon [9 10] But we arenot even left with this possibility since Hawking proved thatquantum effects make the black holes evaporate [11] It wasalready expected that black holes should radiate after the

realization that they have entropy and temperature [11 12]and these should be part of an extension of thermodynamicswhich includes matter as well This evaporation is thermaland after the black hole reaches a planckian size it explodesand reveals to the exterior world that the information isindeed lost In addition if the quantum state prior to theformation of the black hole was pure the final state is mixedincreasing the drama even more Moreover a problem seemsto occur long before the complete evaporation since the blackhole entropy seems to increase during evaporation until thePage time is reached [13] Some consider this to be the realblack hole information paradox [14]

Mainly for general relativists the information loss seemedto be definitive and yet not a big problem [15] positioninitially endorsed by Hawking too On the other hand forhigh energy physicists loss of unitarity was considered aproblem and various proposals to fix it appeared (see eg[16ndash18] and references therein) For example remnants wereproposed containing the information remaining in the blackhole after evaporation The remnant is in a mixed state buttogether with the Hawking radiation forms a pure state Apossible cause for remnants is the yet unknown quantumcorrections expected to occur when the black hole becomes

HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 4130417 16 pageshttpsdoiorg10115520184130417

2 Advances in High Energy Physics

too small comparable to the Plank scale and the usual anal-ysis of Hawking radiation no longer applies [19ndash21] Thereare other possibilities some being discussed in the above-mentioned reviews For example it was proposed that theinformation leaks out of the black hole through evaporationincluding by quantum tunneling that it escapes at the finalexplosion or that it leaks out of the universe in a babyuniverse [22 23] Another possibility is that the informationescapes as Hawking radiation by quantum teleportation [24]which actually happens as if the particle zig-zags forward andbackward in time to escape without exceeding the speed oflight This is not so unnatural if we assume that the finalboundary condition at the future singularity of the black holeforces the maximally entangled particles to be in a singletstate There are also bounce scenarios [25] or by using localscale invariance to avoid singularities [26] Some bouncescenarios are based on loop quantum gravity like [27 28]as well as black hole to white hole tunneling scenarios inwhich quantum tunneling is supposed to break the Einsteinequation and the apparent horizon is prevented to evolve intoan event horizon [29 30] It would take a long review to dojustice to the various proposals and this is beyond the scopeof this article

The dominating proposed solution was for two decadesblack hole complementarity [31ndash33] This was later challengedby the firewall paradox [34] The debate is not settled downyet but the dominant opinion seems to be that we have togive up at least one principle considered fundamental so farand the unlucky one ismost likely the principle of equivalencefrom general relativity One of the objectives of the presentarticle is to show that we can avoid this radical solution whilekeeping unitarity

The problems related to the black hole information lossare considered important being seen as a benchmark for thecandidate theories of quantum gravity which are expected tosolve these problems

The main purpose of this discussion is to identify themain assumptions and see if it is possible to solve theproblem in a less radical way I argue that some of the usuallymade assumptions are unnecessary that there are less radicalpossibilities and that the black hole information problemis not a decisive test for candidate theories of quantumgravity New counterarguments to some popular modelsproposed in relation to the black hole information problemare the following Black hole complementarity is discussedin Section 3 in particular the fact that an argument bySusskind aiming to prove that no-cloning is satisfied bythe black hole complementarity does not apply to mostblack holes (Section 31) the fact that its main argumentthe ldquono-omnisciencerdquo proposal does not really hold forblack holes in general (Section 33) and the fact that blackhole complementarity is also at odds with the principle ofequivalence (Section 32) As for the firewall proposal inSection 41 I explain why the tacit assumption that unitarityshould apply only to the exterior of the black hole and that weshould ignore the interior is not justified and anyway if takenas true it imposes boundary conditions to the field which iswhy the firewall seems to emerge Section 5 is dedicated toblack hole entropy In Section 51 I present an argument based

on time symmetry that the true entropy is not necessarilyproportional to the area of the event horizon and at best inthe usual cases is bounded This has negative implications tothe various proposals that the event horizon would containsome bits representing the microstates of the black holediscussed in Section 54 This may also explain the so-calledldquoreal black hole information paradoxrdquo discussed in Section 6Section 52 contains an explanation of the fact that if thelaws of black hole mechanics should be connected with thoseof thermodynamics this happens already at the classicallevel so they are not necessarily indications of quantumgravity or tests of such approaches Section 53 containsarguments that one should not read too much in the so-called no-hair theorems in particular they do not constraincontrary to a widespread belief neither the horizon nor theinterior of a black hole A major motivation invoked for thetheoretical research of the black hole information and entropyis that these may provide a benchmark to test approachesto quantum gravity but in Section 55 I argue that thesefeatures appear merely by considering quantum fields onspacetime Consequently any approach to quantum gravitywhich includes both quantum field theory and the curvedspacetime of general relativity as aminimal requirement willalso satisfy the consequences derived from them

To my knowledge the above-mentioned arguments pre-sented in more detail in the following are new and inthe cases when I was aware of other results seeming topoint in the same direction I gave the relevant referencesWhile most part of the article may look like a review of theliterature it is a critical review aiming to point out someassumptions which in my opinion drove us too far fromthe starting point which is just the most straightforwardand conservative combination of quantum field theory withthe curved background of general relativity The entirestructure of arguments converges therefore towards a moreconservative picture than that suggested by the more popularproposals The counterarguments are meant to build up thewillingness to consider the less radical proposal that I madewhich follows naturally from my work on singularities instandard general relativity ([35] and references therein) andis discussed in Section 7The background theory is presentedin Section 71 and a new enhanced version of the proposal ismade in Section 72

2 Black Hole Evaporation

Hawkingrsquos derivation of the black hole evaporation [1 11]has been disputed and checked many times and redone indifferent settings and it turned valid at most allowing someimprovements of the unavoidable approximations as well asmild generalizations But the result is correct the radiationis as predicted and thermal in the Kubo-Martin-Schwingersense [36 37] Moreover it is corroborated via the principleof equivalence with the Unruh radiation which takes placein the Minkowski spacetime for accelerated observers [38]Hawkingrsquos derivation is obtained in the framework of quan-tum field theory on curved spacetime but since the blackhole is considered large and the time scale is also large thespacetime curvature induced by the radiation is ignored

Advances in High Energy Physics 3

The derivation as well as the discussion surroundingblack hole information requires the framework of quantumfield theory on curved spacetime [39ndash41] Quantum fieldtheory on curved spacetime is a good effective limit ofthe true but yet unknown theory of quantum gravity Oncurved background there is no Poincare symmetry to selecta preferred vacuum so there is no canonical Fock space con-struction of the Hilbert space The stress-energy expectationvalue of the quantum fields ⟨푎푏(119909)⟩ is connected with thespacetime geometry via Einsteinrsquos equation

119877푎푏 minus 12119877119892푎푏 + Λ119892푎푏 =

81205871198661198884 ⟨푎푏 (119909)⟩ (1)

where 119877푎푏 is the Ricci tensor 119877 is the scalar curvature 119892푎푏is the metric tensor Λ is the cosmological constant 119866 isNewtonrsquos gravitational constant and 119888 is the speed of lightconstant

But in the calculations of the Hawking radiation thegravitational backreaction is ignored being very small Tohave well behaved solutions the spacetime slicing is suchthat the intrinsic and extrinsic curvatures of the spacelikeslices are considered small compared to the Plank lengththe curvature in a neighborhood of the spacelike surfaceis also taken to be small The wavelengths of particles areconsidered large compared to the Plank length The energyandmomentum densities are assumed small compared to thePlank density The stress-energy tensor satisfies the positiveenergy conditions The solution evolves smoothly into futureslices that also satisfy these conditions

The canonical (anti)commutation relations at distinctpoints of the slice are imposed A decomposition intopositive and negative frequency solutions is assumed towhich the Fock construction is applied to obtain the Hilbertspace The renormalizability of the stress-energy expectationvalue ⟨푎푏(119909)⟩ and the uniqueness of the 119899-point function⟨120601(1199091) 120601(119909푛)⟩ are ensured by imposing the Hadamardcondition to the quantum states [41]This condition is neededbecause when two of the 119899-points coincide there is noinvariant way to define the 119899-point function on curvedspacetime The Hadamard condition is imposed on theWightman function119866(119909 119910) = ⟨120601(119909)120601(119910)⟩ and it is preservedunder time evolution This condition is naturally satisfied inthe usual quantum field theory in Minkowski spacetime Itensures the possibility to renormalize the stress-energy tensorand to prevent it from diverging

The Fock space construction of the Hilbert space can bemade in many different ways in curved spacetime since thedecomposition into positive and negative frequency solutionsdepends on the choice of the slicing of spacetime intospacelike hypersurfaces

Suppose that a basis of annihilation operators is (119886])and they satisfy the canonical commutation relations if theyare bosons and the canonical anticommutation relations ifthey are fermions Another observer has a different basisof annihilation operators (휔) assuming that the spacetimeis curved or that one observer accelerates with respect to

the other The two bases are related by the Bogoliubovtransformations

휔 = 12120587 intinfin

0(120572휔]119886] + 120573휔]119886dagger] ) d] (2)

where 120572휔] and 120573휔] are the Bogoliubov coefficientsThe Bogoliubov transformation preserves the canonical

(anti)commutation relations and expresses the change ofbasis of the Fock space allowing us to move from oneconstruction to another The Bogoliubov transformations arelinear but not unitary They are symplectic for bosons andorthogonal for fermions though The number of particles isnot preserved so there is no invariant notion of particles

This is in fact the reason for both the Unruh effectnear a Rindler horizon and the Hawking evaporation neara black hole event horizon Because of the nonunitarityof the Bogoliubov transformation relating the Fock spacerepresentations of two distinct observers particles can beproduced [38ndash40] including for black holes [11] This meansthat what is a vacuum state for an inertial observer is a statewith many particles for an accelerated one This is true inthe Minkowski spacetime if one observer is accelerated withrespect to the other but also for two inertial observers if thecurvature is relevant as in the case of infalling and escapingobservers near a black holeMoreover themany-particle statein which the vacuum of one observer appears to the otheris thermal The particle and the antiparticle created in pairduring the evaporation are maximally entangled

3 Black Hole Complementarity

While Hawkingrsquos derivation of the black hole evaporation isrigorous and the result is correct the implication that theinformation is definitively lost can be challenged In factmost of the literature on this problem is trying to find aworkaround to restore the lost information and the unitarityThe most popular proposals like black hole complementarityand firewalls do not actually dispute the calculations butrather they add the requirement that the Hawking radiationshould contain the complete information

