A New Look at the $C^{0}$-formulation of the Strong Cosmic Censorship Conjecture

We examine the $C^{0}$-formulation of the strong cosmic censorship conjecture (SCC) from a quantum complexity-theoretic perspective and argue that for generic black hole parameters as initial conditions for the Einstein equations, corresponding to the expected geometry of a hyperbolic black hole, the metric is $C^{0}$-extendable to a larger Lorentzian manifold across the Cauchy horizon. To demonstrate the pathologies associated with a hypothetical validity of the $C^{0}$ SCC, we prove it violates the"complexity=volume"conjecture for a low-temperature hyperbolic AdS$_{d+1}$ black hole dual to a CFT living on a ($d-1$)-dimensional hyperboloid $H_{d-1}$, where in order to preserve the gauge/gravity duality we impose a lower bound on the interior metric extendability of order the classical recurrence time.


I. INTRODUCTION
Black holes have large interiors [1,2]. The interior volume of a black hole of mass M is ∼ M 5 , while its lifespan is ∼ M 3 . As was shown in [1] for the case of a oneparameter static spherically-symmetric solution to the Einstein vacuum equations R µν = 0 (i.e. Schwarzschild black hole), even though the horizon area is time translation invariant for an asymptotic observer, its interior volume, defined in a geometrically covariant way as the maximal spacelike hypersurface bounded by a two-sphere foliating the horizon, is dynamical, and keeps growing. Where according to classical general relativity, the interior volume continues to grow unbounded. For the simplest case of a Schwarzschild metric, expressed in advanced Eddington-Finkelstein coordinates 2M r dv 2 +2dvdr +r 2 sin 2 θdφ 2 +dθ 2 (1) where v = t + dr 1−2M/r , and t is Schwarzschild time, the interior volume grows linearly in the advanced time v, where up to a prefactor of 3 √ 3π it reads * Electronic address: aditya.iyer@physics.ox.ac.uk † Electronic address: alexanderyyosifov@gmail.com ‡ Electronic address: vlatko.vedral@physics.ox.ac.uk The maximum of Eq. (2) is obtained by extremizing the integral which maximizes at r max = 3M/2 and S 2 denotes the two-sphere foliating the horizon. This non-trivial relation between horizon area and interior volume was further examined in Ref. [3] where it was argued that for a Schwarzschild black hole the volume of the maximal bounded spacelike hypersurface behind the horizon continues to grow until the black hole reaches the Planck scale. Without loss of generality, the results were expanded [4] for the rotating a = 0 subextremal |a| < M two-parameter family of stationary axisymmetric solutions in asymptotically flat spacetime (i.e. Kerr black hole). Where assuming M 1, and the metric curvature is weak at both r = 2M and r = 3M/2, then similar reasoning to that in a = 0 Schwarzschild geometry was shown to be valid. Namely, for a subextremal Kerr black hole the leading contribution to the interior volume is due to the maximal Cauchy development of initial data defined on a spacelike hypersurface which lies just barely within the black hole. This growth of the interior region was later quantified in arXiv:2206.08716v1 [gr-qc] 17 Jun 2022 the context of AdS/CFT, where the volume of a black hole in d-dimensional asymptotically AdS spacetime was conjectured to be dual to the complexity of a thermal state CFT living on the (d − 1)-dimensional AdS boundary, hence "complexity=volume" (CV). 1 Lately, extendability of the interior metric of dynamical black holes with inner horizon has attracted much attention as it has been demonstrated [6,[9][10][11] that for a maximal Cauchy development of suitable initial data, considered on a spacelike hypersurface Σ inside a subextremal Kerr (or Reissner-Nordstrom) black hole, the metric is C 0 -stable, meaning it is extendable to a larger Lorentzian manifoldM \ M across a non-trivial part of the Cauchy horizon. Hence, invalidating the C 0 form of Penrose's celebrated SCC. In the main part of this paper, building on Ref. [6], we revisit the C 0 -formulation of the SCC from a quantum complexity-theoretic perspective. We investigate the C 0 SCC in the low-temperature limit of a hyperbolic AdS d+1 black hole dual to a CFT living on a (d − 1)-dimensional hyperboloid to demonstrate that if the metric terminates at the Cauchy horizon, then the CV-duality (i.e. gauge/gravity correspondence) will be violated at late times. In light of this, we propose a novel conjecture relating the C 0 -stability of the metric to the complexification of the system, and suggest the extendability of the black hole interior is bounded from below by the classical recurrence time. The paper is organized as follows. In Section II we review the Kerr geometry, and focus on the role of the Cauchy horizon. We then look at the C 0 -formulation of Penrose's SCC and its implications for the interior Kerr black hole region. In Section III we examine the implications of the C 0 -instability of the metric for Kerr-AdS spacetime. Section IV is dedicated to the study of the inconsistencies in the bulk/boundary duality yielded by a hypothetical inextendability of the interior metric beyond the Cauchy horizon.

