JHEP05(2021)132 Springer May 17, 2021 April 20, 2021 : March 23, 2021 : : January 21, 2021 : , Revised Published Accepted c,d,e Received Published for SISSA by [email protected] https://doi.org/10.1007/JHEP05(2021)132 , Stefano Liberati, b,c , [email protected] g , Francesco Di Filippo, . 3 a and Matt Visser f 2101.05006 The Authors. c Black Holes, Spacetime Singularities

Regular black holes with nonsingular cores have been considered in several , [email protected] [email protected] Piazzale Aldo Moro 5, 00185,School Roma, of Italy Mathematics andPO Statistics, Box Victoria 600, University Wellington of 6140, Wellington, E-mail: New Zealand [email protected] Via Bonomea 265, 34136 Trieste,IFPU Italy — Institute forVia Fundamental Beirut Physics 2, of 34014 the Trieste,INFN Universe, Italy Sezione di Trieste, Via Valerio 2, 34127 Trieste,Dipartimento Italy di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, Florida Space Institute, University12354 of Research Central Parkway, Florida, Partnership 1,Center Orlando, for FL, Gravitational U.S.A. Physics,Kyoto Yukawa 606-8502, Institute Japan for Theoretical Physics,SISSA Kyoto — University, International School for Advanced Studies, b c e g d a f Open Access Article funded by SCOAP regularization mechanism, and that nonperturbative backreationaccount effects in must order be to taken provide into of a gravitational consistent stellar description of collapse. the quantum-gravitational endpoint Keywords: ArXiv ePrint: regular black holes havegeneralization in of the fact geometrical stable framework, originally cores. appliedholes to by Reissner-Nordtsröm To black Ori, reconcile these andSharp arguments, show mass we that at discuss regular thenonsingular a black inner black holes horizon. hole have spacetimes This an are result exponentially not can growing the be Misner- definitive taken endpoint as of an a indication quantum that gravity stable Abstract: approaches to quantumproblem gravity, and and as Hawking’s agnosticthe information frameworks inner paradox. to core address While is the destabilized in singularity by a linear recent perturbations, work we opposite argued claims that were raised that regular black holes Raúl Carballo-Rubio, Costantino Pacilio Inner horizon instability and the unstable cores of JHEP05(2021)132 ]. Reg- 13 ]. ]. It is known that 19 , 13 – ], a consideration that 18 ]. 11 10 , 12 9 ]. While the latter analysis ] are so far in perfect agree- , 8 25 – 11 4 ], a fact which is not necessarily 22 – 20 ], while they have also been proposed as 17 , 16 – 1 – 10 5 ]. Recently, we produced arguments supporting the 24 , 23 2 ] demonstrate that, within the framework of standard classical 7 2 , 1 1 12 ]. Observational tests of black hole spacetimes coming from the detection 3 ] and loop quantum gravity [ 15 , 14 In fact, such inner-core instabilities have been derived in the context of a particular It can be shown that regular black holes, even when they are non-rotating, possess A conservative approach consists of replacing the singularity with a regular core, with- while at the same timeregular it models. would call for increased theoretical efforts tomodel produce of consistent regulargeneric black conclusion hole that regular [ black hole cores are unstable [ an inner horizon besides theinner more horizons conventional in trapping general horizon relativitygive [ are rise generically to unstable the underproblematic so small given called perturbations that mass-inflation and general instabilityother [ relativistic hand, black this holes opens arealso to unstable. known the If possibility to confirmed, that be this the would singular. make inner their physical horizons On justification of less the straightforward, regular black holes are out necessarily introducing substantial long-range modificationsular to black the geometry holes [ safety emerge [ in severalan approaches agnostic framework to to quantum address Hawking’s gravity information such paradox as [ asymptotic ment with the predictionsfor of alternatives to classical the general classicalbecomes black relativity. holes more of Nonetheless, pertinent general there relativity giventamed is [ once that still quantum it room gravity effects is are reasonable taken to into account assume [ that singularities will be Singularity theorems [ general relativity, singular black holestional are collapse unavoidably [ formed asof the gravitational end-state waves emitted of by gravita- binary black hole mergers [ 6 Conclusions 1 Introduction 3 Instability of regular black4 holes Numerical analysis 5 Role of the cosmological constant Contents 1 Introduction 2 Generalized Ori model JHEP05(2021)132 ) = (2.3) (2.1) (2.2) v, r ( f , namely . r 6 ]. 26 such that we numerically solve the ) . 4 we show how this equation . At this point in the dis- v 2 ) ( 3 Ω ]. In particular, we revisit the m d v, r 2 26 ( r , . ) ) M + , r ) v, r v ( we present an extended version of Ori’s ] and [ ( dvdr 2 M 25 m ( ∞ + 2 + M ]. 