On the Viability of Regular Black Holes JHEP07(2018)023 ]— 7

Home , Hayward metric

JHEP07(2018)023 Springer July 4, 2018 May 16, 2018 : June 25, 2018 , : : Published Received a,b Accepted Published for SISSA by https://doi.org/10.1007/JHEP07(2018)023 Stefano Liberati, a,b [email protected] [email protected] , , c Francesco Di Filippo, [email protected] . 3 and Matt Visser , a,b a,b 1805.02675 The Authors. c Black Holes, Models of Quantum Gravity, Spacetime Singularities

The evaporation of black holes raises a number of conceptual issues, most of , [email protected] Via Valerio 2, 34127 Trieste,School Italy of Mathematics andPO Statistics, Box Victoria 600, University Wellington of 6140, Wellington, E-mail: New Zealand [email protected] SISSA, International School forVia Advanced Studies, Bonomea 265, 34136 Trieste,INFN Italy Sezione di Trieste, b c a Open Access Article funded by SCOAP ArXiv ePrint: also discuss how toat the overcome price these of difficulties,stages accepting of highlighting that evaporation. that these this models cannot can be be fully doneKeywords: predictive only regarding the final geometric description is available. It hasbut been complete, argued description that these of models the providedisappearance. evaporation an of effective, However, black here we holes point at outregular all that times core known models up is fail to exponentially to theirthe be unstable eventual self-consistent: evaporation against time the perturbations is infinite, with therefore a making finite the timescale, instability impossible while to prevent. We Abstract: them related to thesingularity and final Hawking radiation stages cannot of bethe ignored. evaporation, central Regular where singularity models the with of black a interplay holes nonsingular replace between spacetime the region, central in which an effective classical RaúlCarballo-Rubio, Costantino Pacilio On the viability of regular black holes JHEP07(2018)023 ]— 7 – 3 ]. Regular black hole 12 , 11 13 7 8 10 ], being capable to provide at least an effective 4 – 1 – 10 – 8 4 12 10 7 ]. It is only in the last stages of the evaporation that the 2 , 2 1 1 14 4.3.1 Adiabatic4.3.2 condition Quasi-static condition The nonsingular core in these geometries replaces a region in which the spacetime In this sense, a rather conservative approach is to take seriously the hypothesis that 4.2 Computation of4.3 the evaporation time Adiabatic and quasi-static conditions 4.4 Beyond the quasi-static or adiabatic regimes 3.1 Mass inflation at the inner horizon 4.1 Problem setup and working assumptions models are in this senseclassical a black hole useful geometries test-bed and for understandingof exploring if these the minimal, new localized, classical aspects departures and can from lead quantum to questions resolution mentioned above. curvature takes Planckian values or larger, and therefore in which the classical dynamics general relativity and quantumor mechanics simply — in as determining in the the final information fate loss of black problemany holes. successful [ theory of quantum gravitybe should singular be regions able in to general relativityand remain [ predictive nonsingular at geometric the description would- of the spacetime there [ description of physics outside blackevolution, even holes. in This a semiclassical remainsof realm true, where for Hawking black most holes radiation do ofexistence evaporate [ their due of dynamical to the singularities emission cannotcrucial role be in ignored. determining the However, answer these to very important last questions about stages the do compatibility play of a 1 Introduction Black hole spacetimestheory in ceases general to relativity beHowever, contain predictive these singularities, singularities and are i.e., no hidden notion regions by event of where horizons, classical therefore the spacetime being seems harmless for to the be possible. 5 Conclusion 3 Instability of the inner horizon 4 Infinite evaporation time Contents 1 Introduction 2 Geometric setting JHEP07(2018)023 3 ]. r ) = 27 r – (2.1) (2.2) ( 0 25 m , where 2 , ) πr ]. 2 4 / 20 φ ) d ) vanishes at least as r θ ( r 0 ( 2 m m . we show that the inner core 5 + sin 3 2 θ proves that the evaporation time 4 (d . 2 ) r r ( r + we describe in generality the geometries ) is given by m ) 2 2 2 ] r 2.1 r ( − d 24 F = 0 if and only if – – 2 – r (0) = 1. On the other hand, asymptotic flatness + 17 2 F ], regular black holes provide an in principle com- , ) = 1 t r 11 ( )d 16 , r ]). In principle, it is not guaranteed that a complete F ( 11 15 F . When convenient, we will alternatively use the notation ) r 4 ( φ 2 − = 0 requires e r − = 2 ) = 1. It follows (counting the multiplicity of the roots) that the func- ) are two real functions. We will introduce the time-dependent geome- s r r ], by a region in which curvature is bounded (this is usually known as ( d ) corresponds to the Misner-Sharp-Hernandez quasi-local mass [ ( 0. Under the dominant energy condition, the regularity of the effective r F ( F 14 , → m →∞ 13 r r ) and . This quantity can be finite at r r ) must have an even number of zeros, with these zeros corresponding to different ( r d φ ( / ) F Hence, regularity at Regular black holes provide therefore a framework in which long-standing problems As discussed for instance in [ r ( m in the limit energy density also implies the regularity of the effectiveenforces pressures lim [ tion The function Using the Einstein fielddensity associated equations, with it the isd geometry straightforward in to eq. see ( that the effective energy where tries at the beginning of section 2 Geometric setting Without loss ofregular generality, black locally hole can the be geometry written as of [ a static and spherically symmetric complete and self-contained models.that In we section will beof using regular in black the holesin rest is these of necessarily models the unstable. is paper. genericallycan Section infinite be In and evaded. discusses section Our under conclusions which are conditions this summarized conclusion in section can be analyzed withoutfollows introducing we dramatic shall deviations analyze from in moreto standard details two physics. this crucial class In aspects of what that geometries,oration paying were time, particular so attention discussing far their overlooked in implication the on literature, the instabilities interpretation and of evap- regular black holes as This claim is motivated by twoimplies observations: the (i) the existence regularization of of bothtion, the central are an singularity naturally outer characterized and by ancloser a inner dynamical until horizon evolution, the wherein which, they due blackblack become to hole hole gradually Hawking is is radia- regularized, extremal,the there subsequent and is disappearance (ii) no of as the manifest two obstruction the horizons to singular in describing a core the regular of manner. merging the and evaporating the limiting curvature principlequantum [ gravity theoryworking allows assumption. for such an effective description,plete so and self-consistent that picture for this the evaporation is of a just black hole a up to its disappearance. is modified [ JHEP07(2018)023 ). r ( (2.7) (2.8) (2.3) (2.4) (2.5) (2.6) F (2.12) (2.10) (2.11) ) r ( φ − r/e d . ) − 2 t φ d ); (2.9) = d θ 2 u r − . ) , r ) + sin − ( + r o 2 r . θ ) − − r (d ) + r r 2 ( r = , ( r r o r