Additional motivation for unitarity comes from theAdSCFT correspondence [43] The AdSCFT is not yetrigorously proven and it is in fact against the currentcosmological observations that the cosmological constant ispositive [44 45] but it is widely considered true or standingfor a correct gauge-gravity duality and it is likely that itconvinced Hawking to change his mind about informationloss [46]

The favorite scenario among high-energy physicists wasfor two decades the idea of black hole complementarity [31ndash33] which supposedly resolves the conflict between unitarityessential for quantum theory and the principle of equivalencefromgeneral relativity Susskind and collaborators framed theblack hole information paradox as implying a contradictionbetween unitarity and the principle of equivalence Theyproposed a radical solution of this apparent conflict byadmitting two distinct Hilbert space descriptions for theinfalling matter and the escaping radiation [31]


Assuming that unitarity is to be restored by evaporationalone the infalling information should be found in theHawking radiation or should somehow remain above theblack hole event horizon forming the stretched horizon [31]similar to the membrane paradigm [47] But since this infor-mation falls in the black hole it would violate the no-cloningtheorem of quantum mechanics [48ndash50] If the cloning doesnot happen either the information is not recovered (andunitarity is violated) or no information can cross the horizonwhichwould violate the principle of equivalence fromgeneralrelativity which implies that nothing dramatic should happenat the event horizon assuming that the black hole is largeenough The black hole complementarity assumes that bothunitarity and the principle of equivalence hold true by allow-ing cloning but the cloning cannot be observed becauseeach observer sees only one copy The infalling copy of theinformation is accessible to an infalling observer only (usuallynamed Alice) and the escaping one to an escaping observer(Bob) Susskind and collaborators conjectured that Alice andBob can never meet to confirm that the infalling quantuminformation was cloned and the copy escaped the black hole

At first sight it may seem that the black hole comple-mentarity solves the contradiction by allowing it to exist aslong as no experiment is able to prove it Alice and Bobrsquoslightcones intersect but none of them is included in theother and they cannot be made soThis means that whateverslicing of spacetime they choose in their reference framesthe Hilbert space constructions they make will be differentSo it would be impossible to compare quantum informationfrom the interior of the black hole with the copy of quantuminformation escaping it And it is impossible to conceive anobserver able to see both copies of informationmdashthis wouldbe the so-called omniscience condition which is rejected bySusskind and collaborator to save both unitarity and theprinciple of equivalence

31 No-Cloning andTimelike Singularities Anearly objectionto the proposal that Alice and Bob can never compare thetwo copies of quantum information was that the escapingobserver Bob can collect the escaping copy of the informationand jump into the black hole to collect the infalling copyThisobjection was rejected because in order to collect a singlebit of infalling information from the Hawking radiation Bobshould wait until the black hole loses half of its initial massby evaporationmdashthe time needed for this to happen is calledthe Page time [13] So if Bob decides to jump in the black holeto compare the escaping information with the infalling one itwould be too late because the infalling information will havejust enough time to reach the singularity

The argument based on the Page time works well but itapplies only to black holes of the Schwarzschild type (moreprecisely this is an Oppenheimer-Snyder black hole [51])whose singularity is a spacelike hypersurface For rotating orelectrically charged black holes the singularity is a timelikecurve or cylinder In this case Alice can carry the infallinginformation around the singularity for an indefinitely longtime without reaching the singularity So Bob will be able toreach Alice and confirm that the quantum information wascloned

This objection is relevant because for the black holeto be of Schwarzschild type two of the three parametersdefining the black hole the angular momentum and theelectric charge have to vanish which is very unlikely Thethings are even more complicated if we take into account thefact that during evaporation or any additional particle fallingin the black hole the type of the black hole changes Usuallyparticles have nonvanishing electric charges and spin andeven if an infalling particle is electrically neutral and has thespin equal to 0 most likely it will not collide with the blackhole radially This continuous change of the type of the blackholemay result in changes of type of the singularity renderingthe argument based on the Page time invalid

In Section 33 we will see that even if the black holesomehow manages to remain of Schwarzschild type thecloning can be made manifest to a single observer

32 No-Cloning and the Principle of Equivalence Becauseof the principle of equivalence Susskindrsquos argument shouldalso hold for Rindler horizons in Minkowski spacetimeThe equivalence implies that Bob is an accelerated observerand Alice is an inertial observer who crosses Bobrsquos Rindlerhorizon Because of the Unruh effect Bob will perceive thevacuum state as thermal radiation while for Alice it would bejust vacuum Bob can see Alice being burned at the Rindlerhorizon by the thermal radiation but Alice will experiencenothing of this sort But since they are now in the Minkowskispacetime Bob can stop and go back to check the situationwith Alice and he will find that she did not experience thethermal bath he saw her experiencing While we can just saythat the complementarity should be applied only to blackholes to rule it out for the Rindler horizon and still maintainthe idea of stretched horizon only for black holes this wouldbe at odds with the principle of equivalence which black holecomplementarity is supposed to rescue

33 The ldquoNo-Omnisciencerdquo Proposal The resolution pro-posed by black hole complementarity appeals to the fact thatthe Hilbert spaces constructed by Alice and Bob are distinctwhichwould allowquantumcloning as long as the two copiesbelong to distinct Hilbert spaces and there is no observerto see the violation of the no-cloning theorem This meansthat the patches of spacetime covered by Alice and Bob aredistinct such that apparently no observer can cover both ofthem If there was such an ldquoomniscientrdquo observer he or shewould see the cloning of quantum information and see thatthe laws of quantum theory are violated

Yet there is such an observer albeit moving backwardsin time (see Figure 1) Remember that the whole point oftrying to restore the loss information and unitarity is becausequantum theory should be unitary This means not onlydeterministic but also that the time evolution laws have tobe time symmetric as quantum theory normally is so thatwe can recover the lost information So everything quantumevolution does forward in time should be accessible bybackwards in time evolution An observer going backwardsin time Charlie can then in principle be able to perceive bothcopies of the information carried by Alice and Bob so he isldquoomniscientrdquo


(a) (b)

Figure 1 (a) The Penrose diagram of black hole evaporation depicting Alice and Bob and their past lightcones (b) The Penrose diagram ofa backwards in time observer Charlie depicting how he observes Alice and Bob and the quantum information each of them caries even ifthis information is cloned therefore disclosing a violation of quantum theory

One can try to rule Charlie out on the grounds thathe violates causality or more precisely the second law ofthermodynamics [52] But from the point of view of quantumtheory the von Neumann entropy is preserved by unitaryevolution and the quantum evolution is reversible anywayso it is irrelevant that if in our real universe there is athermodynamic arrow of time this does not invalidate aprincipial thought experiment like this one

4 The Firewall Paradox

After two decades since the proposal of black hole comple-mentarity this solution was disputed by the firewall paradox[34] which suggested that the equivalence principle shouldbe violated at the event horizon where a highly energetic cur-tain or a singularity should form to prevent the informationfalling inside the black hole

The firewall argument takes place in the same settingsas the black hole complementarity proposal but this time itinvolves the monogamy of entanglement More precisely it isshown that the late radiation has to be maximally entangledwith both the early radiation and the infalling counterpartof the late radiation Since the monogamy of entanglementforbids this it is proposed that one of the assumptions has togo most likely the principle of equivalence The immediatereaction varied from quick acceptance to arguments that theparadox is solved too by the black hole complementarity[53 54] After all we can think of the late radiation asbeing entangled with the early one in Bobrsquos Hilbert spaceand with the infalling radiation in Alicersquos Hilbert space Butit turned out that unlike the case of the violation of theno-cloning theorem the violation of monogamy cannot beresolved by Alice and Bob having different Hilbert spaces[55]

One can argue that if the firewall experiment is per-formed it creates the firewall and if it is not performed Alice

sees no firewall so black hole complementarity is not com-pletely lost Susskind and Maldacena proposed the ER=EPRsolution which states that if entangled particles are thrownin different black holes then they become connected by awormhole [56] also see [57]The firewall idea also stimulatedvarious discussions about the relevance of complexity ofquantumcomputation and error correction codes in the blackhole evaporation and decoding the information from theHawking radiation using unitary operations (see [54 58 59]and references therein)

Various proposals to rescue both the principle of equiv-alence and unitarity were made for example based on theentropy of entanglement across the event horizon in [60 61]Hawking proposed that the black hole horizons are onlyapparent horizons and never actual event horizons [62]Later Hawking proposed that supertranslations allow thepreservation of information and further expanded the ideawith Perry and Strominger [63ndash65]

Having to give up the principle of equivalence or unitarityis a serious dilemma so it is worth revisiting the argumentsto find a way to save both

41 The Meaning of ldquoUnitarityrdquo In the literature about blackhole complementarity and firewalls by the assumption orrequirement of ldquounitarityrdquo we should understand ldquounitarityof the Hawking radiationrdquo or more precisely ldquounitarity ofthe quantum state exterior to the black holerdquo Let us call thisexterior unitarity to emphasize that it ignores the interior ofthe black hole It is essential to clarify this because whenwe feel that we are forced to choose between unitarity andthe principle of equivalence we are in fact forced to choosebetween exterior unitarity and the principle of equivalenceThis assumption is also at the origin of the firewall proposalSo no choice between unitarity and the principle of equiva-lence is enforced to us unless by ldquounitarityrdquo we understandldquoexterior unitarityrdquo


The idea that unitarity should be restored from theHawking radiation alone ignoring the interior of the blackhole was reinforced by the holographic principle and theidea of stretched horizon [31 32 66] a place just abovethe event horizon which presumably stores the infallinginformation until it is restored through evaporation and itwas later reinforced even more by the AdSCFT conjecture[43] But it is not excluded to solve the problem by takinginto consideration both the exterior and interior of the blackhole and the corresponding quantum states A proposalaccounting for the interior in the AdSCFT correspondencebased on the impossibility to localize the quantum operatorsin quantum gravity in a background-independent mannerwas made in [67] A variation of the AdSCFT leading to aregularization was made in [68]

In fact considering both the exterior and the interior ofthe black hole is behind proposals like remnants and babyuniverses But we will see later that there is a less radicaloption