2 has an even number of zeroes, and that all its → – 2 – r 23 ) dv ) ) = v, r ) = lim ( v, r v, r v ( ( f ( f m M − = 2 ds ] — which is based on a single perturbing null shell, and takes 26 ] robust, while in contrast it rejects the claims in [ enters through a single-variable function in terms of the Misner-Sharp mass ], we are able to derive a key equation for its evolution in terms of the regular black hole has an unstable core. Therefore, our work makes the v 25 ] to regular spacetimes and claims that regular black holes have stable cores. /r 28 , ) 21 we analyze how the presence of a non-zero cosmological constant affects the coincides with the value of the Misner-Sharp mass at large values of 27 5 v, r ) ( v generic ( ] Ori developed a model to describe the dynamical evolution in the neighborhood of M m 2 The paper is organized as follows. In section The aim of the present work is to clarify the issue of regular black hole instabilities, 21 − dependence on and Most of the1 derivation can becussion we worked do out not needbut only to only assume using that any the its specificthis radial functional parameterization is form dependence equivalent for is to the such assuming Misner-Sharp that that mass, the geometry contains an inner horizon: the inner horizon ofto a explicitly Reissner-Nordström analyze black the hole singularity underOri’s generated linear by model the perturbations, can well-known in mass-inflation bemetric order instability. easily of generalized the form to any (singular or non-singular) spacetime with a In section conclusions of the previous sections. Concluding remarks are2 made in section Generalized Ori model In [ governing the growth ofgenerically the Misner-Sharp predicts mass. an indefinite Inwe section discuss growth the for functional bothdifferential form singular equations of and the governing regular growth thespacetimes, rate. black showing growth holes, In that of and section the instability solutions for are several in specific agreement black with hole the analytical treatment. initial perturbation. We conductthat both a an analytical andconclusions a of numerical [ study, and we show perturbation model to generic spherically symmetric metrics, and we derive the equation black hole considered in the appendix of [ by a reanalysis ofextension the of contradictory results Ori’s inequations model [ of to motion a inquantity generic terms [ spherically of symmetric the Misner-Sharp metric. mass, which By is expressing a the physically motivated of two non-interactingresults, null a shells recent on workthe top [ backreaction of of the a metricblack static into holes [ account background. — extendsThis In Ori’s claim work contradicts contrast for the to result Reissner-Nordström found these under the same assumptions for the specific regular captures the main physical ingredients, it considers a somewhat idealized scenario consisting JHEP05(2021)132 ) + − v v , as ( ) = (2.4) − − (2.5a) (2.5b) divides , r can be I ) m Σ v + ( v denotes the m by the same ( in such a way C − M and λ R which divides the − v Σ ] to be 31 , , 11 30 is the same on both regions, ≥ is on the same side of r ]. We will define − , p R 29 [ p − . β λ v − 0 ) = 0 , m , the shell crosses the corresponding inner 0 denotes the event horizon while r . Additionally, the radius of the shell can be = λ, R , such that ( = – 3 – H + . Therefore we can parametrize − → ∞ f R R δm 1 . → ∞ 0 0 + + − − → → v R lim lim 0 , while we shall choose a distinct null coordinate is dictated by Price’s law [ v λ λ and − ) m − − I and v R ( ) = depends on the region, so that we shall distinguish when − of − of the affine parameter. − ) , or the Hayward spacetime in which 0 R ) v m v − r . When ( ]. λ ( ) v 2 ( − → / m 19 2 R [ m λ e ) 2 → ∞ − ` ) ) . By continuity, the radial coordinate v v v ( + ], we now consider a dynamical model of a null shell ( ( − R in terms of a single affine coordinate m 21 m m Σ , which is implicitly defined by . 0 + 2 ) = r ) = 3 on + r 0 . Relevant sections of the Penrose diagram of a regular black hole. The shell v , r ( ( ) / m + v 3 ( r m It can be shown that the matching conditions imply that both The functional form of Following [ ) m v ( ( horizon at where expressed that it is negativeexpressed and as a function black hole into twoshown subregions by the Penrose diagramadvanced in null figure coordinate to parametrize while the mass function and Well-known examplesM include them Vaidya-Reissner-Nordström spacetime in which Figure 1 the spacetime inCauchy two horizon. regions JHEP05(2021)132 , − is in in − v i ) R v (2.8) (2.9) (2.6) v Z (2.7c) ( (2.12) (2.10) (2.11) (2.7a) in the (2.7b) ]. Fur- 0 and r δm 21 ) because as + = ] generalize, R v . It is conve- 2.7b R ) , otherwise 21 ,R ) = 0 , and we neglected and 0  r 0 − . ,R k = ) , . The quantity Z v m ,r λ  λ k | 0 1 ( )  0 = 3 ) − m κ − | = − m v − ( ( ) that m λ, R , m 1)! ( O ) | . M i v . ( − ) . 1)! . ( − + 2 ) 2.7b f ∂r (for the two regions ) ∂f )) λ k λ − 0 ( in eqs. (7)–(9) of [ 2 v δR ( ≡ R | λ ( v p − R + /∂m ( 0 O ( 2 ) = + 1) Σ | − p κ r O 0 ,R (