− o ) 0 . r + r − ( ) ( ) + v . ) r r o ) + F ) ( − r ( d r , ) is given by + ( r r r d r F ( r , r d ) r 2 ) + φ m ( ) − r r = ( r 2 − − ( r r m φ ) = 0 e r = 0

φ ( r 0 = − , ( | − v – 3 – ) e − | r v + r e r − d ( = 2 ( ) + t r κ φ r 2 = F κ ( | ) reads − )( + 2 r r −| r v e = d m 2 d d = r 2.1 2 v = − v ( ) 0 0. As a consequence, at the outer horizon we have an d ) )d + = F ) around ) r − r ) r ( < r ( v v φ − 2 2.8 F r d κ − − ( ) d − 2 r r φ κ e ( r − φ d( ) = 0, these expressions reduce to the standard ones in the 2 e − d( r − ( e = = φ − 0 and r v d d = > and 2 s + d κ M , are then defined by − ) = r r ( m Ingoing and outgoing radial null curves are determined respectively by the equations and + while at the inner horizon we have an exponential focusing of outgoing null rays, In particular, exponential peeling of outgoing null rays, we see that the surface gravities at the outer and inner horizons are given by The Taylor expansion of eq. ( Schwarzschild geometry. and in terms of which the line element ( The outgoing Eddington-Finkelstein coordinate is insteadFor given by d When studying the causalingoing Eddington-Finkelstein properties coordinate of the geometry, it is convenient to introduce the our results are independent ofr this assumption. The outer and inner horizons, respectively or equivalently by horizons. In the following we assume for simplicity that there are only two horizons, but all JHEP07(2018)023 ] ] – 38 32 43 , ]. 42 31 , 30 ]). ]. 45 , 40 44 ], and also on the rotating case [ 29 ], where outgoing and ingoing perturbations 36 – 4 – ] that the introduction of a cosmological constant may cure 28 ], the presence of the two horizons provides in principle 16 , 11 ), the main aspects were settled more than two decades ago [ 1 ] 31 – 28 ]. However, here we see that it can be applied in complete generality to all 41 ] for a review). In brief terms, the central conclusion of these works is that the 39 The process can be modelled following [ Let us start by explaining why it is natural to expect the existence of outgoing and In particular, it has been shown in [ 1 ], though formal proofs of a number of technical aspects were only available later [ of what would generallyphysical be features. continuous On streams, the other and hand, therefore the should approximations preserve ofsuch the spherical instability in symmetry main charged and blackpresent holes. null once However, one subsequent takes works into have account proved charged that perturbations this [ instability is still 3.1 Mass inflationHere, at we the discuss inner the horizon phenomenongeometries. of mass In inflation this indescription discussion, the of these framework we perturbations. of take This regular spherically shell black model symmetric hole can be null understood shells as the as discretization an idealized black hole” [ regular black hole geometries. are described in termscalculations analytically, of and null exploits the shells. Dray-’t(which Hooft-Redmount is This (DTR) also simplification relation useful [ allows to to describe the perform unstable all nature necessary of white holes [ initially outgoing radiation ishand, strongly there lensed will be back ingoingof to perturbations gravitational the that radiation. collapsing come, Interestingly, star). for thisclassical instance, argument Reissner-Nordström(-deSitter) On or from has the the Kerr(-de been backscattering Sitter) other previouslywork black applied of holes to regular but, the black in the holes, frame- it has been analyzed only for the specific case of the “loop is natural to expect the existence of semiclassical instabilitiesingoing as perturbations well in [ realistiction scenarios. of A an regular ordinary blackstar. hole, black The being hole, star just would would a form emit regulariza- radiation after even the after gravitational crossing collapse the of outer a horizon massive (after this, the the recent works [ 37 (see also [ inner horizon is unstable in theare presence expected of to both ingoing exist and in outgoing realistic perturbations, which collapse scenarios. While this is a classical instability, it 3 Instability of theThe inner potentially horizon unstable nature ofrays the was inner previously horizon noticed, due to andcharged the thoroughly and exponential studied, rotating focusing in black of holes. the null different While but still being related an context active of research area (see, for instance, a complete and self-consistentradiation picture is of taken black into holethey account, evaporation. merge the into Indeed, outer eachdisappear, when other: and leaving Hawking inner a at nonsingular horizons this remnantgeneral, get with stage this a closer the picture finite and black is nonzero hole too closer mass. simplistic is until We and shall extremal. a see that, more The in careful horizons analysis then is required. As discussed for instance in [ JHEP07(2018)023 ) ), of 3.1 rr 2.1 (3.1) g between , so that 0 v r are depicted. D this parameter ] makes precise. C , the ingoing and and 43 0 , r C , 42 for a given moment of ) with different geome- B , 0 D r A 0 (this value is arbitrary but i . and 0 | u ) + − ). In fact, we are interested in 0 C r , = ( I I B u, v D u , F ) A 0 r ( , respectively), in order to satisfy certain C in

+ . | − r 0 r m

= = = = ]. r – 5 – | C r u ) | and 0 ) A 46 B r v , ( ( ] D out 0 B 36 r 36 F m ) 0

