Inner horizon instability and the unstable cores of regular black holes
Raúl Carballo-Rubio, Francesco Di Filippo, Stefano Liberati, Costantino Pacilio, Matt Visser
YYITP-21-02
Inner horizon instability and the unstable cores of regular blackholes
Ra´ul Carballo-Rubio, Francesco Di Filippo,
2, 3
StefanoLiberati,
3, 4, 5
Costantino Pacilio, and Matt Visser Florida Space Institute, University of Central Florida,12354 Research Parkway, Partnership 1, Orlando, FL, USA Center for Gravitational Physics,Yukawa Institute for Theoretical Physics,Kyoto University, Kyoto 606-8502, Japan SISSA - International School for Advanced Studies,Via Bonomea 265, 34136 Trieste, Italy IFPU - Institute for Fundamental Physics ofthe Universe, Via Beirut 2, 34014 Trieste, Italy INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy Dipartimento di Fisica, “Sapienza” Universit`a di Roma & Sezione INFN Roma1,Piazzale Aldo Moro 5, 00185, Roma, Italy School of Mathematics and Statistics, Victoria University of Wellington,PO Box 600, Wellington 6140, New Zealand
Abstract
We discuss the generalization of the Ori model to black holes with generic inner horizons (witharbitrary location and surface gravity), analyzing the behavior of the metric around these innerhorizons under the influence of both a decaying ingoing flux of energy satisfying Price’s law andan outgoing flux described by a null shell. We show that the exact solution for the late-time evo-lution obtained by Ori is valid for all spacetimes in which the Misner–Sharp mass is assumed tobe linearly proportional to the time-dependent metric perturbations introduced. We then considergeneric situations in which this linear proportionality is not assumed, obtaining asymptotic analyt-ical expressions and performing numerical integrations that show how the exponential divergenceassociated with mass inflation is generically replaced at late times by a power-law divergence.We emphasize that all these geometries initially experience a first phase in which mass inflationproceeds exponentially, and describe the physical implications that follow for generic regular blackholes. The formalism used also allows us to make some remarks regarding the early-time transientsassociated with a positive cosmological constant, known to modify the late-time behavior of ingo-ing perturbations from Price’s law to an exponential decay. Finally we compare our analysis withthat in arXiv:2010.04226v1, and illustrate specific shortcomings in the latter work that explain thediffering results and, in particular, make it impossible for that analysis to recover the well-knownOri solution for Reissner–Nordstr¨om backgrounds. a r X i v : . [ g r- q c ] J a n . INTRODUCTION Singularity theorems [1–3] demonstrate that, within the framework of standard generalrelativity, singular black holes are unavoidably formed as the end state of gravitationalcollapse. Observational tests of black hole spacetimes coming from the detection of gravita-tional waves emitted by the merger of two black holes [4–8] are so far in perfect agreementwith the predictions of general relativity. However, there is still room for alternatives toclassical black holes [9, 10], a consideration that becomes more pertinent given that it is rea-sonable to assume that singularities will be tamed once quantum gravity effects are takeninto consideration, which naturally leads to some of these alternatives [11, 12].Among the different classes of black hole mimickers, a very simple approach consistsof replacing the singularity with a regular core without necessarily introducing substantiallong-range modifications to the geometry. The price to pay for such a conservative approachis the introduction of an inner horizon [11, 12] whose stability is not guaranteed. In fact, itis well known that inner horizons in general relativity are generically unstable under smallperturbations (see e.g. [13–15]). Such instability has indeed been found in the context ofregular black holes, first for a particular model [16] and then for a generic regular black holemetric [17]. While these analyses capture the main physical ingredients, they consider asomewhat idealized scenario consisting of two non-interacting null shells on top of a staticbackground.Recent work [18], which aimed to extend the result of Ori’s work for Reissner–Nordstr¨omblack holes [14] to regular black hole spacetimes, claims that a less idealized configuration ofperturbations would lead to a stable inner horizon. However, we point out a technical issuein the analysis of [18], and that the inner horizon of a regular black hole is indeed genericallyunstable. To this end, in sec. II we extend Ori’s calculation to a generic spherically symmetricmetric. In sec. III we repeat the analysis using the formalism of [18], as this formalism mayprovide a more straightforward physical interpretation and it is better suited to numericallysolve the exact time evolution of the geometry. In sec. IV we consider the role of thecosmological constant, showing that also in this case the instability is present, and thatit could become sub-exponential at most at late times (when the backreaction would bealready non-negligible anyway). Finally, in sec. V we solve numerically the differentialequations governing the growth of instability for many specific choices of spacetimes, andwe show that the solutions are in agreement with the analytical treatment of the previoussections. Our conclusions are summarized in sec. VI.