II. THE KERR METRIC AND
C 0 -INEXTENDABILITY The subextremal |a| < M Kerr family of solutions describes a metric whose line element, in Eddington-Finkelstein coordinates, reads where 1 Interestingly, it was then argued in Ref. [5] that complexity, similar to entropy, admits a Second law.
Here, a = J/M is the rotation parameter which gives the spin-to-mass ration of the black hole, where the extremal case corresponds to a ∼ O(1). The metric admits two horizons, respectively at r ± = M ± √ M 2 − a 2 , where r − :={r = CH + } is the inner (Cauchy) horizon and r + :={r = H + } is the outer (event) horizon. 2 Namely the Cauchy horizon will be the focus of the rest of the paper. In Kerr geometry the maximal Cauchy development of generic asymptotically flat initial data, defined on a spacelike hypersurface Σ within the black hole (i.e. the non-empty complement of M \ J − (I + )), is non-uniquely extendable as a solution to the vacuum Einstein equations. This means lack of future determinism for all infalling timelike geodesics γ which reachM \ M. Thus, the region beyond the Cauchy horizon CH + cannot be uniquely determined, implying a breakdown of global hyperbolicity of the metric, as predicted by classical general relativity.
Upon maximal evolution of the initial data on the spacelike hypersurface Σ to a larger Lorentzian manifoldM \ M, Σ will no longer be the Cauchy hypersurface forM which extends across CH + . This globally non-hyperbolic future development of the initial data found in Kerr (and also Reissner-Nordstrom) solutions is in sharp contrast to the simple one-parameter Schwarzschild subfamily of solutions, where the maximal Cauchy evolution is future inextendable as a C 0 Lorentzian manifold across CH + . In Schwarzschild the future is uniquely determined, namely for r < r + all timelike geodesics γ are incomplete, and asymptote to a spacelike boundary in finite proper time.
In an attempt to preserve the global hyperbolicity of the metric Penrose proposed [7] the so-called strong cosmic censorship conjecture which, generally, reads Conjecture 1 (the C 0 -form of the SCC): The maximal Cauchy evolution of generic asymptotically flat (AdS) vacuum initial data is inextendable to a larger C 0 Lorentzian manifoldM across the Cauchy horizon CH + .
The role of the C 0 -formulation of the SCC is to bring uniqueness to the solutions of the vacuum Einstein equations, and hence make the future of all infalling timelike geodesics deterministic. The C 0 -inextendability conjecture restores the predictability of the vacuum equations inside dynamical black holes by introducing a spacelike finite boundary across which the metric cannot be extended even as a (continuous) C 0 .
Recent advancements, however, see Refs. [6,8], suggest Conjecture 1 is false, where the main results relevant to us can be summarized by the following theorem Theorem 1 (Dafermos-Luk): Consider general vacuum initial data corresponding to the expected induced geometry of a dynamical black hole settling down to Kerr (with parameters 0 < |a| < M ) on a suitable spacelike hypersurface Σ 0 in the black hole interior. Then the maximal future development of the spacetime (M, g) corresponding to Σ 0 is globally covered by a double null foliation and has a non-trivial Cauchy horizon CH + across which the metric is continuously extendable.