Exterior unitarity or the proposal that the full infor-mation and purity are restored from Hawking radiationalone simply removes the interior of the black hole from thereference frame of an escaping observer consequently fromhis Hilbert space This type of unitarity imposes a boundarycondition to the quantum fields which is simply the fact thatthere is no relevant information inside the black hole So it isnatural that at the boundary of the support of the quantumfields which is the black hole event horizon quantum fieldsbehave as if there is a firewall This is what the variousestimates revealing the existence of a highly energetic firewallor horizon singularity confirm Note that since the boundarycondition which aims to rescue the purity of the Hawkingradiation is a condition about the final state sometimesits consequences give the impression of a conspiracy assometimes Bousso and Hayden put it [69]

While I have no reason to doubt the validity of the firewallargument [34] I have reservations about assuming unitarityas referring only to quantum fields living only to the exteriorof the black hole while ignoring those from its interior

42 Firewalls versus Complementarity The initial Hilbertspaces of Alice and Bob are not necessarily distinct Even ifthey and their Fock constructions are distinct each state fromone of the spaces may correspond to a state from the otherThe reason is that a basis of annihilation operators in Alicersquosframe say (119886]) is related to a basis of annihilation operatorsin Bobrsquos frame (휔) by a Bogoliubov transformation (2) TheBogoliubov transformation is linear although not unitary

Thus one may hope that the Hilbert spaces of Aliceand Bob may be identified even though through a veryscrambled vector space isomorphism so that black holecomplementarity saves the day However exterior unitarityimposes that the evolved quantum fields from the Hilbertspaces have different supporting regions in spacetime Whilebefore the creation of the black hole they may have thesame support in the spacelike slice they evolve differentlybecause of the exterior unitarity condition Bobrsquos systemevolves so that his quantum fields are constrained to the

exterior of the black hole while Alicersquos quantumfields includethe interior too Bobrsquos Hilbert space is different becausewhen the condition of exterior unitarity was imposed itexcluded the interior of the black hole So even if the initialunderlying vector space is the same for both the Hilbertspace constructed by Alice and that constructed by Bob theircoordinate systems diverged in time so the way they slicespacetime became different While normally Alicersquos vacuumis perceived by Bob as loaded with particles in a thermalstate this time in Bobrsquos frame Alicersquos vacuum energy becomessingular at the horizon This makes the firewall paradox aproblem for black hole complementarity A cleaner argumentbased on purity rather than monogamy is made by Bousso[70]

An interesting issue is that Bob can infer that if the modeshe detects passed very close to the event horizon they wereredshifted So evolving the modes backwards in time it mustbe that the particle passes close to the horizon at a very highfrequencymaybe evenhigher than the Plank frequencyDoesthis mean that Alice should feel dramatically this radiationThere is the possibility that for Alice Bobrsquos high frequencymodes are hidden in her vacuum stateThis is also confirmedby acoustic black holes [71] Only if thesemodes are somehowdisclosed for example if Bob being accelerated performssome temperature detection nearby Alice these modes maybecome manifest due to the projection postulate otherwisethey remain implicit in Alicersquos vacuum

It seems that the strength of the firewall proposal comesfrom rendering black hole complementarity unable to solvethe firewall paradoxThey are two competing proposals bothaiming to solve the same problem While one can logicallythink that proposals that take into account the interior ofblack holes to restore unitarity are good candidates aswell andthat they may have the advantage of rescuing the principle ofequivalence sometimes they are dismissed as not addressingthe ldquorealrdquo black hole information paradox I will say moreabout this in Section 6

5 Black Hole Entropy

The purposes of this section are to prepare for Section 6 andto discuss the implications of black hole entropy for the blackhole information paradox and for quantum gravity

The entropy bound of a black hole is proportional to thearea of the event horizon [12 72 73]

119878퐵퐻 = 119896퐵1198604ℓ2푃 (3)

where 119896퐵 is the Boltzmann constant 119860 is the area of the eventhorizon and ℓ푃 is the Plank length

The black hole entropy bound (3) was suggested byHawkingrsquos result that the black hole horizon area neverdecreases [74] as well as the development of this result intothe four laws of black hole mechanics [72]

51 The Area of the Event Horizon and the Entropy It istempting to think that the true entropy of quantum fields inspacetime should also include the areas of the event horizons


In fact there are computational indications that the blackhole evaporation leaks the right entropy to compensate thedecrease of the area of the black hole event horizon

But there is a big difference between the entropy ofquantum fields and the areas of horizons First entropy isassociated with the state of the matter (including radiation ofcourse) If we look at the phase space we see that the entropyis a property of the state alone so it is irrelevant if the systemevolves in one direction of time or the opposite the entropycorresponding to the state at a time 119905 is the same The same istrue for quantumentropy associatedwith the quantum stateswhich in fact is preserved by unitary evolution and is the samein either time direction

On the other hand the very notion of event horizon ingeneral relativity depends on the direction of time By lookingagain at Figure 1(b) this time without being interested inblack hole complementarity we can see that for Charlie thereis no event horizon But the entropy corresponding to matteris the same independently of his time direction So even ifwe are able to put the area on the event horizon in the sameformulawith the entropy of the fields and still have the secondlaw of thermodynamics the two terms behave completelydifferently So if the area of the event horizon is requiredto compensate for the disappearance of entropy beyond thehorizon and for its reemergence as Hawking radiation forCharlie the things are quite different because he has fullclearance to the interior of the black hole which for him iswhite In other words he is so omniscient that he knows thetrue entropy of thematter inside the black hole and not amerebound given by the event horizon

This is consistent with the usual understanding of entropyas hidden information indeed the true information aboutthe microstates is not accessible (only the macrostate) andthis is what entropy stands for But it is striking neverthelessto see that black holes do the same yet in a completely time-asymmetric manner This is because the horizon entropy isjust a bound for the entropy beyond the horizon the trueentropy is a property of the state

52 Black Hole Mechanics and Thermodynamics Matter orGeometry The four laws of black hole mechanics are thefollowing [72 75]

(i) 0th law the surface gravity 120581 is constant over theevent horizon

(ii) 1st law for nearby solutions the differences in massare equal to differences in area times the surfacegravity plus some additional terms similar to work

(iii) 2st law in any physical process the area of the eventhorizon never decreases (assuming positive energy ofmatter and regularity of spacetime)

(iv) 3rd law there is no procedure consisting of a finitenumber of steps to reduce the surface gravity to zero

The analogy between the laws of black hole mechanicsand thermodynamics is quite impressive [75] In particularenthalpy temperature entropy and pressure correspondrespectively to the mass of the black hole its surface gravityits horizon area and the cosmological constant

These laws of black hole mechanics are obtained in purelyclassical general relativity but were interpreted as laws ofblack hole thermodynamics [11 76 77]Their thermodynam-ical interpretation occurs when considering quantum fieldtheory on curved spacetime and it is expected to followmoreprecisely from the yet to be found quantum gravity

Interestingly despite their analogy with the laws of ther-modynamics the laws of black hole mechanics hold in purelyclassical general relativity While we expect general relativityto be at least a limit theory of a more complete quantizedone it is a standalone and perfectly selfconsistent theoryThis suggests that it is possible that the laws of black holemechanics already have thermodynamic interpretation in thegeometry of spacetime And this turns out to be true sinceblack hole entropy can be shown to be the Noether chargeof the diffeomorphism symmetry [78] This works exactlyfor general relativity and it is different for gravity modifiedso that the action is of higher order in terms of curvatureIn addition we already know that Einsteinrsquos equation canbe understood from an entropic perspective which has ageometric interpretation [79 80]

This is not to say that the interpretations of the laws ofblack hole mechanics in terms of thermodynamics of quan-tum fields do not hold because there are strong indicationsthat they do My point is rather that there are thermody-namics of the spacetime geometry which are tied somehowwith the thermodynamics of quantum matter and radiationThis connection is probably made via Einsteinrsquos equation orwhatever equation whose classical limit is Einsteinrsquos equation

53 Do Black Holes Have No Hair Classically black holesare considered to be completely described by their massangular momentum and electric charge This idea is basedon the no-hair theorems These results were obtained for theEinstein-Maxwell equations assuming that the solutions areasymptotically flat and stationary While it is often believedthat these results hold universally they are in fact similarto Birkhoff rsquos theorem [81] which states that any sphericallysymmetric solution of the vacuum field equations must bestatic and asymptotically flat hence the exterior solutionmust be given by the Schwarzschild metric Werner Israelestablishes that the Schwarzschild solution is the uniqueasymptotically flat static nonrotating solution of Einsteinrsquosequation in vacuum under certain conditions [2] This wasgeneralized to the Einstein-Maxwell equations (electrovac)[3ndash5] the result being the characterization of static asymptot-ically flat solutions only by mass electric charge and angularmomentum It is conjectured that this result is general butcounterexamples are known [82 83]

In classical general relativity the black holes radiategravitational waves and are expected to converge to a no-hairsolution very fast If this is true it happens asymptoticallyand the gravitational waves carry the missing informationabout the initial shape of the black hole horizon becauseclassical general relativity is deterministic on regular globallyhyperbolic regions of spacetime

Moreover it is not known what happens when quantumtheory is applied If the gravitational waves are quantized


(resulting in gravitons) it is plausible to consider the possi-bility that quantum effects prevent such a radiation like inthe case of the electron in the atom Therefore it is not clearthat the information about the infalling matter is completelylost in the black hole even in the absence of Hawkingevaporation So we should expect at most that black holesconverge asymptotically to the simple static solutions but ifthey would reach them in finite time there would be no timereversibility in GR

Nevertheless this alone is unable to provide a solutionto the information loss paradox especially since spacetimecurvature does not contain the complete information aboutmatter fields But we see that we have to be careful when weuse the no-hair conjecture as an assumption in other proofs

54 Counting Bits While black hole mechanics suggest thatthe entropy of a black hole is limited by the Bekenstein bound(3) it is known that the usual classical entropy of a system canbe expressed in terms of its microstates

119878푄 = minus119896퐵sum푖

119901푖 ln119901푖 (4)

where 119901푖 denotes the number of microstates which cannot bedistinguished because of the coarse grainingmacroscopicallyappearing as the 119894-th macrostate A similar formula givesthe quantum von Neumann entropy in terms of the densitymatrix 120588

119878 = minus119896퐵tr (120588 ln 120588) (5)

Because of the no-hair theorem (see Section 53) itis considered that classical black holes can be completelycharacterized by the mass angular momentum and electriccharge at least from the outside This is usually understoodas suggesting that quantum black holes have to containsomewhere most likely on their horizons some additionaldegrees of freedom corresponding to their microstates sothat (3) can be interpreted in terms of (4)