II. GENERALIZED ORI MODEL
In reference [14] Ori developed a model to describe the dynamical evolution of the neigh-borhood of the inner horizon of a Reissner–Nordstr¨om black hole under perturbations, inorder to explicitly analyze the singularity generated by the mass-inflation instability. TheOri model can be easily generalized to any (singular or non-singular) spacetime with a metricof the form ds = − f ( v, r ) dv + 2 dvdr + r d Ω . (1)2 igure 1: Relevant sections of the Penrose diagram of a regular black hole. The shell Σ divides thespacetime in two regions R − and R + . H denotes the event horizon while C denotes the Cauchyhorizon. Most of the derivation can be worked out only using the parameterization f ( v, r ) = 1 − M ( v, r ) /r in terms of the Misner–Sharp mass M ( v, r ). At this point in the discussionwe do not need to assume any specific form of the Misner–Sharp mass, but only that itsradial dependence is such that the geometry contains an inner horizon or, equivalently, that f ( v, r ) has en even number of zeroes, and that all its dependence on v enters through asingle-variable function m ( v ) which coincides with the value of the Misner–Sharp mass atlarge values of r , namely M ( v, r ) = M ( m ( v ) , r ) , (2)with m ( v ) = lim r → + ∞ M ( v, r ) . (3)Well-known examples include the Vaidya–Reissner–Nordstr¨om spacetime in which M ( m, r ) = m − e / r , or the Hayward spacetime in which M ( m, r ) = mr / ( r + 2 m(cid:96) )[19].Following reference [14], we now consider a dynamical model of a null shell Σ whichdivides the black hole into two subregions R − and R + , such that R − is on the same side of I − , as shown by the Penrose diagram in fig. 1. Therefore we can parametrize R − by thesame advanced null coordinate v − of I − , while we shall choose a distinct null coordinate v + to parametrize R + . By continuity, the radial coordinate r is the same on both regions,while the mass function m ( v ) depends on the region, so that we shall distinguish m − ( v − )and m + ( v + ). On the other hand, the matching conditions imply that both v − and v + canbe expressed on Σ in terms of a single affine coordinate λ [20]. We will define λ in such away that it is negative and λ → v − → ∞ . Additionally, the radius of the shell canbe expressed as a function R ( λ ) of the affine parameter.3he functional form of m − ( v − ) is dictated by Price’s law [21, 22] to be m − ( v − ) = m + δm = m − βv p − . (4)When v − → ∞ , the shell crosses the corresponding inner horizon r , which is implicitelydefined by lim λ → R = r ; (5a)lim λ → f − ( λ, R ) = 0; (5b)where R ≡ R ( λ ) is the radius of the shell and f − ( λ, R ) = f ( M ( m − ( λ ) , R ) , R ). It is conve-nient to define the surface gravity in R − at λ = 0 as κ = 12 lim λ → ∂f − ( λ, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( λ ) . (6)We now set up the basic ingredients for the discussion of the generalized Ori model. It canbe shown that the matching conditions on Σ in eqs. (7)–(9) of [14] generalize, respectively,to z i R (cid:48) = R f i ( v i , R ); (7a) v i ( λ ) = (cid:90) λ dλ Rz i ; (7b) z i ( λ ) = Z i + 12 (cid:90) λ dλ (cid:18) f i ( λ, R ) + R ∂f i ∂r ( λ, R ) (cid:19) ; (7c)where z i = R/v i (cid:48) , the index i takes two values + and − (for the two regions R + and R − ,respectively), and a prime denotes differentiation with respect to λ . The quantity Z i is anintegration constant, while we omitted a similar integration constant in (7b) because it isirrelevant. These equations are valid on both sides of the ingoing null shell. Now, in region R − , we can expand eq. (7c) around λ = 0 to obtain z − ( λ ) = Z − − r | κ | λ + O ( λ ) . (8)However, given that lim λ → v − = ∞ , it follows from eq. (7b) that Z − = 0, otherwise v − would tend to a constant proportional to 1 /Z − , and therefore (hereafter, for simplicity, wewill use v in place of v − ) v ( λ ) = − | κ | ln | λ | + O ( λ ) . (9)Notice that the derivation of (8) and (9) proceeded exactly as in reference [14]. Furthermore,eq. (7a) in region R − can be written as a differential equation in v as dR ( v ) dv = 12 f − ( m − ( v ) , R ( v )) . (10)4e can perform a Taylor expansion of f − around the values m − = m and R = r at v = ∞ to obtain dδR ( v ) dv = − Ar δm ( v ) − | κ | δR ( v )+ O ( δm , δmδR, δR ) , (11)where we have written R ( v ) = r + δR ( v ) and A = ∂M − /∂m − | m − = m ,r = r . The generalsolution to the differential equation above (neglecting subleading orders) is given by δR ( v ) = c e −| κ | v − Ar e −| κ | v (cid:90) d˜ v δm (˜ v ) e | κ | ˜ v , (12)where c is an integration constant. This equation is valid for a generic δm ( v ). If we furtherconsider the functional form of δm ( v ) from eq. (4) we obtain δR ( v ) = c e −| κ | v + Aβ | κ | r v p ∞ (cid:88) k =0 (cid:20) ( p + k − p − | κ | k v k (cid:21) = Aβ | κ | r v p (cid:26) p | κ | v + p ( p + 1) | κ | v + O ( v − ) (cid:27) . (13)Let us now consider the evolution of the metric functions in region R + on its boundaryΣ, hence as functions of λ only (alternatively, v ). One can manipulate eqs. (7a) and (7c)for i = + to obtain df + ( λ, R ) dλ = R (cid:48) ∂f + ( λ, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R + R (cid:48)(cid:48) R (cid:48) f + ( λ, R ) , (14)or equivalently, as a differential equation in v , dM + ( v, R ) dv = R ,v ∂M + ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R − (cid:18) λ ,vv λ ,v − R ,vv R ,v (cid:19) (cid:18) M + ( v, R ) − R (cid:19) = R ,v ∂M + ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R + (cid:18) | κ | − p + 1 v + O ( v − ) (cid:19) (cid:18) M + ( v, R ) − R (cid:19) . (15)In the equation above the v subindices denote differentiation with respect to this variable.In the case of Reissner–Nordstr¨om, ∂M + ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R = e R , (16)so the first term of the right-hand side of eq. (15) does not depend on M + and the equationdisplays (exponentially) inflating solutions with leading behaviour M + ( v, R ( v )) ∝ e | κ | v v p +1 . (17)This is precisely the exact solution obtained by Ori [14]. More generally, we see that thisresult by Ori is recovered unchanged for a wider family of geometries satisfying the linearansatz M ( v, r ) = g ( r ) m ( v ) + g ( r ), which includes for instance the Bardeen metric for aregular black hole. 5ot all metrics potentially of interest are covered by this linear ansatz. For instance, inthe case of the regular black hole described by the Hayward metric [19], the functional formof M ( m, r ) = mr / ( r + 2 m(cid:96) ) implies that ∂M∂r = 6 (cid:96) r M . (18)While these more general situations can be analyzed using the equations above, it is notdifficult to see, either using eq. (15) or eq. (7) directly, that the linear ansatz for M ( v, r ) isnecessary in order to derive Ori’s solution in eq. (17). We will discuss these more generalsituations in more detail, using both analytical and numerical methods, after an equivalentformalism is described in the next section. III. ALTERNATIVE DERIVATION
It is instructive to derive the same results with a slightly different, although equivalent,formalism. Moreover, this will allow us to critically analyze the results presented in reference[18]. The analysis of this section will closely follow the discussion in that work up to thepoint where we correct a technical error therein. We will then discuss how this technicalerror invalidates the conclusions presented in reference [18].The setting is the same as in the Ori model discussed in the previous section, and we willtherefore use the same notation. The main difference in this section is the way in which thegluing of the two spacetime regions along Σ is dealt with. Instead of using the existence ofa common (for both regions) affine parameter λ , we will focus on the junction conditions atthe shell [20] [ T µ ν s µ s ν ] = 0 , (19)where T µ ν is the effective stress energy tensor obtained by imposing Einstein’s equations onboth regions of the spacetime, and s µ = (2 /f ± , , ,
0) (20)is the outgoing null normal to the shell. The relevant components of the stress energy tensorare T v v = − πr ∂M∂r , T v r = 4 πr ∂M∂v , T r r = − πr ∂M∂v . (21)Eq. (19) then becomes1 f ∂M + ( v + , r ) ∂v + (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v + ) = 1 f − ∂M − ( v − , r ) ∂v − (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v − ) . (22)We can eliminate the v + dependence from this equation noting that along a null trajectory dr = 12 f ± dv ± , (23)or, equivalently, dv + dv − = f − f + . (24)6sing this equation in order to relate the total derivatives of M + with respect to v + and v − allows us to write eq. (22) as 1 f + ∂M + ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) = 1 f − ∂M − ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) (25)where, as in the previous section, we use v in place of v − . We emphasize that, whenevaluating eq. (25) on the shell Σ, the partial derivatives with respect to v must be takenbefore imposing r = R ( v ) on Σ (one would be calculating total derivatives otherwise, thuschanging the content of the equation). Crucially, this observation lies at the core of thetechnical flaw in [18], as it will be discussed in more detail at the end of this section: indeed,in deriving their eq. (13), the authors imposed the condition r = R ( v ) inside the partialderivative signs in eq. (25).Given that we are following the evolution of M ( v, r ) on the trajectory r = R ( v ) or, in otherwords, we are solving a coupled system of differential equations for R ( v ) and M ( v, R ( v )),we cannot directly integrate eq. (25) because it involves partial derivatives with respect to v . Instead we need to consider an equivalent equation involving total derivatives. To thisend, let us note that dM + ( v, R ( v )) dv = ∂M + ( v, r ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) + ∂M + ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) dR ( v ) dv . (26)Using eq. (25), the above equation can be rewritten as dM + ( v, R ( v )) dv = f + f − ∂M − ( v, r ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) + R ,v ∂M + ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) . (27)Consider now the identity1 f − ∂M − ( v, r ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) = 12 R ,v ∂M − ( v, r ) ∂m − (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) dm − ( v ) dv (28)which holds true on Σ due to the functional form of the Misner–Sharp mass, M ( v, r ) = M ( m ( v ) , r ), and the equation of motion for the null shell Σ, namely eq. (10). Moreover, ineq. (13) we obtained the solution of the equation of motion for the null shell, which usingthe derivative of eq. (4) can be written as R ,v = − A | κ | r dm − ( v ) dv (cid:18) | κ | + p + 1 v (cid:19) + O ( v − p − ) . (29)It is then straightforward to see that the derivatives of the function m − ( v ) cancel on theright-hand side of eq. (28), while the definition A = ∂M − /∂m − | m − = m ,r = r allow us to furthersimplify the resulting