+ p ) + to obtain ) κ ( ) as  and | + v λ − | p λ, R ∂M | ( | ( ) 0 =0 λ + − ∞ . Hereafter, for simplicity, we will use λ, R λ, r v − | + = 0 X = κ k ( ( ( | = 0 − f ∂r m i 0 p v λ − ln ( A | ) proceeded exactly as in reference [ f r λ v | 0 /Z p − δm 0 around the values  – 4 – 0 ∂f 1 κ f r − 0 1 | 2.9 at | κ A r Aβ 1 2 − 0 | 0 and , − dλ f − κ − ) ) → − | Z λ = lim , λ v R 1 + ( , it follows from eq. ( i = ) Z ,R 2 1 +  ) around R i z v . Taking into account the functional form of ) ) = ) = 1 2 v can be written as a differential equation in ( ) and ( ∞ v p ) δR | v = ( λ λ 2 dv 0 v ( i dλ ( ( − + κ 0 + = 0 2.7c 2.8 dR f v λ 0 r dv − i takes two values R κ | −| r z Aβ − 2 0 R e i Z Z dδR v 1 κ | 0 c ) = = v → 0 ) = ) = = ( λ λ λ R ) = i ( ( R , δmδR, δR i i v z 2 lim ( z v ) in region δR δm , the index ( 0 is the radius of the shell and i O 2.7a ) to obtain λ ( R/v ; therefore R , we can expand eq. ( − = − v ≡ i ), the general solution to the above differential equation is given by → ∞ R z v R 2.4 We now set up the basic ingredients for the discussion of the generalized Ori model. eq. ( limit where we have written terms of order Notice that the derivationthermore, of eq. ( ( We can perform a Taylor expansion of However, given that would tend to aplace constant proportional of to an integration constant, while weit omitted is a irrelevant. similar These integrationregion equations constant in are ( valid on both sides of the ingoing null shell. Now, in where respectively), and a prime denotes differentiation with respect to It can berespectively, shown to that the matching conditions on nient to define the surface gravity in Here JHEP05(2021)132 , ) ) r ( 2 (3.4) (3.1) (3.2) (3.3) 2.7c , i.e., g (2.13) (2.14) + )) and the )+ M . v v ( ) becomes + (  ) and ( + M 2 R v, R m 2.14 ( ]. ) 2.7a on its boundary − r n + ( ) 32 + 1 M g R  v, R +1 2 R ( γ p , ) = + v ) − − ], the functional form of M is polynomial in ) v, r (  19 λ, R 0 + (   2 r v, R ) /∂r + ( M 2 f + + − − 0 00 v , R M R ( . )) 2 ∂M 2 v  O v ) does not depend on R | ( e + 0 2 +1 + . ! κ p 2 R | v e ,v 2.14 = = v, R ,vv r v M ( + 1

R R R 2 , the differential equation ( ∝ + ) 4 p ). One can manipulate eqs. ( ` = + r , r v 6 − )) M

v v M λ, r ) | − ( – 5 – ( = ,v 0 ∂r ,vv   + λ κ λ | v, r  ( 2 ]. More generally, this result is recovered unchanged v, R ∂f  ∂r ∂r ∂M ( + − 0 21 v + − + R  R R ∂M M O = = = r r +