The theorem suggests that general exterior subextremal spacetimes, emerging from initial data sufficiently close to Kerr data, have C 0 -stable non-trivial Cauchy horizons in their interiors. This, together with the global existence statement in the exterior (weak cosmic censorship), are already a strong argument against Conjecture 1. In fact, the weak cosmic censorship (i.e. the completeness of null infinity I + ) makes an even stronger statement as it implies the entire Penrose diagram is nonlinearly stable for solutions to R µν = 0. So given stability in the Kerr exterior J − (I + ), we can infer similar properties to hold in the non-empty complement of M \ J − (I + ) as well. Moreover, C 0 -stability results of CH + have also been derived for a broader class of systems, e.g. for Reissner-Nordstrom solutions to the Einstein-Maxwell-real-scalarfield equations [9][10][11], while in Ref. [12], assuming polynomial decay rate, for suitable initial data of the Einstein-Maxwell-Klein-Gordon equations in a Reissner-Nordstrom background, the spacetime was shown to be continuously extendable. 3 On a linear level, the C 0 SCC, being the strongest inextendability statement, has been shown not to admit a blow up at CH + even for the simpler Kerr or Reissner-Nordstrom solutions in asymptotically flat spacetime, given that the polynomially decaying scalar waves solving the Klein-Gordon equation propagate to the black hole interior. Similarly, boundedness at CH + has also been demonstrated [13] for Kerr (and Reissner-Nordstrom)-dS spacetime with the decay rate here being exponential. Despite the growing interest in the subject of spacetime extendability in black hole interiors, all C 0 SCC formulations rely on classical geometric arguments and scalar wave decay rate estimations, making the SCC one of the few big problems in gravity yet to receive the quantum treatment. As such, Sections III and IV constitute the first application of quantum complexity in this context.

III. QUANTUM COMPLEXITY AND THE
C 0 -STABILITY We begin this Section with a brief review of the CVduality and outline a newly-developed mechanism for measuring it in terms of the flux of a conserved volume current. Then, we discuss what the C 0 -formulation of the SCC (Conjecture 1) implies for the interior of a Kerr-AdS black hole. Although some of the original results were derived for the case of an eternal two-sided Schwarzschild-AdS black hole, Fig. 1, they have been shown to be valid for a large class of solutions.

A. Quantum Complexity
Quantum complexity increases linearly for time exponential in the entropy t max ∼ e S , known as the classical recurrence time. At this point it saturates and remains at its maximum C max = e S for t qr ∼ e e S , known as the quantum recurrence time, and then fluctuates back to its initial low value [14]. The CV-duality [15] suggests the complexity of the boundary CFT is dual to the volume of the maximal spacelike hypersurface (i.e. codimension-one submanifold) which asymptotes to some time, anchored at the conformal AdS boundary, see Fig. 1

C ∼
where l is the characteristic AdS length scale, = h/2π is the reduced Planck constant, and G N denotes Newton's constant, where we assume c = = G N = 1. The CV-duality has been shown to hold for a wide class of theories and in arbitrary number of dimensions. Its validity has been demonstrated for the simplified setting of a (1+1)-dimensional Reissner-Nordstrom background in JT gravity [16], as well as for a d = 3 spherically symmetric BTZ-AdS black hole [17,18]. Recently, a novel way for calculating the volume of maximal spacelike hypersurfaces in the bulk in terms of the flux of the volume current was suggested [19] where is the spacetime volume element, and v denotes the unit timelike vector field orthogonal to the t = constant hypersurfaces anchored at the boundary. The way we think about t in this context is in terms of asymptotic time, i.e. null coordinate which is connected to the Minkowski polar coordinates as t = r + τ at past infinity, where τ denotes coordinate time, and r is the radial coordinate [1]. Here, V (Σ t ) is simply the flux of the volume flow through the hypersurface Σ at boundary time t. The volume current method is valid under the assumption that the whole asymptotically AdS spacetime is foliated (without gaps) by a family of maximal spacelike hypersurfaces (i.e. Cauchy slices Σ ti ) induced by the boundary foliation. As is well known, the CV-duality is intrinsically ill-defined and diverges at asymptotic infinity. This divergence, however, can easily be dealt with by (i) counting only the volume growth within the black hole interior, and (ii) focusing on the volume's time dependence. Following this prescription, one can formulate a pathology-free Second law of complexity and demonstrate that by calculating the volume change between neighboring Cauchy slices behind the horizon in terms of the flow of the volume current, 4 as we go to later slices, the interior volume can only grow, bounded from above by the complexity of the holographic CFT.