It is often suggested that there are some horizonmicrostates either floating above the horizon but not fallingbecause of a brick wall [84ndash86] or being horizon gravitationalstates [87]

Other counting proposals are based on counting stringexcited microstates [88ndash90] There are also proposals ofcounting microstates in LQG for example by using a Chern-Simons field theory on the horizon as well as choosing aparticular Immirzi parameter [91]

Another interesting possible origin of entropy comesfrom entropy of entanglement resulting by the reduced densitymatrix of an external observer [92 93] This is proportionalbut for short distances requires renormalization

But following the arguments in Section 51 I think thatthe most natural explanation of black hole entropy seems tobe to consider the internal states of matter and gravity [94]A model of the internal state of the black hole similar tothe atomic model was proposed in [95ndash97] Models basedon Bose-Einstein condensates can be found in [98ndash100] andreferences therein

Since in Section 51 it was explained that the horizonsjust hide matter and hence entropy and are not in fact thecarriers of the entropy it seems more plausible to me that thestructure of the matter inside the black hole is just boundedby the Bekenstein bound and does not point to an unknownmicrostructure

55 A Benchmark to Test Quantum Gravity Proposals Theinterest in the black hole information paradox and blackhole entropy is not only due to the necessity of restoringunitarity This research is also motivated by testing variouscompeting candidate theories of quantum gravity Quantumgravity seems to be far from our experimental possibilitiesbecause it is believed to become relevant at very small scalesOn the other hand black hole information loss and blackhole entropy pose interesting problems and the competingproposals of quantum gravity are racing to solve them Themotivation is that it is considered that black hole entropy andinformation loss can be explained by one of these quantumgravity approaches

On the other hand it is essential to remember how blackhole evaporation and black hole entropy were derived Themathematical proofs are done within the framework of quan-tum field theory on curved spacetime which is considereda good effective limit of the true but yet to be discoveredtheory of quantum gravity The calculations are made nearthe horizon they do not involve extreme conditions likesingularities or planckian scales where quantum gravity isexpected to take the lead The main assumptions are

(1) quantum field theory on curved spacetime(2) the Einstein equation with the stress-energy ten-

sor replaced by the stress-energy expectation value⟨푎푏(119909)⟩ (see (1))

For example when we calculate the Bekenstein entropybound we do this by throwing matter in a black hole and seehow much the event horizon area increases

These conditions are expected to hold in the effective limitof any theory of quantum gravity

But since both the black hole entropy and the Hawkingevaporation are obtained from the two conditions mentionedabove this means that any theory in which these conditionsare true at least in the low energy limit is also able to implyboth the black hole entropy and the Hawking evaporation Inother words if a theory of quantum gravity becomes in somelimit the familiar quantum field theory and also describesEinsteinrsquos gravity it should also reproduce the black holeentropy and the Hawking evaporation

Nevertheless some candidate theories to quantum grav-ity do not actually work in a dynamically curved spacetimebeing for example defined on flat or AdS spacetime yetthey still are able to reproduce a microstructure of blackhole entropy This should not be very surprising giventhat even in nonrelativistic quantum mechanics quantumsystems bounded in a compact region of space have discretespectrum So it may be very well possible that these resultsare due to the fact that even in nonrelativistic quantummechanics entropy bounds hold [101] In flat spacetime we


can think that the number of states in the spectrum isproportional with the volume However when we plug in themasses of the particles in the formula for the Schwarzschildradius (which incidentally is the same as Michellrsquos formula inNewtonian gravity [102]) we should obtain a relation similarto (3)

The entropy bound (3) connects the fundamental con-stants usually considered to be characteristic for generalrelativity quantum theory and thermodynamics This doesnot necessarily mean that the entropy of the black holewitnesses about quantum gravityThis should be clear alreadyfrom the fact that the black hole entropy bound was notderived by assuming quantum gravity but simply from theassumptions mentioned above It is natural that if we plug theinformation and the masses of the particles in the formula forthe Schwarzschild radius we obtain a relation between theconstants involved in general relativity quantum theory andthermodynamics It is simply a property of the system itselfnot a witness of a deeper theory But of course if a candidatetheory of quantum gravity fails to pass even this test this maybe a bad sign for it

6 The Real Black Hole Information Paradox

Sometimes it is said that the true black hole informationparadox is the one following fromDon Pagersquos article [13] Forexample Marolf considers that here lies the true paradoxicalnature of the black hole information while he calls themere information loss and loss of purity ldquothe straw maninformation problemrdquo [14] Apparently the black hole vonNeumann entropy should increase with one bit for eachemitted photon At the same time its area decreases bylosing energy so the black hole entropy should also decreaseby the usual Bekenstein-Hawking kind of calculation Sowhat happens with the entropy of the black hole Does itincrease or decrease This problem occurs much earlier inthe evolution of the black hole when the black hole area isreduced to half of its initial value (the Page time) so we donot have to wait for the complete evaporation to notice thisproblem Marolf put it as follows[14]

This is now a real problem Evaporation causes theblack hole to shrink and thus to reduce its surfacearea So 119878퐵퐻 decreases at a steady rate On the otherhand the actual von Neumann entropy of the blackhole must increase at a steady rate But the first mustbe larger than the second So some contradiction isreached at a finite time

I think there are some assumptions hidden in thisargument We compare the von Neumann entropy of theblack hole calculated during evaporation with the black holeentropy calculated by Bekenstein and Hawking by throwingparticles in the black hole While the proportionality of theblack hole entropy with the area of the event horizon hasbeen confirmed by various calculations for numerous casesthe two types of processes are different so it is natural thatthey lead to different states of the black hole and hence todifferent values for the entropy This is not a paradox it isjust an evidence that the entropy contained in the black hole

depends on the way it is created despite the bound given bythe horizon So it seems more natural not to consider thatthe entropy of the matter inside the black hole reached themaximumbound at the beginning but rather that it reaches itsmaximum at the Page time due to the entanglement entropywith the Hawking radiation Alternatively we may still wantto consider the possibility of having more entropy in theblack hole than the Bekenstein bound allows In fact Rovellimade another argument pointing in the same direction thatthe Bekenstein-Bound is violated by counting the number ofstates that can be distinguished by local observers (as opposedto external observers) using local algebras of observables[103] This argument provided grounds for a proposal of awhite hole remnant scenario discussed in [104]

7 A More Conservative Solution

We have seen in the previous sections that some importantapproaches to the black hole information paradox and therelated topics assume that the interior of the black hole isirrelevant or does not exist and the event horizon plays theimportant role I also presented arguments that if it is torecover unitarity without losing the principle of equivalencethen the interior of the black hole should be considered aswell and the event horizon should not be endowed withspecial properties More precisely given that the originalculprit of the information loss is its supposed disappear-ance at singularities then singularities should be closelyinvestigated The least radical approach is usually consideredthe avoidance of singularity by modifying gravity (ie therelation between the stress-energy tensor and the spacetimecurvature as expressed by the Einstein equation) so that oneor more of the three assumptions of the singularity theorems[6ndash8] no longer hold In particular it is hoped that this maybe achieved by the quantum effects in a theory of quantumgravity However it would be even less radical if the problemcould be solved without modifying general relativity andsuch an approach is the subject of this section

But singularities are accompanied by divergences inthe very quantities involved in the Einstein equation inparticular the curvature and the stress-energy tensor So evenif it is possible to reformulate the Einstein equation in termsof variables that do not diverge remaining instead finite at thesingularity the question remains whether the physical fieldsdiverge or break down In other words what are in fact thetrue fundamental physical fields the diverging variables orthose that remain finiteThis questionwill be addressed soon

An earlier mention of the possibility of changing thevariables in the Einstein equation was made by Ashtekarfor example in [105] and references therein where it isalso proposed that the new variables could remain finite atsingularities even in the classical theory However it turnedout that one of his two new variables diverges at singularities(see eg [106]) Eventually this formulation led to loopquantum gravity where the avoidance is instead achieved onsome toy bounce models (see eg [28 29]) But the problemwhether standard general relativity can admit a formulationfree of infinities at singularities remained open for a while


71 Singular General Relativity In [107 108] the authorintroduced a mathematical formulation of semi-Riemanniangeometry which allows a description of a class of singularitiesfree of infinities The fields that allowed this are invariantand in the regions without singularities they are equivalent tothe standard formulation To understand what the problemis and how it is solved recall that in geometry the metrictensor is assumed to be smooth and regular that is withoutinfinite components and nondegenerate which means thatits determinant is nonvanishing If the metric tensor hasinfinite components or if it is degenerate the metric is calledsingular If the determinant is vanishing one cannot definethe Levi-Civita connection because the definition relies onthe Christoffel symbols of the second kind

Γ푖푗푘 fl 12119892푖푠 (119892푠푗푘 + 119892푠푘푗 minus 119892푗푘푠) (6)

which involve the contraction with 119892푖푠 which is the inverseof the metric tensor 119892푖푗 hence it assumes it to be nonde-generate This makes it impossible to define the covariantderivative and the Riemann curvature (hence the Ricci andscalar curvatures as well) at the points where the metricis degenerate These quantities blow up while approachingthe singularities Therefore Einsteinrsquos equation as well breaksdown at singularities

However it turns out that on the space obtained byfactoring out the subspace of isotropic vectors an inversecan be defined in a canonical and invariant way and thatthere is a simple condition that leads to a finite Riemanntensor which is defined smoothly over the entire spaceincluding at singularities This allows the contraction of acertain class of tensors and the definition of all quantitiesof interest to describe the singularities without runninginto infinities and is equivalent to the usual nondegeneratesemi-Riemannian geometry outside the singularities [107]Moreover it works well for warped products [108] allowingthe application for big bang models [109 110] This approachalso works for black hole singularities [42 111 112] allowingthe spacetime to be globally hyperbolic even in the presenceof singularities [113] More details can be found in [35 114]and the references therein Here I will first describe some ofthe already published results and continuewith new andmoregeneral arguments