) ) ) implies that subindices denote differentiation with respect to this variable. v + 1 ) λ, R v, r v, r is not linear in 2 v ( dλ p ( ( only (alternatively, ∂r ∂r + + + λ m` df | − /∂r 0 ∂M ∂M + + 2 κ | 3 ,v ,v  r ∂M R R ( = / . We then obtain 3 = = n + )) ) v M ( mr to obtain ∝ v, R ( v, R ) = dv dv ( + + /∂r = + However, if Let us now consider the evolution of the metric functions in region + i m, r dM ( dM , hence as functions of We will be more∂M general and consider the case where all the metricsof of a interest regular areM black covered hole by described a by linear the ansatz. Hayward metric For [ instance, in the case This is the solution obtained byfor Ori a [ wider family of geometrieswhich satisfying includes the for linear instance ansatz the Bardeen metric for anonlinear regular and black the hole last [ thus term on modifying the the right-hand exponential side may behavior drive associated the with evolution at mass-inflation. late times, Crucially, not so the firstequation term displays of (exponentially) inflating the solutions right-hand with side leading behaviour of eq. ( In the case of Vaidya-Reissner-Nordström spacetime In the equation above the This is the main equationcomment that its we solutions derive and in show the that paper. it In predicts the unstable next3 cores section for we regular are black Instability going holes. to of regular black holes Σ for or equivalently, as a differential equation in JHEP05(2021)132 , 0 (3.8) (3.9) (3.5) (3.6) (3.7) → ∞ (3.10) γ < . While v , up to a v )) v ( ), from which , v, R | we can solve this 3.3 ( 0 0 + κ ) is given by 2 | 0 M . r 3.7 ) γ > v | − 0 κ )) . v ( 0 −| v p, +1 grows indefinitely with . p v, R −  . v ( + , 2 +1 2
+ Γ( p 0 p , when present, might occur beyond 3 M 0 +1 v γ v . It is possible to show that, for r p | M + 1 v 0 | v γ − γ γ 0 | + M κ p 0 | κ , v 0 | | 2 r γ − κ κ e ` | ≈ | +1 0 p )) p )) ) ≈ v v m 0 v ( 2 )) v ( ∝ − v – 6 – . On the other hand, for ( ( 2 v | v, R )) ,R 0 − ( v 0 κ ( p v + pβ` v, R | ( ( v 6 1 1 ). Therefore, at late times, the growth rate of the M + | c − v, R 0 = n + ( M 3.6 ≈ κ , which depend on the specific metric under consideration. γ 1 M )) n − v ' | n + ( )) M and v ( v, R γ ( + means that we are neglecting subdominant terms in the v, R dv M ( ' + for which dM 0 diverges for a finite value of v )) v ( v, R Before closing this section, we provide an argument for the fact that the transition In the next section, we are going to integrate the perturbative equations of motions ( + no concrete example of such behaviour and we expectfrom it the to exponential be to non-generic. thethe regime polynomial of growth validity of of theIf linear analysis. the corrections For concreteness, to we refer the to Schwarzschild Hayward’s metric. metric come from quantum regularization effects, numerically for several regular blackwill hole show that, metrics for usually all considered thethe metrics in functional under the form consideration, (exponential literature. orform We polynomial) of of the the growth metric, rateTherefore, depends we the find on evidence fact the that specific regular thatwe black cannot inflation holes have exclude occurs unstable the cores; appears existence while, in of to principle, a be fine-tuned a metric generic avoiding this phenomenon. conclusion, we found in perfect agreementinstability with is eq. slowed down, ( and becomes polynomial rather than exponential. Expanding the incomplete Gamma function at late times we obtain where the symbol limit, and By neglecting the last term on the right hand side, the solution of ( As a concrete example,it we follows consider that Hayward’s metric and the relation ( From that point on, the firstand and the the second growth term is of polynomial the right-hand rather side than become comparable exponential, The behaviour of the solution dependsM on the sign of differential equation in different regimes.side dominates At the early dynamics, times, andcritical the leads time first to term an on exponential growth the of right-hand for some given constants JHEP05(2021)132 . )) v ( (4.1) (4.2) (4.3c) (3.11) (3.12) (4.3a) (4.3b) v, R ( + . Therefore, M 0 m ], apart for the and 26  ) ` v ( R +1 pβ p 0 6 v , 2 1
] 3 0 r  29 . [ ) , + . v 0) 2 , Σ +1 ( , 2 , pβ ` p 0 ` + ∂r 0 6 0 ∂v is already exponentially large w.r.t. the v ∂M 0 ∂M , , ) for the variables r ∂v 0 2 m 2 m 1 ∂M + ` ] = 0 , 2 +1 2 m ν pβ πr p πr 0 ± M s 2.14 6 4 2 4 v πr | µ ∼ /f 2 0 0 − − s – 7 – ` ν κ | m )) = = 4 = µ v ( r v = (2 T r ≈ ∼ [ ) and ( v r v 0 µ T T T s )) m v, R v ( 2.10 ( + . In turn, this implies M v, R ` ( can be derived following the treatment in [ + ∼ ) v 1 M ), we find that the transition occurs when ( of the spacetime, and − + | 0 m κ −} 3.10 , + { ∼ | . While the main advantage of the first method is the use of the physically R 0 . Let us stress that this argument reinforces the case for mass-inflation ) r 0 v ) and ( ( m + is the effective stress energy tensor obtained by imposing Einstein’s equations 3.8 m ν µ T and An equation for ) v ( on both regions is the outgoing nulltensor are normal to the shell. The relevant components of the stress energy correction of a mistake, whichstarting explains the point opposite is conclusions the we junction reach condition in at our work. the shell The where R motivated Misner-Sharp mass as onesubtleties of that arise the in variables, the the interpretationis of second the expressed method mass-inflation in will phenomenon when terms clarify the of some metric the unphysical variable In this section, wemotion provide for explicit several numericalliterature. solutions choices for We of can the integrate the eqs. perturbativeHowever, black ( equations we hole will of follow metric an which alternative have route and been integrate considered the in equations for the the variables initial value as it argues that,effectively exponential. in the situations of physical interest,4 the dominant behaviour Numerical will analysis be where we used namely, at the transition point the value of from eqs. ( it is reasonable to expect that the regularization parameter satisfies JHEP05(2021)132 . ) 3 v ( after (4.6) (4.7) (4.4) (4.5) R . 4 diverges. (left). and + 2 ) , as deduced m v ( + + M m are evaluated . v ) is implicitly a function − . v  ( ) 2 + , for which v e R ( v . As expected, the system M = m . r 2