B. C 0 -instability: What if ?
In light of this, the question we address is: What would a hypothetical validity of Conjecture 1 imply from a quantum complexity-theoretic perspective? As is well known, the inextendibility of the metric to a larger Lorentzian manifoldM beyond CH + even as a 4 Note that in the black hole interior r < r + the volume flow is a future-pointing timelike vector, orthogonal to the maximal foliation.  Fig. 1 in [20]) of a maximally-extended two-sided Kerr-AdS black hole, depicting CH + as a migrated singularity. Here, tmax denotes the boundary foliation corresponding to CH + , where "max" refers to the latest slice (i.e. CH + ), assuming Conjecture 1 is true and the spacetime terminates. Note that CH + is depicted as a spacelike boundary for illustrative purposes only; in light of its interpretation as a migrated singularity.
continuous C 0 suggests the Cauchy evolution is spacelike. Namely, the metric terminates at a singular finite boundary, meaning the region beyond CH + does not exist. Essentially, this is the statement that timelike geodesics γ cannot reachM \ M. Loosely speaking, the spacelike finite boundary, corresponding to the maximal Cauchy evolution, could effectively be interpreted as a migrated singularity, where the singularity now lies at some larger r, closer to the outer horizon H + . 5 This way, by migrating the spacelike finite boundary to coincide or be arbitrarily close to the maximal slice Σ max and CH + , the Kerr solution obtains the well-defined determinism of the Schwarzschild subfamily. 6 To be more precise, suppose we have a subextremal Kerr-AdS black hole, where we require M > 0, a = 0 and |a| < 1/l where l 2 = −Λ/3 is the AdS curvature radius for negative cosmological constant Λ < 0, and (r, t, θ, φ) ∈ R 2 ×S 2 denote the coordinates which cover the black hole interior and the near-horizon region just outside the black hole. Here, the polynomial vanishes at H + , where H + is a Killing horizon, and M Ξ 2 is the ADM mass. Assuming the weak cosmic censorship, let (M, a) ∈ R 2 >0 be non-degenerate for |a| < 1/l. Then, (10) has 2 real roots for 0 < r − < r + , and the metric has both axial ∂ φ and timelike ∂ t Killing fields whose relation to the Killing vector field is the standard where Ω H + is the angular velocity of H + . Let us now consider Fig. 3. Given CH +7 lies at some macroscopic distance from the ringlike curvature singularity r s , we get 5 In Ref. [21], examining AMPS' firewall argument, Susskind proposed that black holes which have evaporated less than half of their initial entropy retain their regular horizon structures but have non-standard singularities instead. In particular, it was suggested that the growing von Neumann entropy of the Hawking radiation is what drives the singularity to migrate, and at late times coincide with the event horizon. 6 For practical purposes we can assume that Σmax and CH + coincide. 7 Note that the Cauchy horizon is the surface which separates the space-from timelike character of maximal hypersurfaces Σt.
where r c denotes the region between CH + and the ringlike singularity. Then, assuming the validity of Conjecture 1, a non-negligible macroscopic region of the black hole interior will, essentially, be cut off.
As we show in Section IV below, this interior region cut off is very problematic.