An essential difficulty related to singularities is givenby the fact that despite the Riemann tensor being smoothand finite at such singularities the Ricci tensor 119877푖푗 fl 119877푠푖푠푗usually continues to blow up The Ricci tensor and its tracethe scalar curvature 119877 = 119877푠푠 are necessary to define theEinstein tensor 119866푖푗 = 119877푖푗 minus (12)119877119892푖푗 Now here is the partwhere the physical interpretation becomes essential In theEinstein equation the Einstein tensor is equated to the stres-energy tensor So they both seem to blow up and indeedthey do Physically the stress-energy tensor represents thedensity of energy andmomentum at a point However what isphysically measurable is never such a density at a point but itsintegral over a volume The energy or momentum in a finitemeasure volume is obtained by integrating with respect tothe volume element And the quantity to be integrated for

example the energy density 11987900dV표푙 where 11987900 = 119879(119906 119906) fora timelike vector 119906 and dV표푙 fl radicminusdet119892d1199090 andd1199091 andd1199092 andd1199093is finite even if 11987900 997888rarr infin since dV표푙 997888rarr 0 in the properway The mathematical theory of integration on manifoldsmakes it clear that what we integrate are differential formslike11987900dV표푙 and not scalar functions like11987900 So I suggest thatwe should do in physics the same as in geometry because itmakesmore sense to consider the physical quantities to be thedifferential forms rather than the scalar components of thefields [109] This is also endorsed by two other mathematicalreasons On one hand when we define the stress-energy 119879푖푗we do it by functional derivative of the Lagrangian withrespect to the metric tensor and the result contains thevolume element which we then divide out to get 119879푖푗 Shouldwe keep it we would get instead 119879푖푗dV표푙 Also when we derivethe Einstein equation from the Lagrangian density 119877 we infact vary the integral of the differential form 119877dV표푙 and not ofthe scalar 119877 And the resulting Einstein equation has again afactor dV표푙 which we leave out of the equation on the groundsthat it is never vanishing Well at singularities it vanishes sowe should keep it because otherwise we divide by 0 and weget infinities The resulting densitized form of the Einsteinequation

119866푖푗dV표푙 + Λ119892푖푗dV표푙 = 81205871198661198884 119879푖푗dV표푙 (7)

is equivalent to Einsteinrsquos outside singularities but as alreadyexplained I submit that it better represents the physicalquantities and not only because these quantities remain finiteat singularities I call this densitized Einstein equation butthey are in fact tensorial as well the fields involved aretensors being the tensor products between other tensors andthe volume form which itself is a completely antisymmetrictensor Note that Ashtekarrsquos variables are also densitiesand they are more different from the usual tensor fieldsinvolved in the semi-Riemannian geometry and Einsteinrsquosequation yet they were proposed to be the real variablesboth for quantization and for eliminating the infinities in thesingularities [105] But the formulation I proposed remainsfinite even at singularities and it is closer as interpretation tothe original fields

Another difficulty this approach had to solve was thatit applies to a class of degenerate metrics but the blackholes are nastier since the metric has components thatblow up at the singularities For example the metric tensorof the Schwarzschild black hole solution expressed in theSchwarzschild coordinates is

d1199042 = minus(1 minus 2119898119903 ) d1199052 + (1 minus 2119898

119903 )minus1

d1199032 + 1199032d1205902 (8)

where119898 is the mass of the body the units were chosen so that119888 = 1 and 119866 = 1 and

d1205902 = d1205792 + sin2120579d1206012 (9)

is the metric of the unit sphere 1198782For the horizon 119903 = 2119898 the singularity of the metric can

be removed by a singular coordinate transformation see for


example [115 116] Nothing of this sort could be done forthe 119903 = 0 singularity since no coordinate transformationcan make the Kretschmann scalar 119877푖푗푘푙119877푖푗푘푙 finite Howeverit turns out that it is possible to make the metric at thesingularity 119903 = 0 into a degenerate and analytic metricby coordinate transformations In [111] it was shown thatthis is possible and an infinite number of solutions werefound which lead to an analytic metric degenerate at 119903 = 0Among these solutions there is a unique one that satisfiesthe condition of semiregularity from [107] which ensures thesmoothness and analyticity of the solution for the interior ofthe black hole This transformation is

119903 = 1205912

119905 = 1205851205914(10)

and the resulting metric describing the interior of theSchwarzschild black hole is

d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)

This is not to say that physics depend on the coordinatesIt is similar to the case of switching from polar to Cartesiancoordinates in plane or like the Eddington-Finkelstein coor-dinates In all these cases the transformation is singular atthe singularity so it is not a diffeomorphism The atlas thedifferential structure is changed and in the new atlas withits new differential structure the diffeomorphisms preserveof course the semiregularity of themetric And just like in thecase of the polar or spherical coordinates and the Eddington-Finkelstein coordinates it is assumed that the atlas in whichthe singularity is regularized is the real one and the problemswere an artifact of the Schwarzschild coordinates whichthemselves were in fact singular

Similar transformations were found for the other types ofblack holes (Reissner-Nordstrom Kerr and Kerr-Newman)and for the electrically charged ones the electromagnetic fieldalso no longer blows up [42 112]

72 Beyond the Singularity Returning to the Schwarzschildblack hole in the new coordinates (11) the solution extendsanalytically through the singularity If we plug this solution inthe Oppenheimer-Snyder black hole solution we get an ana-lytic extension depicting a black hole which forms and thenevaporates whose Penrose-Carter diagram is represented inFigure 2

The resulting spacetime does not have Cauchy horizonsbeing hyperbolic which allows the partial differential equa-tions describing the fields on spacetime to be well posed andcontinued through the singularity Of course there is still theproblem that the differential operators in the field equationsof the matter and gauge fields going through the singularityshould be replaced with the new ones Such formulationsare introduced in [117] and sufficient conditions are to besatisfied by the fields at the singularities so that their evolutionequations work was given in the case of Maxwell and Yang-Mills equations

Figure 2 An analytic extension of the black hole solution beyondthe singularity

It is an open problemwhether the backreaction will makethe spacetime to curve automatically so that these conditionsare satisfied for all possible initial conditions of the fieldThisshould be researched in the future including for quantumfields It is to be expected that the problem is difficult andwhat is given here is not the general solution but rather atoy model Anyway no one should expect very soon an exacttreatment of real case situations so the whole discussion hereis in principle to establish whether this conservative approachis plausible enough

However I would like to propose here a different moregeneral argument which avoids the difficulties given bythe necessity that the field equations should satisfy at thesingularities special conditions like the sufficient conditionsfound in [117] and also the open problem of which arethe conditions to be satisfied by the fermionic fields atsingularities

First consider Fermatrsquos principle in optics A ray of light ingeometric optics is straight but if it passes from one mediumto another having a different refraction index the ray changesits direction and appears to be broken It is still continuousbut the velocity vector is discontinuous and it appears thatthe acceleration blows up at the surface separating the twomedia But Fermatrsquos principle still allows us to know exactlywhat happens with the light ray in geometric optics

On a similar vein I think that in the absence of a proofthat the fields satisfy the exact conditions [117] when crossinga singularity we can argue that the singularities are not athreat to the information contained in the field by using theleast action principle instead

The least action principle involves the integration of theLagrangian densities of the fields While the conditions thefields have to satisfy at the singularity in order to behavewell are quite restrictive the Lagrangian formulation is muchmore general The reason is that integration can be done overfields with singularities also on distributions and the resultcan still be finite

Consider first classical point-like particles falling in theblack hole crossing the singularity and exiting through the


(a)

(b)

Figure 3 (a) The causal structure of the Schwarzschild black hole in coordinates (120591 120585) from (10) (b) The causal structure of the Reissner-Nordstrom black hole in coordinates (120591 120588) playing a similar role (see [42])

white hole which appears after the singularity disappearsThehistory of such a test particle is a geodesic and to understandthe behavior of geodesics we need to understand first thecausal structure In Figure 3 the causal structures of (a) aSchwarzschild black hole and (b) a Reissner-Nordstrom blackhole are represented in the coordinates which smoothen thesingularity (see [118])

If the test particle is massless its path is a null geodesic In[118] I showed that for the standard black holes the causalstructure at singularities is not destroyed The lightcones willbe squashed but they will remain lightcones Therefore thehistory of a massless particle like a photon is if we applythe least action principle just a null geodesic crossing thesingularity and getting out

If the test particle is massive its history is a timelikegeodesic In this case a difficulty arises because in thenew coordinates the lightcones are squashed This allows fordistinct geodesics to intersect the singularity at the samepoint and to have the same spacetime tangent direction Inthe Schwarzschild case this does not happen for timelikegeodesics but in the Reissner-Nordstrom case [42] all ofthe timelike geodesics crossing the singularity at the samepoint become tangent Apparently this seems to imply thata geodesic crossing a timelike singularity can get out of it inany possible direction in a completely undetermined way Tofix this one may want to also consider the second derivativeor to use the local cylindrical symmetry around the timelikesingularity

But the least action principle allows this to be solvedregardless of the specific local solution of the problem atthe singularity The timelike geodesics are tangent only atthe singularity which is a zero-measure subset of spacetimeSo we can apply the least action principle to obtain thehistory of a massive particle and obtain a unique solutionThe least action principle can be applied for classical testparticles because a particle falling in the black hole reachesthe singularity in finite proper time and similarly a finiteproper time is needed for it to get out Moreover the pathintegral quantization will consider anyway all possible paths

so even if there would be an indeterminacy at the classicallevel it will be removed by integrating them all

For classical fields the same holds as for point-likeclassical particles only the paths are much more difficult tovisualize The least action principle is applied in the con-figuration space even for point-like particles and the sameholds for fields the only difference being the dimension ofthe configuration space and the Lagrangian The points fromthe singularity formagain a zero-measure subset compared tothe full configuration space so finding the least action pathis similar to the case of point-like particles The Lagrangiandensity is finite at least at the points of the configuration spaceoutside the singularities which means almost everywhereBut the volume element vanishes at singularities whichimproves the situation So its integral can very well be finiteeven if the Lagrangian density would be divergent at thesingularities It may be the case that the fields have singularLagrangian density at the singularity and that when weintegrate them it is not excluded that even the integral maydiverge but in this case the least action principle will force usanyway to choose the paths that have a finite action densityat the singularities and such paths exist for example thosesatisfying the conditions found in [117]

So far we have seen that the principle of least action allowsdetermining the history of classical point-like particles orfields from the initial and final conditions even if they crossthe singularity This is done so far on fixed background sono backreaction via Einsteinrsquos equation is considered onlyparticles or fields But the Lagrangian approach extends easilyto include the backreaction we simply add the Hilbert-Einstein Lagrangian to that of the fields or point-like particlesSo now we vary not only the path of point-like particlesor fields in the configuration space but also the geometryof spacetime in order to find the least action history Thisadditional variation gives even more freedom to choose theleast action path so even if on fixed background the initialcondition of a particular field will not evolve to become atthe singularity a field satisfying the conditions from [117]because the spacetime geometry is varied as well to include


backreaction the spacetime adjusts itself to minimize theaction and it is not too wild to conjecture that it adjusts itselfto satisfy such conditions