2 ) + ) r ) 2 2 /` v 2 ` v e 2 ) is derived and, crucially, it is ) ( 3 / ) ) ( 3 3 r , r 3 v R v r r ) ( − − − ) ) − ( 2 4.6 = l v, r − v v e m v ) r ( ( 0 − (

( + v 2 m m − ∂v ( +2 − m − R m r M 3 ) and that ( 2 1 r m ∂v  ∂M − ∂M 2.4 ) becomes ) − + v 2 2 − − 1 f f ( 1 f 4.6 f R m = = − = = ) can be solved in terms of v – 8 – ) + − ) 2 v + e ( ≡ v dv dv ] ( R 2.10 ] v ] R = 33 ) + r =

19 v r 32

) ( is dictated by ( + First, as a sanity check, we integrate the system for the ) v exhibit corresponding divergences, see figure + ) ( ∂v , r v 0 + ) and ( + ( ∂M breaks down at a finite value of + f + m − R m v metric + 4.6 1 ( 2 + f m ∂v + Bardeen [ m − Hayward [ and . Dymnikova [ 2 dependence from this equation noting that, along a null trajectory, . This is inconsistent with how ( + + ∂M Reissner-Nordström R Σ + 2 m m + The case of Hayward metric is less trivial. Indeed, as shown in figure v 1 f . We will also show the corresponding evolution of on ) was interpreted as if the partial derivatives in ] reaches the conclusion that regular black holes have stable cores. 1 ) 26 v 4.6 ( R ) becomes . Black hole parametrizations used for the numerical integrations in section = 4.1 ] eq. ( r 26 (right). At the same time, as it is also apparent from the functional form of the , the system of eqs. ( 2 Table 1 + We shall now present some explicit examples of solutions for the metric coefficients m figure metric, the functions Hayward metric. (left), the integration for However, the integration can be continued from the opposite side of the discontinuity, The results of thedevelops numerical a integration mass are inflation instability plotted and in the figure Misner-Sharp mass diverges exponentially, see displayed in table from the solutions of Reissner-Nordström metric. Reissner-Nordström black hole, for which eq. ( Given that the evolution of of In ref. [ imposing the reason why [ and expressing everything in terms of We can eliminate the Then eq. ( JHEP05(2021)132 , . ± of + f ± M f = 12 p , = 1) = 5 is linear . We can v + ( = 1 diverges, as R M β + → ∞ . M v + 1 0 m follows a first phase of + is such that M = 1) = and of the metric functions v + and of the metric functions + ( has a discontinuity and becomes m + + m m m + asymptotically for (right). A direct comparison with ) with the initial conditions m 2 is not a physical variable and so this 4 ` 5 and . (2 + / = 0 m 3 0 r ` , (right). Indeed, = 1) = 5 – 9 – → − v 3 = 10 ( + 0 R m m behave similarly to the Reissner-Nordström case. , . In both plots we have used the parameters + + f M = 12 p , and + = 1 m β . The functional form of the Bardeen metric is such that + 1 0 m with the initial conditions is negative and approaches = 5 = 1) = . Left: numerical evolution of the mass parameter e . Left: numerical evolution of the mass parameter + v , also shows that ( m . Therefore, from the previous section, we expect an exponential inflation of + 2 + is well behaved, as shown in figure m = 10 m + 0 Bardeen metric. in This is validated byfigure