r −

( r

A =

F r | ] for the full diagram). 21 . The ingoing and outgoing shells meet at the radius . Schematic Penrose diagram of a star collapsing to a regular black hole with concentric 1 These relations are independent of the field equations and are formulated purely on If we focus on a local neighbourhood around the crossing point The situation we are discussing is schematically represented in the Penrose diagram this relation takes the simple form [ That is, in sphericallythe symmetric metric situations, in the the constraint four above on spacetime the regions coefficient at the crossing point must be satisfied. Eq. ( to expect that theof mass the in outgoing these and four ingoingconservation regions laws. shells will This ( physical be intuition related is between what the them DTR and relationgeometric the [ grounds. mass In particular, for a spherically symmetric geometry of the form ( outgoing shells divide the spacetimetries in and, four in regions particular, ( differentcorresponds values to of the the Bondi mass mass, parametercorresponding while (in regular region in black the hole others geometry. regions it From is a defined physical by perspective, analogy) of it the is reasonable time. Alternatively, we canunderstanding the use behavior the of the nulloutgoing system coordinates when curve; ( this that crossing is, pointfor is we the displaced will along condition take a that null athe it constant crossing lies value point inside describes the a outer curve horizon), and modify the value of character are known togeneralized to be include inconsequential, these aspects in [ the senseof that figure the DTR relation can be Figure 1 outgoing and ingoing nulloutgoing and shells. ingoing shells. The TheWe only DTR corresponding depict four relation the spacetime relevant is regions quadrants ofhole applied the (see, maximally to e.g., extended Penrose the [ diagram crossing of a point regular black JHEP07(2018)023 , | − and κ (3.4) (3.5) (3.6) (3.2) (3.3) | in 2 , and is − m ). v 0 = r ( − B greater than, κ m r , ) − is constant along ) applies so that, 0 ) ) = 2 r 0 ( increasing the value − r 2.8 out 0. This implies that, ) r ( in 0 0 ( . 0 r m ), one has D r m ) | → ( ) F − m − 0 ) B ) r r κ 0 − F | ( r r 0 / ( − ( ) = r φ 1 r out B 0 . − ( r v F , e . | ( o m ) have a clear physical meaning: . This timescale is Planckian for γ v 2 − | v − v κ + out − 3.2 v − κ ) (note that ) −| | m 0 ) e e − ∝ u 0 γ r r = ) ∝ − ( 0 u − | v ) ) are defined. On the other hand, u ) ). Note that the proportionality constant 0 r = out v u ∝ u 1 ( )( | = m u, v 0 ) ) r u − 0 r | v r ( u ) ( ( – 6 – 2d ), and also 0 v 0 = B ) + 0 ( r u 0 F F | r 0 ( r ) ) ( r ( ] to be given by a power law, namely in − v ( B r in ( ( B 51 m 0 m φ m – r F ( − ) and − e 0 47 A ) u . Taking into account that ) + 0 m = 0 = − r u r r ( | ( v ) C d B v ( m m 0 r ( : this quantity describes how the ingoing tails decay with ) = in ) = 0 : this quantity vanishes on the inner horizon, but we need to determine grows exponentially, on a timescale determined by the surface gravity in 0 r m r B ( ( m A F in 0. A of measured in the ingoing null coordinate | m m v − γ > κ | / gets closer to the location of the inner horizon, along outgoing null curves. Along these trajectories eq. ( 0 v u ): = We just need tobut integrate arbitrarily this close equation to, startingit from follows some that value of Behavior of how fast it approaches thisof value when we displaceclose the to point the inner horizon, one has determined by Price’s law [ where, for the purposesbound of the present discussion, it is enough to consider the lower Behavior of 0 u are the mass of the ingoing and outgoing shells. These three contributions are finite, | u The first three terms on the right-hand side of eq. ( measures the mass of the region between the ingoing and outgoing