Now let us consider quantum fields When moving toquantum fields on curved background since the proper timeof all classical test particles is finite we can apply the pathintegral formulation of quantum field theory [119 120] Sincethe proper time is finite along each path 120593 joining two pointsincluding for the paths crossing a singularity and since theaction 119878(120593 119905) is well defined for almost all times 119905 then119890(푖ℏ)푆(휑푡) is also well defined So at least on fixed curvedbackground even with singularities it seems to exist littledifference from special relativistic quantum field theory viapath integrals

Of course the background geometry should also dependon the quantum fields Can we account for this in theabsence of a theory of quantum gravity We know thatat least the framework of path integrals works on curvedclassical spacetime (see eg [121]) where the Einstein equa-tion becomes (1) To also include quantized gravity is moredifficult because of its nonrenormalizability by perturbativemethods Add to this the fact that at least for the StandardModelwe know that in flat background renormalization helpsand even on curved background without singularities Butwhat about singularities Is not it possible that they makerenormalization impossible In fact quite the contrary maybe true in [122] it is shown that singularities improve thebehavior of the quantum fields including for gravity at UVscales These results are applied to already existing resultsobtained by various researchers who use various types ofdimensional reduction to improve this behavior for quantumfields including gravity In fact some of these approachesimprove the renormalizability of quantum fields so well thateven the Landau poles disappear even for nonrenoramlizabletheories [123 124] But the various types of dimensionalreduction are in these approaches postulated somehow adhoc for no other reason than to improve perturbative renor-malizability On the contrary if the perturbative expansion ismade in terms of point-like particles these behave like blackholes with singularities and some of the already postulatedtypes of dimensional reduction emerge automatically withno additional assumption from the properties of singularities[122] Thus the very properties of the singularities leadautomatically to improved behavior at the UV scale even fortheories thought to be perturbatively nonrenormalizable

The proposal I described in this section is still at thebeginning compared to the difficulty of the remainingopen problems to be addressed First there is obviouslyno experimental confirmation and it is hard to imaginethat the close future can provide one The plausibility restsmainly upon making as few new assumptions as possiblein addition to those coming from general relativity andquantum theory theories well established and confirmed butnot in the regimes where both become relevant For somesimple examples there are mathematical results but a trulygeneral proof with fully developed mathematical steps andno gaps does not exist yet And considering the difficulty ofthe problem it is hard to believe that it is easy to have very

soon a completely satisfying proof in this or other approachesNevertheless I think that promising avenues of research areopened by this proposal

Data Availability

Everything is included no additional data is needed it is ahep-th manuscript

Conflicts of Interest

The author declares that there are no conflicts of interest

References

[1] S W Hawking ldquoBreakdown of predictability in gravitationalcollapserdquo Physical Review D Particles Fields Gravitation andCosmology vol 14 no 10 pp 2460ndash2473 1976

[2] W Israel ldquoEvent horizons in static vacuum space-timesrdquo Phys-ical Review A Atomic Molecular and Optical Physics vol 164no 5 pp 1776ndash1779 1967

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

[4] B Carter ldquoAxisymmetric black hole has only two degrees offreedomrdquo Physical Review Letters vol 26 no 6 pp 331ndash3331971

[5] W K Misner S Thorne and J A Wheeler Gravitation W HFreeman and Company 1973

[6] R Penrose ldquoGravitational collapse and space-time singulari-tiesrdquo Physical Review Letters vol 14 pp 57ndash59 1965

[7] S W Hawking and R Penrose ldquoThe singularities of gravita-tional collapse and cosmologyrdquo Proceedings of the Royal Societyof London vol 314 no 1519 pp 529ndash548 1970

[8] S W Hawking and G F R Ellis The Large Scale Structure ofSpace-Time Cambridge University Press 1995

[9] R Penrose ldquoGravitational Collapse the Role ofGeneral Relativ-ityrdquo Revista del Nuovo Cimento Numero speciale 1 pp 252ndash2761969

[10] R Penrose ldquoThe Question of Cosmic Censorshiprdquo in BlackHoles and Relativistic Stars R M Wald Ed pp 233ndash248niversity of Chicago Press Chicago IL USA 1998

[11] S W Hawking ldquoParticle creation by black holesrdquo Communica-tions in Mathematical Physics vol 43 no 3 pp 199ndash220 1975

[12] J D Bekenstein ldquoBlack holes and entropyrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 7 pp 2333ndash2346 1973

[13] D N Page ldquoAverage entropy of a subsystemrdquo Physical ReviewLetters vol 71 no 9 pp 1291ndash1294 1993

[14] D Marolf ldquoThe black hole information problem Past presentand futurerdquo Reports on Progress in Physics vol 80 no 9 2017

[15] W G Unruh and R M Wald ldquoInformation lossrdquo Reports onProgress in Physics vol 80 no 9 p 092002 2017

[16] J Preskill ldquoDo black holes destroy informationrdquo inBlackHolesMembranes Wormholes and Superstrings vol 1 p 22 WorldScientific River Edge NJ USA 1993

[17] S B Giddings ldquoThe black hole information paradoxrdquo 1995httpsarxivorgabshep-th9508151


[18] S Hossenfelder and L Smolin ldquoConservative solutions to theblack hole information problemrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 81 no 6 Article ID064009 13 pages 2010

[19] S W Hawking ldquoThe unpredictability of quantum gravityrdquoCommunications inMathematical Physics vol 87 no 3 pp 395ndash415 198283

[20] S B Giddings ldquoConstraints on black hole remnantsrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 49no 2 pp 947ndash957 1994

[21] S B Giddings ldquoWhy arenrsquot black holes infinitely producedrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 51 no 12 pp 6860ndash6869 1995

[22] M A Markov ldquoProblems of a perpetually oscillating universerdquoAnnals of Physics vol 155 no 2 pp 333ndash357 1984

[23] M K Parikh and FWilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000

[24] S Lloyd ldquoAlmost certain escape from black holes in finalstate projection modelsrdquo Physical Review Letters vol 96 no 6061302 4 pages 2006

[25] V P Frolov ldquoInformation loss problem and a lsquoblack holersquo modelwith a closed apparent horizonrdquo Journal of High Energy Physicsvol 2014 no 5 2014

[26] D P Prester ldquoCuring Black Hole Singularities with Local ScaleInvariancerdquoAdvances inMathematical Physics vol 2016 ArticleID 6095236 9 pages 2016

[27] A Ashtekar V Taveras and M Varadarajan ldquoInformation isnot lost in the evaporation of 2D black holesrdquo Physical ReviewLetters vol 100 no 21 211302 4 pages 2008

[28] A Ashtekar F Pretorius and F M Ramazanoglu ldquoEvaporationof two-dimensional black holesrdquo Physical Review D vol 83 no4 Article ID 044040 2011

[29] C Rovelli and F Vidotto ldquoPlanck starsrdquo International Journal ofModern Physics D vol 23 no 12 Article ID 1442026 2014

[30] HMHaggard andC Rovelli ldquoQuantum-gravity effects outsidethe horizon spark black to white hole tunnelingrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 92no 10 104020 11 pages 2015

[31] L Susskind LThorlacius and J Uglum ldquoThe stretchedhorizonand black hole complementarityrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 48 no 8 pp 3743ndash37611993

[32] C R Stephens G rsquot Hooft and B F Whiting ldquoBlack holeevaporation without information lossrdquo Classical and QuantumGravity vol 11 no 3 pp 621ndash647 1994

[33] S Leonard andL JamesTheholographic universe ndash An introduc-tion to black holes information and the string theory revolutionWorld Scientific 2004

[34] A Almheiri D Marolf J Polchinski and J Sully ldquoBlack holesComplementarity or firewallsrdquo Journal of High Energy Physicsvol 2013 no 2 pp 1ndash19 2013

[35] O C Stoica Singular General Relativity [PhD Thesis]Minkowski Institute Press 2013

[36] R Kubo ldquoStatistical-mechanical theory of irreversible pro-cesses I general theory and simple applications to magneticand conduction problemsrdquo Journal of the Physical Society ofJapan vol 12 no 6 pp 570ndash586 1957

[37] P C Martin and J Schwinger ldquoTheory of many-particlesystems Irdquo Physical Review A Atomic Molecular and OpticalPhysics vol 115 no 6 pp 1342ndash1373 1959

[38] W G Unruh ldquoNotes on black-hole evaporationrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 14no 4 pp 870ndash892 1976

[39] S A Fulling ldquoNonuniqueness of canonical field quantizationin riemannian space-timerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 7 no 10 pp 2850ndash2862 1973

[40] P C Davies ldquoScalar production in Schwarzschild and Rindlermetricsrdquo Journal of Physics A Mathematical and General vol 8no 4 pp 609ndash616 1975

[41] R M Wald Quantum Field Theory in Curved Space-Time andBlack HoleThermodynamics University of Chicago Press 1994

[42] O Stoica ldquoAnalytic ReissnerndashNordstrom singularityrdquo PhysicaScripta vol 85 no 5 p 055004 2012

[43] M Maldacena ldquoThe large-N limit of superconformal fieldtheories and supergravityrdquo International Journal of TheoreticalPhysics vol 38 no 4 pp 1113ndash1133 1999

[44] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 pp 1009ndash1038 1998

[45] S Perlmutter G Aldering and G Goldhaber ldquoMeasurementsofΩ and Λ from 42 High-Redshift SupernovaerdquoThe Astrophys-ical Journal vol 517 no 2 pp 565ndash586 1999

[46] S W Hawking ldquoInformation loss in black holesrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 72Article ID 084013 2005

[47] R H Price and K S Thorne ldquoMembrane viewpoint onblack holes properties and evolution of the stretched horizonrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 33 no 4 pp 915ndash941 1986

[48] J L Park ldquoThe concept of transition in quantum mechanicsrdquoFoundations of Physics vol 1 no 1 pp 23ndash33 1970

[49] W K Wootters and W H Zurek ldquoA single quantum cannot beclonedrdquoNature vol 299 no 5886 pp 802-803 1982

[50] D Dieks ldquoCommunication by EPR devicesrdquo Physics Letters Avol 92 no 6 pp 271-272 1982