the numerical results in figure expected. While it mightnegative, seem odd the that reader the shouldbehaviour variable take is in an mind artifactM of that the parametrization. Onexponential the expansion, other followed hand, byagreement the with a physical the variable phase analysis of of the polynomial previous growth. section. This is in complete and where see from the form of the metric that the limiting value of Figure 3 a Hayward regular black hole.we Right: picked the numerical parameters evolution of the Misner-Sharp mass. In both plots m Figure 2 of a Reissner-Nordström blackevolution hole. of the Right: Misner-Sharp comparison mass between the numerical and the analytical JHEP05(2021)132 of of a ± f ± f = 1) = 5 = 1) = 5 v v ( ( R R ) is modified — it exhibits 2.4 and of the metric functions is not a polynomial, we cannot and of the metric functions + + m m /∂r with the initial conditions with the initial conditions + 5 . ∂M = 1 = 0 diverges exponentially. On the other hand, ` ` , + , M – 10 – = 10 = 10 0 0 m m , , = 12 = 12 p p ) shows that , , 5 ]. Since in this case = 1 33 = 1 As our final example, we consider the regular black hole metric β β . . + 1 + 1 0 0 m m = 1) = = 1) = . Left: numerical evolution of the mass parameter . Left: numerical evolution of the mass parameter ). Again, we stress that this is an artifact of the parametrization. v v ( ( 3 + + presents a change-of-sign discontinuity analogous to the case of the Hayward metric m m + So far we ignoredthe the fact role that of corePlanckian instabilities the and the happen cosmological role at constant.the of the presence the of Intuitively, inner cosmological a this horizon, non-zero constant is cosmological is where constant, justified completely Price’s the law negligible. by energy ( However, density in is numerical integration (figure m (figure 5 Role of the cosmological constant and Dynmikova metric. proposed by Dymnikova [ apply the reasoning of the previous section and we have no prior expectations. A direct Figure 5 Dymnikova regular black hole. Right:we numerical picked evolution the of parameters the Misner-Sharp mass. In both plots a Bardeen regular black hole.we picked Right: the numerical parameters evolutionand of the Misner-Sharp mass. In both plots Figure 4 JHEP05(2021)132 I . + ) v ( M r (5.7) (5.3) (5.4) (5.5) (5.6) (5.1) (5.2) R > ω ), the ∂ = | ) r 0 v

( 5.1 κ ) | R . ) = v r v,r (

( R ) ∂r + = r

v,r ) ( . When ∂M then the last term is replaced by the I ∂r + − 2 + v v, r f ( < ω ∂M M ∂r + . First, using ( + | v 1 | 0 3 0 − κ , v κ ∂M | I v ! ] I −| ω v ) − 2 ω e I and I − | f ω − e 35 0 ω and , 2 ) − . . κ + I | I αe v 34 |− ) 2 1) is linear in ω 0 I 1) − − − ω κ Aαe + n | 0 > ω v I ( ( − | − ) |− | 0 ω 0 M 0 0 Aα m 0 v/ | 0 r r κ 2 I κ κ | | κ ∂ ( ω /r | ( + e −| e 0 ) = I )) v r | ω v v ∝ 0 , if ∝ ( ( ( κ + − 3 )) )) −| – 11 – v v δm | v e ( 0 ( v, R 1 , we still find that the inflation is governed by an ( κ Aαe c + n | + + 2 −| 0 0 1) v, R e v, R κ M M ( ( m 1 | − c 2 + + 0 ∝