shell and, therefore, ) v a) b) ( B out = 0 This is the equation thateter characterizes the in phenomenon the of region massof inflation: the the mass inner param- horizon.timescale 1 In other words,most the of the inner models horizon in is the unstable literature with (see a table characteristic Combining these ingredients, we see that at late times ( in order to understand thebehavior evolution of with the system at lateu times, we need to understand the m the original mass ofmoreover the the region in regular which black them coordinates hole ( before thebut ingoing the shell last is contribution absorbed.r has This to is be analyzed carefully. The reason is that, as the point where we have can be manipulated in order to obtain JHEP07(2018)023 C and (4.1) (4.2) (4.3) + . How- κ A , is always . ) A M 2 φ m d . Hence, from v θ 2 . This can be seen . , + sin M ) 2 v θ ) and taking into account ( 4.3 2 + (d r 3.2 2 ) r v ( ) can be directly translated into + 4 + v 3.6 d ). The reason is that Cκ r d − 3.6 . )) is the Stefan-Boltzman constant and v ( + π ) = 2 κ ], where the evaporation is shown to happen in v SB r,M ( 52 σ ]. ( . = + φ 53 1 − T A is directly the physical mass in region ) e – 7 – v A ( 4 m + 2 T 2 v SB σ In order to address the problem, we start from the most ))d ). v − 0 2 ( r ( The evaporation is a quasi-static process, meaning that the = B and, as a consequence, eq. ( ) The only relevant dynamical process during the evaporation F r, M v ( ( v M d F M )) and d v ( in m r,M ( , φ 2 out − e m ) is the area of the outer horizon, − v ) is positive, which can be realized by recalling eq. ( ( = + 3.6 2 A s Quasi-static condition. black hole passes continuously through a sequence of equilibrium states. Adiabatic condition. is Hawking’s radiation which,temperature at given each by moment of the evaporation, is thermal with d is generically given by a positive numerical coefficient times According to our hypothesis, the mass loss rate is determined by the Stefan-Boltzmann In the Reissner-Nordströmmetric, The only exception being the “loop black hole” [ have been promoted to dynamical functions of the evaporation time 2. 1. 2 A + now on we will be dealing implicitly with geometries of the form an infinite time, leavingsolutions no in remnant. the presence Infinite of evaporation higher-curvature times terms [ have also been found in specific singular where is a positive constant,r whose precise value is for the moment irrelevant. Here We will comment on the validity of these assumptionslaw, in section would be (in thelarity). classical case, To our the knowledge, internalits the horizon finiteness evaporation becomes is time always an has assumed. effectiveconservative not viewpoint, spacelike been which singu- can studied be systematically, but summarized in the following two assumptions: 4.1 Problem setupThe and instability working assumptions discussed previouslyevaporation raises time a is longer potential thanperturbations inconsistency the on of typical the timescale the regular of model. blackthe the hole instability, geometry geometry If the dramatically; increases backreaction the exponentially, of it hence is modifying not clear what the outcome of this dynamical evolution the unbounded growth of thism parameter. In particular,explicitly on in the the inner examples horizon discussed the in function table 4 Infinite evaporation time the signs of ever, in the geometries wealthough are this studying does in this not paper,proportional change to this the the would meaning mass not of be generally eq. the ( case, in eq. ( JHEP07(2018)023 ) v = 4.2 (4.9) (4.8) (4.4) (4.6) , and M ? , M ? . At the M to M ? ). In order to M M ) diverges (this ? 4.4 + ∆ ,M ? from ? 0. These cannot be r M ( M M φ M ∆ 1 (4.7) , ) σ > ? ∆ as functions of σ ]). This would be the worst r ,M ) from : ( + ? 54 ∆ ? r o , r