expression to1 f − ∂M − ( v, r ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) = − r | κ | (cid:18) | κ | + p + 1 v (cid:19) − + O ( v − )= − r (cid:18) | κ | − p + 1 v (cid:19) + O ( v − ) . (30)7s a consequence we obtain dM + ( v, R ( v )) dv = (cid:18) | κ | − p + 1 v + O (cid:0) v − (cid:1)(cid:19) (cid:110) M + ( v, R ( v )) − r (cid:111) + R ,v ∂M + ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) . (31)If ∂M + /∂r is proportional to M + , namely M + ( v, r ) = g ( r ) m + ( v ) + g ( r ), the last termbecomes subdominant at late times due to being multiplied by R ,v = O ( v − p − ) terms andwe have M + ( v, R ( v )) = c e | κ | v v p +1 (cid:2) O (cid:0) v − (cid:1)(cid:3) + O ( v ) , (32)where c is an integration constant. Hence, we reproduce Ori’s result. However, if ∂M + /∂r is not linear in M + , the differential equation (31) becomes nonlinear and the last term onits right-hand side may drive the evolution at late times, thus modifying the exponentialbehavior associated with mass inflation. Hence, the analysis must proceed in a case-by-casebasis for these more general situations.For instance let us consider the case where ∂M + /∂r is polynomial in M + . We would get dM + ( v, R ( v )) dv = (cid:18) | κ | − p + 1 v + O (cid:0) v − (cid:1)(cid:19) (cid:110) M + ( v, R ( v )) − r (cid:111) − γv p +1 M n + ( v, R ( v )) . (33)for some given constants γ and n depending on the specific metric under consideration. Thebehavior of the solution of this differential equation depends on the sign of γ . It is possibleto show, for γ <
0, that M + ( v, R ( v )) diverges for a finite value of v . On the other hand,for γ > M + ( v, R ( v )), up to a critical time v for which M n − ( v , R ( v )) ≈ | κ | γ v p +10 . (34)From that point on the first and the second term of the right hand side assume comparablevalues and the growth is polynomial rather than exponential, M n − ( v, R ( v )) ∝ v p +1 , v (cid:29) v . (35)As a concrete example, let us consider Hayward’s metric for which the relation (18) allowsus to write dM + ( v, R ( v )) dv (cid:39) | κ | M + ( v, R ( v )) − γ M ( v, R ( v )) v p +1 − r | κ | , (36)where the symbol (cid:39) means that we are neglecting subdominant terms in the v → ∞ limit,and γ = 6 pβ(cid:96) | κ | r (2 m (cid:96) + r ) . (37)By neglecting the last term on the right hand side, the solution of (36) is given by M + ( v, R ( v )) ≈ e | κ | v v p c v p − ( − v ) p | κ | p − γ Γ( − p, −| κ | v ) . (38)8xpanding the incomplete Gamma function at late times we obtain M + ( v, R ( v )) ≈ | κ | γ v p +1 , (39)in perfect agreement with eq. (35). Therefore, the growth rate of the instability is sloweddown, becoming polynomial rather than exponential at late times. However, it is reasonableto expect that this polynomial behaviour will not have any physical consequence, as itoccurs beyond the validity of our linear analysis and after a certain timescale during whichthe Misner–Sharp mass has already become substantially large in an exponential fashion.Indeed, assuming that the regularization scale (cid:96) satisfies the condition (cid:96) (cid:28) m , (as wouldbe the case if, for instance, (cid:96) is given by the Planck length), and substituting (37) into (34)we see that M + grows exponentially up to the critical value M + ( v, R ( v )) ≈ | κ | (2 m (cid:96) + r ) r (cid:96) v p +10 pβ ∼ m (cid:96) v p +10 pβ , (40)where we have used r ∼ | κ | − ∼ (cid:96). (41)Eq. (40) implies M + ( v, R ( v )) m ∼ m (cid:96) v p +10 pβ (cid:29) . (42)So the transition between exponential and polynomial regimes happens when the backreac-tion of the perturbation on the background geometry is already very large due to an initialphase of exponential growth of the Misner–Sharp mass, and at this stage we expect nonlin-ear effects to become relevant and supersede our linear treatment. It is also important tokeep in mind that, in any case, M + ( v, R ( v )) is always divergent.Let us now pause for a brief moment to understand why reference [18] fails to capture thedivergent nature of M + ( v, R ( v )) that we have described above. As previously mentioned, inderiving eq. (13) of reference [18], the condition r = R ( v ) was imposed inside the derivativesof eq. (25). That is, the partial derivative is effectively replaced with a total derivative.Therefore, eq. (27) is (incorrectly) modified into dM + ( v, R ( v )) dv = f + f − (cid:32) ∂M − ( v, r ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) + ∂M − ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) dR ( v ) dv (cid:33) = f + f − (cid:32) ∂M − ( v, r ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) + f − ∂M − ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) (cid:33) . (43)In order to extract the leading behavior encapsulated in this equation let us note that thedefinition of the surface gravity of the inner horizon, ∂ [ M − ( v, r ) /r ] ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) = | κ | + O (cid:0) v − p (cid:1) , (44)9mplies that ∂M − ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) = M − ( v, R ( v )) r + r | κ | + O (cid:0) v − p (cid:1) . (45)Using the relation above, together with eqs. (28) and (29), we can therefore write ∂M − ( v, r ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) + f − ∂M − ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) = ∂M − ( v, r ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) − r | κ | ∂M − ( v, r ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) (cid:26) M − ( v, R ( v )) r + r | κ | + O (cid:0) v − p (cid:1)(cid:27) = − r | κ | ∂M − ( v, r ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) (cid:8) M − ( v, R ( v )) + O (cid:0) v − p (cid:1)(cid:9) . (46)This implies that the leading behavior of the incorrect expression (43) is encapsulated inthe equation dM + ( v, R ( v )) dv (cid:39) M − ( v, R ( v ))2 r f + = M − ( v, R ( v ))2 r − M − ( v, R ( v )) r M + ( v, R ( v )) . (47)This differential equation does not have growing exponential solutions, but, due to the signof the second term, decaying ones. In fact, its general solution at leading order is M + ( v, R ( v )) (cid:39) r c e − M v/r , (48)where we have defined the constant M = M − ( m , r ). In particular, M + ( v, R ( v )) tends tothe finite value r / v (cid:29) r /M . Finally, let us note, as a further evidence of somethingwrong with eq. (43), that, exactly for the aforementioned reasons (and also due to thefact that for the discussion above to work it is not necessary to select an ansatz for theMisner–Sharp mass M ( m, r )), it also fails to reproduce the well known instability of theinner horizon of a standard Reissner–Nordstr¨om black hole. IV. ROLE OF THE COSMOLOGICAL CONSTANT
In the discussion so far we have ignored the role of the cosmological constant. There isa very simple and intuitive reason behind this decision. The instability we have discussedoriginates close to the inner horizon, where the energy density is Planckian and the role ofthe cosmological constant is completely negligible. However, in the presence of a non-zerocosmological constant, the late time behavior of the perturbation is no longer described byPrice’s law (4), but has an exponential decay [23, 24] characterized by m − ( v ) = m + δm ( v ) = m − αe − ω I v , (49)where ω I > δR ( v ) = c e −| κ | v + Aαr ( | κ | − ω I ) e − ω I v (50)where c is an integration constant. This modifies the rate at which f − approaches zerowhich, let us recall, due to eq. (10) is given by the derivative of the expression right above.Eq. (27) can now be written as dM + ( v, R ( v )) dv (cid:39) f + Aαω I e − ω I v − | κ | c e −| κ | v − ω I Aαe − ω I v /r ( | κ | − ω I ) + f − ∂M + ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) . (51)We have to analyze the cases | κ | > ω I and | κ | < ω I independently. For | κ | > ω I we have dM + ( v, R ( v )) dv (cid:39) r ( | κ | − ω I ) (2 M + ( v, R ( v )) /r − f − ∂M + ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) . (52)If ∂ r M + is linear in M + the last term can be neglected leading to M + ( v, R ( v )) ∝ e ( | κ |− ω I ) v . (53)This result has a simple physical interpretation. We have the same blueshift factor e | κ | v thatwe had in the absence of the cosmological constant, while the polynomial decay is replacedby the exponential decay described in equation (49).Let us now study the case in which ∂ r M + is polynomial in M + . We have dM + ( v, R ( v )) dv (cid:39) ( | κ | − ω I ) M + ( v, R ( v )) − γe − ω I v M n + ( v, R ( v )) − r ( | κ | − ω I ) . (54)The late-time behavior of the solution of this differential equation describes an exponentialinstability M + ( v, R ( v )) ∝ e ω I v/ ( n − . (55)Finally, for | κ | < ω I we have dM + ( v, R ( v )) dv (cid:39) (2 M + ( v, R ( v )) /r − | κ | Aαe − ( ω I −| κ | ) v − | κ | e −| κ | v ∂M + ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) . (56)The exact form of the solution depends on the specific choice of the metric. However, theasymptotic behavior is generically regular (that is, the Misner–Sharp mass does not divergeasymptotically). The only scenario left to consider is the special case | κ | = ω I . While thisscenario requires some fine tuning to be realized, let us study it for the sake of completeness.The general solution in eq. (12) now reads δR = c e −| κ | v + αr e −| κ | v v, (57)11eading to dM + ( v, R ( v )) dv ≈ Aαv M + ( v, R ( v )) − Aαr v + f − ∂M + ( v, r ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = R ( v ) (58)If ∂ r M + is linear in M + we get M + ( v, R ( v )) ∝ v αA . (59)Therefore, in this case M + ( v, R ( v )) exhibits a polynomial divergence. V. NUMERICAL ANALYSIS
In this section we provide some explicit numerical solutions of the equations discussedabove for some particular choices of the function f ( v, r ). We can use two equivalent methodsfor the numerical integrations below. The first one is solving eqs. (10) and (25) for thevariables R ( v ) and m + ( v ). The other method considers instead the variables R ( v ) and M + ( r, R ( v )), solving the system of differential equations given by eqs. (10) and (27). Themain advantage of the second method is that it uses the Misner–Sharp mass as one of thevariables, while the function m + ( v ) used in the first method has no clear physical meaning.We will nevertheless use the first method in order to highlight some of the issues that canarise in specific examples due to this choice of variables. Reissner–Nordstr¨om black hole
In order to connect our discussion with Ori’s original work, let us start by analyzing thecase of a Reissner-Nordstr¨om black hole. The metric function reads f ± ( v, r ) = 1 − m ± ( v ) r + e r . (60)Then, eq. (25) becomes m (cid:48) + ( v ) R − R m + ( v ) + q = m (cid:48)− ( v ) R − R m − ( v ) + q . (61)The asymptotic time evolution of the mass parameter in the past of the shell is determinedby Price’s law (4). Eq. (61) can now be solved numerically. The results of the numericalintegration are plotted in fig. 2. As expected, the system develops a mass inflation instabilityand the Misner–Sharp quasilocal mass diverges exponentially. Regular black hole: Hayward