) (2 M M grows exponentially with leading behaviour ) = /r I v + ) = I v ω ), in which the polynomial fall-off in )) | ( + ω v M ) becomes 0 ) for a very long time and, therefore, the eventual exponential v ( − − r ( κ M e 3.2 | ∂ | − m I 5.4 ). On the other hand, at variance with the flat case, if 2.4 2 δR 0 κ v,R | ( 5.1 ( 0 + Aαω , i.e., r ) becomes , eq. ( + + M I f ' ) can be ignored. This is certainly the case in our universe, in which (2 M )) 2.10 v ' < ω 5.1 ' − + ( | is the imaginary part of the least damped quasinormal mode of oscillation of )) 0 M )) 0 v κ v ( | v, R dv ( ( > is an integration constant. This, in turn, results in the following equation for the + I 1 v,R dv c ω v,R ( dv ( + dM When For theoretical completeness, we investigate the scenario in which a non-zero cosmo- If the cosmological constant is small, the fall-off of the perturbations follow the poly- + dM dM This is analogousexponential to decay eq. of ( ( is polynomial in exponential growth Following the same reasoning asbecomes subdominant in and section It is convenient to analyzewe have separately the cases where evolution of logical constant does playthe a core significant instability. role, We tosolution parallel inspect to the if eq. discussion it ( in can sections be fine-tuned to suppress nomial Price’s law of eq.fall-off ( in eq. ( therefore core instabilities willany manifest significant well role. before the cosmological constant could play where the black hole. an exponential decay of the perturbations at late time [ JHEP05(2021)132 ]. While certainly 44 , 43 , 17 is a property of the single – ]. We extended Ori’s pertur- I 14 26 , < ω 25 | 0 ]. Therefore, in order to address the κ | 42 , 12 – 12 – ]; therefore, the avoidance of mass-inflation will not be a 41 – 36 A rigorous approach would require to specify the dynamical field equations of the the- We stress that the instability of the inner horizon does not necessarily imply the for- When a nonvanishing cosmological constant is allowed, we found that in principle the We found that the Misner-Sharp mass always experiences a period of exponential a theoretically consistent framework. explored in the literature arethese asymptotic theories, safety several and studies loop haveholes argued quantum is replaced that gravity. by the Indeed, a singularity within nonsingulara core of with general demanding an relativistic inner task, horizon black [ wecommunity to hope address that the our effects of work nonperturbative serves backreaction on as regular a spacetimes stimulus in for the quantum gravity might expect that, into a a full different theory class ofendpoint of quantum of non-singular gravity, the spacetimes backreaction inner [ drives core the instability, geometry a geometrical analysisory is leading not to informative the enough. formation of regular black holes. Possible quantum gravity frameworks universe due to the very small value of themation cosmological of constant. a physical singularity.has What a we have huge shown effect is onmass. that the an geometry initial While and linear in leads perturbation general to an relativity unbounded this growth usually of the implies Misner-Sharp the formation of a singularity, we inner core might becomefeature stable. of However, this the occurs theory.they case-by-case require and the Furthermore, it late-time even tail iscosmological when of not constant) the the a to perturbation conditions generic (which be for is physically sensitive stability relevant, to which are the is presence realized, not of the going to be the case in our instability which can either go onby indefinitely a (e.g., with period the Bardeengrowth of metric) does or polynomial not be growth stop. followed (e.g., Additionally,is we likely with argued to that occur the the when HaywardTherefore, onset the approximation of metric). exponential of the mass-inflation linear polynomial In perturbations appears instability regular has to both black already be hole broken cases, down. cores. the the leading instability mechanism of bative model to a genericthe inner spherically horizon symmetric is spacetime, a robust showingof and that very mass-inflation mass general might prediction. inflation be at In obscured particular,evident while by when the the the occurrence choice equations of are parameterization, expressed the in phenomenon terms is of the Misner-Sharp mass. generic feature. 6 Conclusions In this paper, we have reanalyzedhole the spacetimes, issue in of light the of inner recent horizon contradictory instability results for [ regular black is regular. 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