and ? 4.2 < M r ,M β 0 ) + = ? and r ∂F σ r ∂ M

and ) + β . = 0 (4.5) + κ ? F , r ? r 2 r ) r, M ∂ ... ∆ = ∂r (1 + ( ) in order to emphasize the dependence of ∂r∂M r ) = 0 ?

? + ) + 2 ∂F ,M ? M ? (1 + ,M r, M 2 – 8 – r ) ( ? M r

+ = r ) around r F r ∆ ∆ r, M ∂r ( ( are related to each other by eq. ( 2 ( ∂ r ? φ ? = ∂F M 4.4 F ). This in turn means that the evaporation rate ( − , so that the condition + r ,M ,M ? e r ∂F ? ? r r ) + + ∆ ) and M

4.4 = ? ? 2 + ∆ 2 F , and also that these functions can depend on the time F = and ? 2 2 with two real constants M ,M r κ ∂r r, M ∂ M ∂ ). ? ∂M ( M = r M 1 2 v = φ 1 2 ( ( M ) requires knowledge of + F + + M M r and 4.2 of the extremal black hole vanishes, unless ? r 0 = ? κ = + ? r r . Our strategy then consists of integrating eq. ( M Consider a configuration in which the outer horizon radius is just an arbitrarily small Integration of eq. ( away from + ∆ ? r where we parametrizedarbitrarily ∆ chosen, because find their relation, let us expand eq. ( ∆ and correspondingly the mass has an arbitrarily small deviation ∆ 4.2 Computation ofWe the start by evaporation assuming time that theup evaporation has to proceeded adiabatically a andM quasi-statically point at whichshowing the that the mass corresponding is time interval arbitrarily is close infinite. to the extremal value the case for near-extremal Reissner-Nordströmblackpossible holes scenario [ given that,timescale no will matter be how trivially slowcondition smaller the under than which mass the this inflation evaporation happens. time. instability grows, Let its us therefore explore the the surface gravity specific case is analyzed invanishes section at the endclue of that the the evaporation evaporation time process. may be infinite An (in infinitely fact, this slow has evaporation been rate shown to is be a precisely At extremality must hold. Since the surface gravity is these functions on theonly mass implicitly, through outer horizon we have We are using the notation JHEP07(2018)023 ) and is of 4.15 α (4.16) (4.10) (4.14) (4.15) (4.11) (4.12) (4.13) . Given M n 3 , while the ? − ) and ( M = 4 4.14 γ must be a positive . The leading term ), we can integrate = 0 and, therefore, and 4 , + ), ( ? γ r j ? | − 4.2 κ r , 4.9 ) ), M σ 1 ∆ − i n . 4.15 i ? ∂F/∂r ? ( r o 1 and . ? ,M 1 ? + − r ,M −

1 ? = 0 with two real constants γ n r 4 n − F

? n 6= 0. Eqs. ( − = n κ , σ ∂r ) evaluated at ∂ ) , M 1 ? γ ) F γ − j 1 . ( ,M n ? 4.4 d ) is ﬁnite, we have ( − 0 ? 6= 0 r o +1) 0 from r 0 r∂ | j ! ? ? ? + + + Z i ,M r, M ,M : ( κ γ > +1) 0. But, from eq. ( γ ,M ? 4 ( ? i v ∂ 4 r ? r ( φ α