Having reproduced Ori’s results for the case of a Reissner–Norstr¨om black hole, we are nowconfident in extending the computation for different choices of regular black hole geometries.12 igure 2: Left: Numerical evolution of the mass parameter m + and of the metric functions f ± ofa Reissner–Nordstr¨om black hole. Right: Comparison between the numerical and the analyticalevolution of the Misner–Sharp mass M + . In both plots we have used the parameters β = 1, p = 12, m = 10, q = 5 with the initial conditions R ( v = 1) = 5 and m + ( v = 1) = m + 1. The first metric that we are going to analyze is the Hayward regular black hole metricintroduced in [19], f ± ( v, r ) = 1 − m ± ( v ) r r + 2 m ± ( v ) (cid:96) . (62)The analog of eq. (61) now reads (notice the difference with respect to the incorrect eq. (13)in [18]) m (cid:48) + ( v )(2 (cid:96) m + ( v ) + R ) (2 ( (cid:96) − R ) m + ( v ) + R ) = m (cid:48)− ( v )(2 (cid:96) m − ( v ) + R ) (2 ( (cid:96) − R ) m − ( v ) + R )(63)We now impose Price’s law (4) for m − ( v ), and solve numerically for m + ( v ). The result ofthe integration is shown in fig. 3. The mass parameter m + ( v ) has a vertical asymptote and,therefore, diverges for finite value of v . This mathematical divergence does not have anyphysical implications as the Misner–Sharp mass stays finite. After the asymptote, m + ( v )changes sign and its late time limit is given bylim v → + ∞ m + ( v ) = − r (cid:96) . (64)As a consequence, we can see from eq. (62) that the Misner–Sharp mass diverges forlarge values of v . However, as shown in the right panel of fig. 3 this divergence is polynomialrather than exponential. As discussed above, an alternative method that can be used toobtain the same results without dealing with the (unphysical) divergence of m + ( v ) at finite v is changing the variables used for the numerical integration to the physical variables R ( v )and the Misner–Sharp mass M + ( r, R ( v )), and solving the system of differential equationsgiven by eqs. (10) and (27). This leads directly to the right plot in fig. 3.13 igure 3: Left: Numerical evolution of the mass parameter m + and of the metric functions f ± of aHayward regular black hole. Right: Numerical evolution of the Misner–Sharp mass. In both plotswe picked the parameters β = 1, p = 12, m = 10, (cid:96) = 0 . R ( v = 1) = 5and m + ( v = 1) = m + 1. Regular black hole: Bardeen
As our next example, let us analyze what happens if we consider the Bardeen metric f ± = 1 − m ± ( v ) r ( r + (cid:96) ) / . (65)With this choice we obtain m (cid:48) + ( v )( (cid:96) + R ) / − R m + ( v ) = m (cid:48)− ( v )( (cid:96) + R ) / − R m − ( v ) (66)As before, we numerically integrate the system of equations for m + and R , and we reportthe results in fig. 4. Once again, the numerical integration is in perfect agreement with theanalytic result: in this case, the mass parameter m + ( v ) diverges exponentially leading to anexponential divergence of the Misner–Sharp mass M + ( v, R ( v )). It is important to remarkthat the different behaviors of m + ( v ), a variable that has no direct physical meaning, inthe Hayward and Bardeen cases is simply caused by the different functional forms of theMisner–Sharp M ( m, r ) as a function of m ( v ). Indeed, in the Hayward case m ( v ) appearsboth in the numerator and denominator of M ( m, r ), while in the Bardeen case m ( v ) onlyappears in the numerator. The Misner–Sharp mass M + ( v, R ( v )), which has a clear physicalmeaning, displays a divergent behavior in both cases regardless of this difference; in terms ofthis quantity, the difference between both cases manifests in its different rate of divergence. Regular black hole: Dymnikova
Next, let us consider a Dymnikova black hole [25] f ± = 1 − m ± (cid:16) − e − r /(cid:96) m ± (cid:17) r . (67)14 igure 4: Left: Numerical evolution of the mass parameter m + and of the metric functions f ± of aBardeen regular black hole. Right: Numerical evolution of the Misner–Sharp mass. In both plotswe picked the parameters β = 1, p = 12, m = 10, (cid:96) = 1 with the initial conditions R ( v = 1) = 5and m + ( v = 1) = m + 1.Figure 5: Left: Numerical evolution of the mass parameter m + and of the metric functions f ± of a Dymnikova regular black hole. Right: Numerical evolution of the Misner–Sharp mass. Inboth plots we picked the parameters β = 1, p = 12, m = 10, (cid:96) = 0 . R ( v = 1) = 5 and m + ( v = 1) = m + 1. This is a particularly interesting regular black hole metric to consider as the derivative of theMisner–Sharp mass with respect to the radial coordinate is not a polynomial. Therefore,we cannot apply the reasoning of the previous section to analytically determine the latetime behavior of M + ( r, R ( v )). As depicted in fig. (5), the numerical integration shows anexponential divergence also for this geometry. Regular black hole: Cosmological constant
Finally, let us consider the presence of a non-zero cosmological constant. For this examplewe consider again the Hayward metric (62) and we assume Price’s law (4) up to a given15 igure 6: Comparison of the evolution of the Misner–Sharp mass for a Hayward regular black hole,with and without a non-zero cosmological constant. In both plots we picked the parameters β = 1, p = 12, m = 10, (cid:96) = 0 . R ( v = 1) = 5, and m + ( v = 1) = m + 1, leading to a surface gravity | κ | ≈ .