[51] J R Oppenheimer and H Snyder ldquoOn continued gravitationalcontractionrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 56 no 5 pp 455ndash459 1939

[52] L S Schulman Timersquos arrows and quantum measurementCambridge University Press 1997

[53] R Bousso ldquoObserver complementarity upholds the equivalenceprinciplerdquo 2012 httpsarxivorgabs12075192

[54] DHarlow and P Hayden ldquoQuantum computation vs firewallsrdquoJournal of High Energy Physics vol 6 no 85 2013

[55] R Bousso ldquoComplementarity is not enoughrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 87 no 122013

[56] J Maldacena and L Susskind ldquoCool horizons for entangledblack holesrdquo Fortschritte der PhysikProgress of Physics vol 61no 9 pp 781ndash811 2013

[57] K L H Bryan and A J M Medved ldquoBlack holes andinformation a new take on an old paradoxrdquo Advances in HighEnergy Physics vol 2017 Article ID 7578462 8 pages 2017

[58] D Stanford and L Susskind ldquoComplexity and shock wavegeometriesrdquoPhysical ReviewD Particles Fields Gravitation andCosmology vol 90 no 12 2014

[59] S Aaronson ldquoThe complexity of quantum states and trans-formations from quantum money to black holesrdquo 2016httpsarxivorgabs160705256


[60] S L Braunstein S Pirandola and K Zyczkowski ldquoBetter latethan never Information retrieval from black holesrdquo PhysicalReview Letters vol 110 no 10 Article ID 101301 2013

[61] A Y Yosifov and L G Filipov ldquoEntropic EntanglementInformation Prison Breakrdquo Advances in High Energy Physicsvol 2017 Article ID 8621513 7 pages 2017

[62] S W Hawking ldquoInformation preservation and weather fore-casting for black holesrdquo 2014 httpsarxivorgabs14015761

[63] SW Hawking ldquoThe information paradox for black holesrdquo TechRep DAMTP-2015-49 2015

[64] S W Hawking M J Perry and A Strominger ldquoSoft Hair onBlack Holesrdquo Physical Review Letters vol 116 no 23 Article ID231301 2016

[65] S W Hawking M J Perry and A Strominger ldquoSuperrotationcharge and supertranslation hair on black holesrdquo Journal of HighEnergy Physics vol 5 p 161 2017

[66] L Susskind ldquoTheworld as a hologramrdquo Journal ofMathematicalPhysics vol 36 no 11 pp 6377ndash6396 1995

[67] K Papadodimas and S Raju ldquoBlack Hole Interior in theHolographic Correspondence and the Information ParadoxrdquoPhysical Review Letters vol 112 no 5 2014

[68] Z-L Wang and Y Yan ldquoBulk Local Operators ConformalDescendants and Radial Quantizationrdquo Advances in HighEnergy Physics vol 2017 Article ID 8185690 11 pages 2017

[69] A Gefter ldquoComplexity on the horizonrdquo Nature 2014[70] R Bousso ldquoFirewalls from double purityrdquo Physical Review D

Particles Fields Gravitation and Cosmology vol 88 no 8 2013[71] S Weinfurtner E W Tedford M C Penrice W G Unruh

and G A Lawrence ldquoMeasurement of Stimulated HawkingEmission in an Analogue Systemrdquo Physical Review Letters vol106 no 2 2011

[72] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973

[73] R Bousso ldquoThe holographic principlerdquo Reviews of ModernPhysics vol 74 no 3 pp 825ndash874 2002

[74] S W Hawking ldquoGravitational radiation from colliding blackholesrdquo Physical Review Letters vol 26 no 21 pp 1344ndash13461971

[75] R B Mann Black Holes Thermodynamics Information AndFirewalls Springer New York NY USA 2015

[76] L Parker ldquoQuantized fields and particle creation in expandinguniverses Irdquo Physical Review A Atomic Molecular and OpticalPhysics vol 183 no 5 pp 1057ndash1068 1969

[77] B P Dolan Where is the pdv term in the first law of black holethermodynamics 2014

[78] R MWald ldquoBlack hole entropy is the Noether chargerdquoPhysicalReview D Particles Fields Gravitation and Cosmology vol 48no 8 pp R3427ndashR3431 1993

[79] T Jacobson ldquoThermodynamics of spacetime the Einsteinequation of staterdquo Physical Review Letters vol 75 p 1260 1995

[80] E Verlinde ldquoOn the origin of gravity and the laws of NewtonrdquoJournal of High Energy Physics vol 4 p 29 2011

[81] G D Birkhoff and R E Langer Relativity and Modern Physicsvol 1 Harvard University Press Cambridge 1923

[82] M Heusler ldquoNo-hair theorems and black holes with hairrdquoHelvetica Physica Acta Physica Theoretica Societatis PhysicaeHelveticae Commentaria Publica vol 69 no 4 pp 501ndash5281996

[83] N E Mavromatos ldquoEluding the no-hair conjecture for blackholesrdquo 1996 httpsarxivorgabsgr-qc9606008

[84] W H Zurek and K S Thorne ldquoStatistical mechanical origin ofthe entropy of a rotating charged black holerdquo Physical ReviewLetters vol 54 no 20 pp 2171ndash2175 1985

[85] G rsquot Hooft ldquoOn the quantum structure of a black holerdquoNuclearPhysics B vol 256 no 4 pp 727ndash745 1985

[86] R B Mann L Tarasov and A Zelnikov ldquoBrick walls for blackholesrdquo Classical and Quantum Gravity vol 9 no 6 pp 1487ndash1494 1992

[87] S Carlip ldquoEntropy from conformal field theory at Killinghorizonsrdquo Classical and Quantum Gravity vol 16 no 10 pp3327ndash3348 1999

[88] A Strominger and C Vafa ldquoMicroscopic origin of theBekenstein-Hawking entropyrdquo Physics Letters B vol 379 no1ndash4 pp 99ndash104 1996

[89] G T Horowitz and A Strominger ldquoCounting States of Near-Extremal Black Holesrdquo Physical Review Letters vol 77 no 12pp 2368ndash2371 1996

[90] A Dabholkar ldquoExact counting of supersymmetric black holemicrostatesrdquo Physical Review Letters vol 94 no 24 241301 4pages 2005

[91] A Ashtekar J Baez A Corichi and K Krasnov ldquoQuantumgeometry and black hole entropyrdquo Physical Review Letters vol80 no 5 pp 904ndash907 1998

[92] L Bombelli R K Koul J Lee and R D Sorkin ldquoQuantumsource of entropy for black holesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 34 no 2 pp 373ndash3831986

[93] M Srednicki ldquoEntropy and areardquo Physical Review Letters vol71 no 5 pp 666ndash669 1993

[94] V Frolov and I Novikov ldquoDynamical origin of the entropy of ablack holerdquo Physical Review D Particles Fields Gravitation andCosmology vol 48 no 10 pp 4545ndash4551 1993

[95] C Corda ldquoEffective temperature hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 2012

[96] C Corda ldquoBlack hole quantum spectrumrdquo The EuropeanPhysical Journal C vol 73 p 2665 2013

[97] C Corda ldquoBohr-like model for black-holesrdquo Classical andQuantum Gravity vol 32 no 19 article 5007 2015

[98] G Dvali and C Gomez ldquoQuantum compositeness of gravityblack holes AdS and inflationrdquo Journal of Cosmology andAstroparticle Physics no 1 023 front matter+46 pages 2014

[99] R Casadio A Giugno OMicu and A Orlandi ldquoBlack holes asself-sustained quantum states and Hawking radiationrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 90no 8 2014

[100] R Casadio A Giugno OMicu and A Orlandi ldquoThermal BECblack holesrdquo Entropy vol 17 no 10 pp 6893ndash6924 2015

[101] J D Bekenstein ldquoHow does the entropyinformation boundworkrdquo Foundations of Physics An International Journal Devotedto the Conceptual Bases and Fundamental Theories of ModernPhysics vol 35 no 11 pp 1805ndash1823 2005

[102] S Schaffer ldquoJohn michell and black holesrdquo Journal for theHistory of Astronomy vol 10 no 1 pp 42-43 1979

[103] C Rovelli ldquoBlack holes have more states than those givingthe Bekenstein-Hawking entropy a simple argumentrdquo 2017httpsarxivorgabs171000218


[104] E Bianchi M Christodoulou F DrsquoAmbrosio H M Haggardand C Rovelli ldquoWhite holes as remnants A surprising scenariofor the end of a black holerdquo 2018 httpsarxivorgabs180204264

[105] A Ashtekar Lectures on Non-Perturbative Canonical GravityWorld Scientific Singapore 1991

[106] G Yoneda H-a Shinkai and A Nakamichi ldquoTrick for passingdegenerate points in the Ashtekar formulationrdquo Physical ReviewD Particles Fields Gravitation andCosmology vol 56 no 4 pp2086ndash2093 1997

[107] O C Stoica ldquoOn singular semi-Riemannian manifoldsrdquo Inter-national Journal of Geometric Methods in Modern Physics vol11 no 5 1450041 40 pages 2014

[108] O C Stoica ldquoThe geometry of warped product singularitiesrdquoInternational Journal of Geometric Methods in Modern Physicsvol 14 no 2 1750024 16 pages 2017

[109] O C Stoica ldquoThe Friedmann-Lemaıtre-Robertson-Walker BigBang Singularities are Well Behavedrdquo International Journal ofTheoretical Physics vol 55 no 1 pp 71ndash80 2016

[110] O C Stoica ldquoBeyond the Friedmann-Lemaıtre-Robertson-Walker Big Bang singularityrdquo Communications in TheoreticalPhysics vol 58 pp 613ndash616 2012

[111] O C Stoica ldquoSchwarzschild singularity is semi-regularizablerdquoTheEuropeanPhysical Journal Plus vol 127 no 83 pp 1ndash8 2012

[112] O C Stoica ldquoKerr-Newman solutions with analytic singularityand no closed timelike curvesrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 77 no 1 pp 129ndash138 2015

[113] O C Stoica ldquoSpacetimes with singularitiesrdquo Analele stiintificeale Universitatii Ovidius Constanta vol 20 no 2 pp 213ndash2382012

[114] O C Stoica ldquoThe geometry of singularities and the black holeinformation paradoxrdquo Journal of Physics Conference Series vol626 Article ID 012028 2015

[115] A S Eddington ldquoA Comparison of Whiteheadrsquos and EinsteinrsquosFormulaeligrdquo Nature vol 113 no 2832 p 192 1924