βσ – 9 – r

α ∂

− n ! n F ? F C γ > j = ∂F/∂M n σ n 1 ! 2 ? 4 + ∂r M ∂r i ∂ ∂F ∂ r r ∂M +

! ). Let us start considering the simplest case in which − ) in the Stefan-Boltzmann law ( κ 1 − n

=0 ). It follows that ∞ ! ∂r ) ∂F 4.5 X = 1 i,j ? n 4.12 r ) ) v ( − 2 4.10 + + φ r r ∆ ( ( − 2 2 = φ φ e − − β ) vanishes due to eq. ( e e ) and ( = ) becomes 1 4.9 + = = 4.8 κ + 2 is the first natural number for which ) cannot be satisfied. 4.14 κ ≥ is finite if and only if 4.14 n v Eq. ( and 6= 0. The most general situation is analyzed below. In this case ∆ ), under the assumption that ? n 4.5 ,M = ? 2, eq. ( is the same as in eq. ( r = 1. , where | σ ≥ n n σ integer. Hence a contradiction. n 0. Plugging eqs. ( Let us now relax the assumption are completely generic, anddistinguish sufficient three that cases: the evaporation time isCase infinite. Indeed, let us where that where in the second linein we have the used expansion the is Taylor expansion of However, We see that ∆ where we have parametrized the deviationγ of > the latter to obtain the evaporation time ∆ so that From eq. ( second term vanishes due to∂F/∂M eq. ( order The first term of eq. ( JHEP07(2018)023 ` ) − ), r t 4.9 ∞ ∆ 1 are 4.10 0, where ≥ time 1 γ . 2 n < Evaporation σ I,J 4.1 3 ) − − 0 are appropriate r )] terms. 4 ( ` ` ` ` 1 1 1 − φ ≤ κ − − − − 0, which contradicts e . On the other hand, `/M J > γ , where ( n − ) would not change the 4 O is a positive integer and Jσ ≤ − is defined as in eq. ( σ < + 3 γ M I 3 σ ` σ r, M − 2 n ` ` ` 0 and ( r − Inner horizon ) cannot be satisfied. φ q ≥ ∼ I | 4.14 J ) γ < r ∞ 4 0 0 0 ( M φ ), where < | − ∆ I where σ are the same. However, from eq. ( 4.9 r ). Therefore i ) γ 2 . This implies 2 Jσ / 3 4.16 M` ] Jσ + ) ) 2 3 2 2 / / and I in eq. ( 2 3 + ) , hence ) r 3 3 3 2. Hence, eq. ( – 10 – r 2 Metric σ M` M` n I σ = − ( ( ≥ Mr Mr Mr +2 +2 M` γ e = m 3 3 ≥ ∼ n r r ( ( n − +(2 n γ 2 1 r r h [ M ) there are two subcases. Either ] 24 ). 4.15 2 ] ] ] 20 , 18 , 11 19 17 From eq. ( The term of order ∆ ). ) is trivially violated, or 1 for the same reasoning of eq. ( Model − summarizes the properties of several models presented in the literature, and 1. 1. 4.14 4.14 n 1 Hayward [ . The most relevant properties of the examples that we have considered. The parameter = is provided, so that these quantities must be multiplied by [1 + Bardeen [ ) is tuned so that some of its derivatives vanish. σ > σ < Dymnikova [ γ the last inequality follows from integers. In theeq. last ( subcase can be canceledappropriate only integers by such that terms of order ∆ eq. ( − Table κ r, M ( Hayward-Frolov-Zelnikov [ In this section we discuss the validity of4.3.1 the two hypotheses listed Adiabatic in condition section Hawking radiation is derivedevolving assuming slow that enough the in geometry a certain is sense. static or, The at precise the meaning of very the least, word “slow” in this that, in all theseone models, can read the that exponents F these coefficients are in general model independent,4.3 unless the function Adiabatic and quasi-static conditions straightforward to realize that aconclusion. generic, but This non-divergent, infinite timescalethe spacetime has (see important figure implications for the global structuremakes explicit of that these models have an infinite evaporation time. It may seem surprising Therefore, we conclude that thetries. evaporation time Intuitively, is the infinite resultgravity for goes follows analytic to from spacetime zero the geome- sufficiently fact fast that, and makes in the the evaporation extremal rate indefinitely limit, slow. the It surface is Case is a length scale,and which is usually identified with the Planck length. OnlyCase the leading order of Table 1 JHEP07(2018)023 (4.17) (4.18) (4.19) (4.20) (4.21) ) leads to 4.2 , one can easily show that ), it follows that 4.2 0 i 4.12 . γ 4 . , σ . v − . + − 1 ∝ d d γ 0