8. Left: ω I = 1 < | κ | . Right: ω I = 3 > | κ | time v , and an exponential decay after v . The parameter α in eq. (49) is picked so that m − is continuous, while we have considered two choices of ω I , one for which | κ | > ω I , andone for which | κ | < ω I . As shown in fig. 6 the numerical integration agrees with what weexpect from the analysis of sec. IV. For | κ | > ω I the quasilocal mass grows exponentiallywhile, in the case | κ | < ω I after an exponential phase, the Misner–Sharp mass saturates toa constant value. VI. CONCLUSIONS
In this paper, we have analyzed the stability of inner horizons in both singular and regularblack holes by extending Ori’s model to a generic spherically symmetric spacetime. We havereproduced Ori’s results and recovered an exponential mass inflation in the case of Reissner–Norstr¨om black holes, but we have also extended our analysis to several families of regularblack hole geometries, showing that mass inflation at the inner horizon is a robust and verygeneral prediction.Our findings contradict the conclusions of [18], the discrepancy being due to a technicalflaw in the derivation therein. In particular, while in reference [18] the Misner–Sharp masswas found to reach finite values asymptotically in time, we have shown that, once thistechnical flaw is accounted for, the Misner–Sharp mass becomes divergent for the all thecases analyzed there.A novel result of our analysis is that the late-time instability can become polynomialinstead of exponential. In fact, this happens for well-known regular black hole metrics suchas the Hayward metric. However, this regime is always preceded by a period of exponentialgrowth, and we quantified that the onset of the polynomial instability likely occurs when thelinear approximation has already broken down. In all the cases that we have considered, there16s always an initial phase in which mass inflation proceeds exponentially, and which persistsfor large enough times so that the Misner–Sharp mass has already grown considerably.Therefore, the polynomial regime is unlikely to be physically relevant. However, at themoment we cannot completely discard the possibility that it might be possible to cook upa metric for which this transition happens at very early stages, so that the transition tothe polynomial regime takes place before the initial exponential growth of the Misner–Sharpbecomes significant. The slower polynomial rate of growth would therefore make the innerhorizon stable for a longer time.Similar conclusions apply when a nonvanishing cosmological constant is allowed. While inprinciple a cosmological constant could stabilize the inner horizon, it requires the imaginarypart of the least damped quasinormal mode to be larger than the surface gravity of the innerhorizon for every type of perturbation and for every choice of the mass. This is definitely notthe case for the Reissner–Norstr¨om–de Sitter case [26–29]. Furthermore, even if this condi-tion is realized, the regularization of the instability will only arise after a very long transientin which the curvature invariants grow exponentially. Hence, even if in these hypotheticalscenarios the Misner–Sharp mass would be finite, it would reach substantially large valuesthat make imprescindible to consider the backreaction due to the initial exponential phaseof mass inflation.Finally, let us stress that the instability of the inner horizon does not necessarily imply theformation of a singularity. We have proved that a small perturbation has a huge effect on thegeometry and leads to the unbounded growth of the Misner–Sharp mass. While in generalrelativity this usually implies the formation of a singularity, in a full theory of quantumgravity, we might expect the backreaction (perhaps dominated by semiclassical effects [30])to drive the geometry to a different class of non-singular spacetime [12]. However, in order toaddress this problem, a geometrical analysis cannot be enough, and we would need to specifythe dynamical field equations of a specific theory. Possible quantum gravity frameworksto understand this issue are given by asymptotic safety and loop quantum gravity. Infact, within these theories, there are works predicting regular black holes with an innerhorizon [31–36]. These works are mathematically self-consistent as they only deal withvacuum spacetimes, showing that quantum gravity effects can regularize the singularity.However, we have shown in this paper that these analyses are incomplete as the presence ofperturbations cannot be ignored. Taking into account the perturbations in a consistent waywould be a very demanding but essential computation.
Acknowledgments
FDF acknowledges financial support by Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 17H06359SL acknowledges funding from the Italian Ministry of Education and Scientific Research(MIUR) under the grant PRIN MIUR 2017-MB8AEZ.CP acknowledges: financial support provided under the European Union’s H2020 ERC,Starting Grant agreement no. DarkGRA–757480; support under the MIUR PRIN and FAREprogrammes (GW-NEXT, CUP: B84I20000100001); support from the Amaldi Research Cen-17er funded by the MIUR program “Dipartimento di Eccellenza” (CUP: B81I18001170001).MV was supported by the Marsden Fund, via a grant administered by the Royal Society ofNew Zealand. [1] R. Penrose, “Gravitational collapse and space-time singularities,”
Phys. Rev. Lett. (1965)57–59.[2] S. Hawking, “The occurrence of singularities in cosmology. III. Causality and singularities,” Proc. Roy. Soc. Lond. A
A300 (1967) 187–201.[3] S. W. Hawking and R. Penrose, “The Singularities of gravitational collapse and cosmology,”
Proc. Roy. Soc. Lond.