[116] D Finkelstein ldquoPast-future asymmetry of the gravitational fieldof a point particlerdquo Physical Review Journals Archive vol 110 p965 1958

[117] O C Stoica ldquoGauge theory at singularitiesrdquo 2014 httpsarxivorgabs14083812

[118] O C Stoica ldquoCausal structure and spacetime singularitiesrdquo2015 httpsarxivorgabs150407110

[119] PAM Dirac ldquoThe Lagrangian in quantum mechanicsrdquoPhysikalische Zeitschrift der Sowjetunion vol 1 no 3 1933

[120] R P Feynman and A R Hibbs Quantum Mechanics and PathIntegrals Emended Edition Dover Publications Incorporated2012

[121] HKleinertPath integrals in quantummechanics statistics poly-mer physics and financial markets World Scientific Singapore2009

[122] O C Stoica ldquoMetric dimensional reduction at singularitieswithimplications to quantum gravityrdquoAnnals of Physics vol 347 pp74ndash91 2014

[123] P P Fiziev and D V Shirkov ldquoSolutions of the Klein-Gordonequation on manifolds with variable geometry includingdimensional reductionrdquo Theoretical and Mathematical Physicsvol 167 no 2 pp 680ndash691 2011

[124] D V Shirkov ldquoDream-land with Classic Higgs field Dimen-sional Reduction and all thatrdquo in Proceedings of the SteklovInstitute of Mathematics vol 272 pp 216ndash222 2011

Hindawiwwwhindawicom Volume 2018

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too small comparable to the Plank scale and the usual anal-ysis of Hawking radiation no longer applies [19ndash21] Thereare other possibilities some being discussed in the above-mentioned reviews For example it was proposed that theinformation leaks out of the black hole through evaporationincluding by quantum tunneling that it escapes at the finalexplosion or that it leaks out of the universe in a babyuniverse [22 23] Another possibility is that the informationescapes as Hawking radiation by quantum teleportation [24]which actually happens as if the particle zig-zags forward andbackward in time to escape without exceeding the speed oflight This is not so unnatural if we assume that the finalboundary condition at the future singularity of the black holeforces the maximally entangled particles to be in a singletstate There are also bounce scenarios [25] or by using localscale invariance to avoid singularities [26] Some bouncescenarios are based on loop quantum gravity like [27 28]as well as black hole to white hole tunneling scenarios inwhich quantum tunneling is supposed to break the Einsteinequation and the apparent horizon is prevented to evolve intoan event horizon [29 30] It would take a long review to dojustice to the various proposals and this is beyond the scopeof this article

The dominating proposed solution was for two decadesblack hole complementarity [31ndash33] This was later challengedby the firewall paradox [34] The debate is not settled downyet but the dominant opinion seems to be that we have togive up at least one principle considered fundamental so farand the unlucky one ismost likely the principle of equivalencefrom general relativity One of the objectives of the presentarticle is to show that we can avoid this radical solution whilekeeping unitarity

The problems related to the black hole information lossare considered important being seen as a benchmark for thecandidate theories of quantum gravity which are expected tosolve these problems

The main purpose of this discussion is to identify themain assumptions and see if it is possible to solve theproblem in a less radical way I argue that some of the usuallymade assumptions are unnecessary that there are less radicalpossibilities and that the black hole information problemis not a decisive test for candidate theories of quantumgravity New counterarguments to some popular modelsproposed in relation to the black hole information problemare the following Black hole complementarity is discussedin Section 3 in particular the fact that an argument bySusskind aiming to prove that no-cloning is satisfied bythe black hole complementarity does not apply to mostblack holes (Section 31) the fact that its main argumentthe ldquono-omnisciencerdquo proposal does not really hold forblack holes in general (Section 33) and the fact that blackhole complementarity is also at odds with the principle ofequivalence (Section 32) As for the firewall proposal inSection 41 I explain why the tacit assumption that unitarityshould apply only to the exterior of the black hole and that weshould ignore the interior is not justified and anyway if takenas true it imposes boundary conditions to the field which iswhy the firewall seems to emerge Section 5 is dedicated toblack hole entropy In Section 51 I present an argument based

on time symmetry that the true entropy is not necessarilyproportional to the area of the event horizon and at best inthe usual cases is bounded This has negative implications tothe various proposals that the event horizon would containsome bits representing the microstates of the black holediscussed in Section 54 This may also explain the so-calledldquoreal black hole information paradoxrdquo discussed in Section 6Section 52 contains an explanation of the fact that if thelaws of black hole mechanics should be connected with thoseof thermodynamics this happens already at the classicallevel so they are not necessarily indications of quantumgravity or tests of such approaches Section 53 containsarguments that one should not read too much in the so-called no-hair theorems in particular they do not constraincontrary to a widespread belief neither the horizon nor theinterior of a black hole A major motivation invoked for thetheoretical research of the black hole information and entropyis that these may provide a benchmark to test approachesto quantum gravity but in Section 55 I argue that thesefeatures appear merely by considering quantum fields onspacetime Consequently any approach to quantum gravitywhich includes both quantum field theory and the curvedspacetime of general relativity as aminimal requirement willalso satisfy the consequences derived from them

To my knowledge the above-mentioned arguments pre-sented in more detail in the following are new and inthe cases when I was aware of other results seeming topoint in the same direction I gave the relevant referencesWhile most part of the article may look like a review of theliterature it is a critical review aiming to point out someassumptions which in my opinion drove us too far fromthe starting point which is just the most straightforwardand conservative combination of quantum field theory withthe curved background of general relativity The entirestructure of arguments converges therefore towards a moreconservative picture than that suggested by the more popularproposals The counterarguments are meant to build up thewillingness to consider the less radical proposal that I madewhich follows naturally from my work on singularities instandard general relativity ([35] and references therein) andis discussed in Section 7The background theory is presentedin Section 71 and a new enhanced version of the proposal ismade in Section 72

2 Black Hole Evaporation

Hawkingrsquos derivation of the black hole evaporation [1 11]has been disputed and checked many times and redone indifferent settings and it turned valid at most allowing someimprovements of the unavoidable approximations as well asmild generalizations But the result is correct the radiationis as predicted and thermal in the Kubo-Martin-Schwingersense [36 37] Moreover it is corroborated via the principleof equivalence with the Unruh radiation which takes placein the Minkowski spacetime for accelerated observers [38]Hawkingrsquos derivation is obtained in the framework of quan-tum field theory on curved spacetime but since the blackhole is considered large and the time scale is also large thespacetime curvature induced by the radiation is ignored



119877푎푏 minus 12119877119892푎푏 + Λ119892푎푏 =

81205871198661198884 ⟨푎푏 (119909)⟩ (1)








0(120572휔]119886] + 120573휔]119886dagger] ) d] (2)



















(a) (b)
























119878퐵퐻 = 119896퐵1198604ℓ2푃 (3)


























119901푖 ln119901푖 (4)


119878 = minus119896퐵tr (120588 ln 120588) (5)






































119903 )minus1

d1199032 + 1199032d1205902 (8)


d1205902 = d1205792 + sin2120579d1206012 (9)





119903 = 1205912

119905 = 1205851205914(10)


d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)













(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







119877푎푏 minus 12119877119892푎푏 + Λ119892푎푏 =

81205871198661198884 ⟨푎푏 (119909)⟩ (1)








0(120572휔]119886] + 120573휔]119886dagger] ) d] (2)



















(a) (b)
























119878퐵퐻 = 119896퐵1198604ℓ2푃 (3)


























119901푖 ln119901푖 (4)


119878 = minus119896퐵tr (120588 ln 120588) (5)






































119903 )minus1

d1199032 + 1199032d1205902 (8)


d1205902 = d1205792 + sin2120579d1206012 (9)





119903 = 1205912

119905 = 1205851205914(10)


d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)













(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery
















(a) (b)
























119878퐵퐻 = 119896퐵1198604ℓ2푃 (3)


























119901푖 ln119901푖 (4)


119878 = minus119896퐵tr (120588 ln 120588) (5)






































119903 )minus1

d1199032 + 1199032d1205902 (8)


d1205902 = d1205792 + sin2120579d1206012 (9)





119903 = 1205912

119905 = 1205851205914(10)


d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)













(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






(a) (b)
























119878퐵퐻 = 119896퐵1198604ℓ2푃 (3)


























119901푖 ln119901푖 (4)


119878 = minus119896퐵tr (120588 ln 120588) (5)






































119903 )minus1

d1199032 + 1199032d1205902 (8)


d1205902 = d1205792 + sin2120579d1206012 (9)





119903 = 1205912

119905 = 1205851205914(10)


d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)













(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery


















119878퐵퐻 = 119896퐵1198604ℓ2푃 (3)


























119901푖 ln119901푖 (4)


119878 = minus119896퐵tr (120588 ln 120588) (5)






































119903 )minus1

d1199032 + 1199032d1205902 (8)


d1205902 = d1205792 + sin2120579d1206012 (9)





119903 = 1205912

119905 = 1205851205914(10)


d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)













(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery



























119901푖 ln119901푖 (4)


119878 = minus119896퐵tr (120588 ln 120588) (5)






































119903 )minus1

d1199032 + 1199032d1205902 (8)


d1205902 = d1205792 + sin2120579d1206012 (9)





119903 = 1205912

119905 = 1205851205914(10)


d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)













(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery










119901푖 ln119901푖 (4)


119878 = minus119896퐵tr (120588 ln 120588) (5)






































119903 )minus1

d1199032 + 1199032d1205902 (8)


d1205902 = d1205792 + sin2120579d1206012 (9)





119903 = 1205912

119905 = 1205851205914(10)


d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)













(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery




























119903 )minus1

d1199032 + 1199032d1205902 (8)


d1205902 = d1205792 + sin2120579d1206012 (9)





119903 = 1205912

119905 = 1205851205914(10)


d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)













(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery
















119903 )minus1

d1199032 + 1199032d1205902 (8)


d1205902 = d1205792 + sin2120579d1206012 (9)





119903 = 1205912

119905 = 1205851205914(10)


d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)













(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







119903 = 1205912

119905 = 1205851205914(10)


d1199042 = minus 412059142119898 minus 1205912 d120591

2 + (2119898 minus 1205912) 1205914 (4120585d120591 + 120591d120585)2

+ 1205914d1205902(11)













(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






(a)

(b)















Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery











Data Availability




References



































































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






















































































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery











































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






























Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





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