3 1

I I + r v − M γ d ∝ d

+ σ >

− κ d + r ∝ ∝ − v – 11 – d κ d γ v d + 2 + v 1 3 d d κ κ d 2 + 1 ) and using the evaporation law ( 1 d κ core − inner σ go to zero. From eq. ( 4.8 − + i i 2 + ]. For the purposes of the present discussion, the thermality κ , so that the adiabatic condition becomes = 0 v 56 d r , and / 55 v d / + κ ) for the formation and evaporation in an infinite time of a regular black hole. . The would-be Penrose diagram (ignoring the instability of the inner horizon discussed 3 Hence, it follows that the adiabatic condition is satisfied if and only if Repeating the same analysis performedthis at condition the is end always of satisfied section for spacetime geometries that are analytic. This permits to solve for d ration, when both d while taking the derivative in eq. ( of Hawking radiation is guaranteed if the following adiabatic condition is satisfied: For regular black holes, this condition might only fail during the final stages of the evapo- Figure 2 in section framework was defined in [ JHEP07(2018)023 (4.23) (4.24) (4.22) , it must → ∞ it is sufficient to M through an adiabatic . peak ? M M M and at ) = 0 ? ) has a global minimum for ) (4.25) M r, M : we see that there is only one M ( 4.23 = r, M M ( M . . ˜ λF with ∆ 2 + 1 + ` r ) M − 3 − , φ of the black hole mass. If the corresponding r 2 2 r 3 M r, M + ∆ ∂r 2 ( on the mass 4 ? M` – 12 – 2 peak . The function ( π = M ∂F ` Mr M 2 = + 2 2 / + 3 + κ → M r ]. Hayward’s metric is specified by ) = κ − − 11 for a given mass are identified by the relation r r, M ( r is high enough, then the evaporation process can enter in ˜ and F ) = 1 4, which corresponds to the extremal mass. The surface gravity . In order to find the exact value of M `/ peak 2 3 r, M T peak , reaches a mass ( T ? √ → F M M + = 3 = r ? . Qualitative plot of the Hawking temperature as a function of the mass. , M M → ∞ and shows the dependence of ` Figure 3 3 M 3 √ Since the surface gravity becomes zero both for = ? Figure maximum at minimize the function When r of the outer horizon is The outer and inner horizons a regime in whichapproximation most no of longer its holds.case. mass For concreteness, Therefore, is we we emitted referoriginally to must in proposed what address by a is Hayward probably whether short [ the or time, simplest regular not so black this that hole model, is the the quasi-static much greater than and quasi-static process driven by the Hawking radiation. have a global maximum forHawking some value temperature 4.3.2 Quasi-static condition We have studied explicitly only theis late stages arbitrarily of the small evaporation, inHowever, so which the we that temperature implicitly the assumed quasi-static that approximation the is black surely hole, valid starting in from this a stage. macroscopic mass JHEP07(2018)023 , ) 6= 17 ) 4.16 (4.29) (4.26) (4.28) (4.27) we find r, M ( φ the system is planck 0). Therefore, L peak = σ > M ` is not zero but it is ? 16. The corresponding κ `/ 0. . = 27 < 2 − , . 4 peak . Indeed, returning to eqs. ( ` − δ 1 . Identifying ? peak for the “Bardeen black hole” [ κ 10 − ` πr ? − M M 4 r ≈ ≈ n ∝ → = peak − ) 4 peak ≤ r π` ) ensures that the adiabatic condition is and M 1 M T M δ peak ` in the latter one. A similar conclusion was 18 from diverging. Notice also that, when the t SB +∆ ≤ 4.29 ? ? σ = and = 3 κ 3 . We interpret this result as an indication that σ − peak – 13 – ], finding that the mass emitted at the peak is when r,M π − M ? = 19 peak 2 peak 5 peak r peak +∆ 0 while lim κ r 1) ? − r 1, we are in a special case: in eq. ( to be positive that M r = ( 6 > − peak = δ φ γ −