A314 (1970) 529–548.[4]
Virgo, LIGO Scientific
Collaboration, B. P. Abbott et al. , “Observation of GravitationalWaves from a Binary Black Hole Merger,”
Phys. Rev. Lett. no. 6, (2016) 061102, arXiv:1602.03837 [gr-qc] .[5]
Virgo, LIGO Scientific
Collaboration, B. P. Abbott et al. , “GW151226: Observation ofGravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence,”
Phys. Rev. Lett. no. 24, (2016) 241103, arXiv:1606.04855 [gr-qc] .[6]
VIRGO, LIGO Scientific
Collaboration, B. P. Abbott et al. , “GW170104: Observation ofa 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2”
Phys. Rev. Lett. no. 22,(2017) 221101, arXiv:1706.01812 [gr-qc] .[7]
Virgo, LIGO Scientific
Collaboration, B. P. Abbott et al. , “GW170814: AThree-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence,”
Phys. Rev. Lett. no. 14, (2017) 141101, arXiv:1709.09660 [gr-qc] .[8]
Virgo, LIGO Scientific
Collaboration, B. Abbott et al. , “GW170817: Observation ofGravitational Waves from a Binary Neutron Star Inspiral,”
Phys. Rev. Lett. no. 16,(2017) 161101, arXiv:1710.05832 [gr-qc] .[9] R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Visser, “Phenomenological aspects ofblack holes beyond general relativity,”
Phys. Rev. D no. 12, (2018) 124009, arXiv:1809.08238 [gr-qc] .[10] V. Cardoso and P. Pani, “Testing the nature of dark compact objects: a status report,” Living Rev. Rel. no. 1, (2019) 4, arXiv:1904.05363 [gr-qc] .[11] R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Visser, “Opening the Pandora’s box atthe core of black holes,” Class. Quant. Grav. no. 14, (2020) 145005, arXiv:1908.03261[gr-qc] .[12] R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Visser, “Geodesically complete blackholes,” Phys. Rev. D (2020) 084047, arXiv:1911.11200 [gr-qc] .[13] E. Poisson and W. Israel, “Inner-horizon instability and mass inflation in black holes,”
Phys.Rev. Lett. (1989) 1663–1666.[14] A. Ori, “Inner structure of a charged black hole: An exact mass-inflation solution,” Phys.Rev. Lett. (1991) 789–792.[15] A. J. Hamilton and P. P. Avelino, “The Physics of the relativistic counter-streaming nstability that drives mass inflation inside black holes,” Phys. Rept. (2010) 1–32, arXiv:0811.1926 [gr-qc] .[16] E. G. Brown, R. B. Mann, and L. Modesto, “Mass Inflation in the Loop Black Hole,”
Phys.Rev. D (2011) 104041, arXiv:1104.3126 [gr-qc] .[17] R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, and M. Visser, “On the viability ofregular black holes,” JHEP (2018) 023, arXiv:1805.02675 [gr-qc] .[18] A. Bonanno, A.-P. Khosravi, and F. Saueressig, “Regular black holes have stable cores,” arXiv:2010.04226v1 [gr-qc] .[19] S. A. Hayward, “Formation and evaporation of regular black holes,” Phys. Rev. Lett. (2006) 031103, arXiv:gr-qc/0506126 [gr-qc] .[20] C. Barrab`es and W. Israel, “Thin shells in general relativity and cosmology: The lightlikelimit,” Phys. Rev. D (Feb, 1991) 1129–1142. https://link.aps.org/doi/10.1103/PhysRevD.43.1129 .[21] R. H. Price, “Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar andgravitational perturbations,” Phys. Rev. D5 (1972) 2419–2438.[22] R. H. Price, “Nonspherical Perturbations of Relativistic Gravitational Collapse. II.Integer-Spin, Zero-Rest-Mass Fields,” Phys. Rev. D5 (1972) 2439–2454.[23] A. S. Barreto and M. Zworski, “Distribution of resonances for spherical black holes,” Mathematical Research Letters no. 1, (1997) 103–121.[24] S. Dyatlov, “Asymptotic distribution of quasi-normal modes for kerr–de sitter black holes,”in Annales Henri Poincar´e , vol. 13, pp. 1101–1166, Springer. 2012.[25] I. Dymnikova, “Vacuum nonsingular black hole,”
Gen. Rel. Grav. (1992) 235–242.[26] V. Cardoso, J. a. L. Costa, K. Destounis, P. Hintz, and A. Jansen, “Quasinormal modes andstrong cosmic censorship,” Phys. Rev. Lett. (Jan, 2018) 031103. https://link.aps.org/doi/10.1103/PhysRevLett.120.031103 .[27] O. J. Dias, F. C. Eperon, H. S. Reall, and J. E. Santos, “Strong cosmic censorship in deSitter space,”
Phys. Rev. D no. 10, (2018) 104060, arXiv:1801.09694 [gr-qc] .[28] S. Hod, “Strong cosmic censorship in charged black-hole spacetimes: As strong as ever,” Nucl. Phys.
B941 (2019) 636–645, arXiv:1801.07261 [gr-qc] .[29] V. Cardoso, J. L. Costa, K. Destounis, P. Hintz, and A. Jansen, “Strong cosmic censorshipin charged black-hole spacetimes: still subtle,”
Phys. Rev.
D98 no. 10, (2018) 104007, arXiv:1808.03631 [gr-qc] .[30] C. Barcel´o, V. Boyanov, R. Carballo-Rubio, and L. J. Garay, “Black hole inner horizonevaporation in semiclassical gravity,” arXiv:2011.07331 [gr-qc] .[31] A. Bonanno and M. Reuter, “Renormalization group improved black hole spacetimes,”
Phys.Rev. D (Jul, 2000) 043008. https://link.aps.org/doi/10.1103/PhysRevD.62.043008 .[32] A. Platania, “Dynamical renormalization of black-hole spacetimes,” Eur. Phys. J.
C79 no. 6, (2019) 470, arXiv:1903.10411 [gr-qc] .[33] A. Held, R. Gold, and A. Eichhorn, “Asymptotic safety casts its shadow,”
JCAP (2019)029, arXiv:1904.07133 [gr-qc] .[34] L. Modesto, “Semiclassical loop quantum black hole,” Int. J. Theor. Phys. (2010)1649–1683, arXiv:0811.2196 [gr-qc] .
35] E. Alesci and L. Modesto, “Particle Creation by Loop Black Holes,”
Gen. Rel. Grav. (2014) 1656, arXiv:1101.5792 [gr-qc] .[36] C. Rovelli and F. Vidotto, “Planck stars,” Int. J. Mod. Phys.
D23 no. 12, (2014) 1442026, arXiv:1401.6562 [gr-qc] ..