− + 3 − + n r n ( e κ ) it is evident that it is not possible to satisfy the adiabatic dt − . Therefore we can roughly estimate the mass loss at the peak ? dM r ` = σ T → , finding 4.21 δ with the Planck length, the peak temperature is two orders of + ˜ λ r ` and ) and ( M , 4.14 r ] in the context of asymptotic safety. 57 in the former case and 10 ), it is sufficient for peak ) such that it diverges at 4.21 M It is possible to go outside the quasi-static approximation by simply picking a Evading the adiabatic condition is less trivial. One would need to pick a function We have numerically computed 7 ] and the “Dymnikova black hole” [ − r, M ( In particular, the lowerviolated, bound while of the upperupper bound bound prevents is saturated, discontinuous, namely lim φ and ( where 0 that is verythat large requires at beyond the semiclassicaldescription peak physics in mass. to terms of be the In addressed regular this (in black case hole other the geometry words, breaks system the down). would effective enter in a regime 4.4 Beyond theComparing quasi-static eqs. or ( adiabatic regimes condition and also havethe a only finite way evaporation toquasi-static time or build (this the a follows adiabatic model regimes. from with finite evaporation time is going outside either the 18 10 reached in [ by the characteristic timescalethat the at emitted the mass peak, the is about system 10 is notapproximation is much reasonable. perturbed even at the peak point, implying that the adiabatic this is consistent withdominated the quasi-static only condition, by let thepeak us scale by point multiplying that the at peak luminosity height of the temperature peak is Therefore, if we identify magnitude smaller than the Planck temperature. In order to assess the extent to which with respect to JHEP07(2018)023 ) and )] of = 0, 4.29 ? r r ,M = − r r ( φ ) must satisfy − r, M ( φ 0, while its derivative → r ) approaches zero in the limit − r ( 0 F is an horizon, then the metric is not singular. ). Indeed ? ? r – 14 – M might still be small for values of exp [ M − ) is not the only constraint that κ ), 4.29 goes to a constant in the limit ) 2.10 , i.e., only when ? r,M ( φ M − e 1. Strictly speaking, this condition is necessary only when the black = must vanish in this limit. Furthermore, we also want the instability M r )] ,M , and, from eq. ( − ? r ( M φ = 0. Computing the Ricci and Kretschmann scalar, one can check that, if eq. ( − We have discussed that attaining finite evaporation times necessarily involves breaking If these properties are satisfied, the model can in principle be consistent from a geo- Let us remark that eq. ( → r continue. However, itfreedom in seems describing that theopinion, late there this stages is makes of regular the no blackin evaporation holes general beyond order much less this to guiding appealing critical be principle fromprovide point. viable, a answers. restricting these physical In perspective models the our as, must fail precisely in the regime in which they should either the adiabatic orevaporation law the to quasi-static hold. conditions, Thispoint adds which at some which are degree these of necessary conditions uncertainty. breakin for First, down: the one the this known has additional models standard to information of decide isentire regular the just black evolution. not holes, encoded which Moreover, satisfy one these must conditions throughout also their specify the details of how the evolution would of any modelconsistent of calculation regular of black thepresented hole. evaporation in the time, literature, On showing the complete theIn that, evaporation combination cannot other in take with all place hand, the in thetime we a unstable finite models shows nature have timescale. that currently of also these the models provided nonsingular are a core, not the self- self-consistent. infinite evaporation 5 Conclusion We have analyzed two aspectsof of the the inner physics horizon of andis regular the black unstable evaporation holes, on timescale. namely a We the have finite instability shown timescale, that the and inner that horizon this instability is a completely general feature would become physically relevant, invalidatesof the these original models. motivation and limits the scope order unity or even greater. metric perspective. However, this limitsbecause the one predictability of must the resort late toby stages yet regular of unknown evaporation, black physics holes. beyond the Therefore, effective that description the provided effective description breaks down just when it we must have that with respect to timescale of theexp inner [ horizon tohole be is much macroscopic or longer mesoscopicM than ( the evaporation time, implying in order for thenonsingular. scenario There to are be two places internallyat in consistent. which the Obviously metric could the becomeholds metric singular, only should at when remain Additionally, it can be checked that, in order for the metric to not diverge at JHEP07(2018)023 22 26 , Phys. , (2006) ]. ]. 96 (2015) 33. 3 SPIRE SPIRE , Trieste, Italy, IN [ IN Rept. Prog. Phys. ][ , (2017) 092002 ]. Class. Quant. Grav. (1993) 3743 Class. Quant. Grav. , 80 , Front. Phys. Phys. Rev. Lett. 71 , (1976) 870 ]. , SPIRE IN [ D 14 SPIRE ]. 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