IV. PROBING THE C 0 -INSTABILITY
In this Section we study low-temperature black holes with hyperbolic horizons in asymptotically AdS spacetime, dual to CFTs living on hyperboloids. In this setting, we examine the C 0 -form of the SCC from a quantum complexity-theoretic perspective and argue that if Conjecture 1 is true, then the CV-duality would be violated at late times. We demonstrate that the leading contribution to the interior volume comes from a Cauchy slice which lies in the r c region between r s and CH + , see Eq. (12). Given the generic quasi-periodic behavior of quantum complexity, the evolution of the black hole interior will be non-trivially modified. While a macroscopic part of the interior region is cut off, the corresponding complexity of the holographic CFT is expected to continue to evolve until it saturates, hence yielding a deviation from the standard bulk/boundary duality. Therefore, while the complexity of the boundary CFT continues to grow in time, the dual bulk volume will be prematurely saturated. To preserve the CV-duality, we conjecture that the metric must be C 0 -extendable at least for as long as quantum complexity increases.
The gravitational dual of a CFT on a (d − 1)dimensional hyperboloid H d−1 is an AdS d+1 black hole with hyperbolic horizon geometry. We investigate the C 0 -form of the SCC in the low-temperature limit of these solutions which have been shown to exhibit interesting thermodynamic features. Consequently, we demonstrate that inextendability of the interior metric as a C 0 beyond CH + (as suggested by Conjecture 1) violates the CVduality as it leads to O(1) variation in the bulk/boundary correspondence at late times. The hyperbolic AdS d+1 black hole has R 2 × S d−1 topology and is given by the metric where for the hyperbolic CFT metric and where dH 2 d−1 is the unit metric on the H d−1 hyperboloid. Here it is convenient to use the convention that lengths are measured in units of the curvature radius (i.e. l = 1) for simplicity of notation. The black hole's mass M and temperature T are defined, respectively, as and where for M = 0 we get T = (2π) −1 . In this set of coordinates, r 0 is the horizon, and in the near-horizon region f (r) = 4πT (r − r 0 ), while in the asymptotic limit, f (r) → r 2 /l 2 . We are interested in low-temperature T = 0, hyperbolic black holes for which there are solutions that admit regular r − and r + horizons for a family of −M terms. 8 We should note that in this respect the negative mass family is special since not all hyperbolic solutions have welldefined black hole configurations, and some have even been shown to break down at r + . For instance, the M = 0 solution only covers part of the manifold (the outside black hole region), and hence is not useful for our considerations. For T = 0, in the hyperbolic scenario, there are minimum values for −M and r + which are defined, respectively, as where for d = 4 they read Motivated by the CV-duality, the volume of a maximal spacelike hypersurface in T = 0 hyperbolic black hole 8 Interestingly, T = 0 hyperbolic black holes have global structure whose interior region resembles that of Reissner-Nordstrom, and has two well-defined horizons, similar to (8).
background can be estimated by dividing the hypersurface Σ ri into individual segments, and approximating the local maximum of the radial function f (r) (to be defined below) on each segment. Where in the hyperbolic AdS d+1 background (13), for r r min , the metric asymptotes to an empty AdS d+1 solution, while in the near-horizon region, just outside the black hole, the metric is AdS d+1 × H d−1 . In this setting, each spacelike hypersurface decomposes as 9 Σ ti = Σ in ∪ Σ out (20) As is the standard procedure in the literature, the volume of maximal r = constant spacelike hypersurfaces is generally given as where f (r) is a "placeholder," and depends on the region whose contribution to the volume we wish to estimate. For the interior region of (13), f (r) takes the form [22] r d−1 1 − r 2 + M min r d−2 (22) and is maximized in the region r c , Eq. (12). It is important to note that for massive non-extremal black holes the region r c can be arbitrarily big. Therefore, a significant part of the interior volume of low-temperature hyperbolic black holes is contributed by the region r c , i.e. between CH + and the ringlike singularity. The overall complexity of this family of solutions has been demonstrated to evolve as [23] for small t, and at late times as Evidently, the growth rate of complexity is consistent with the classical predictions.
In this scenario we argue that the local maximum of (22) is very problematic from a complexity-theoretic perspective, where in the context of the SCC, complexity is a measure of the extendability of the metric, i.e. the amount of emergent space behind the horizon. 10 On the asymptotic AdS d+1 boundary the complexity of the holographic CFT, living on a H d−1 hyperboloid, as suggested by (23,24), up to small corrections, grows linearly for time exponential in the entropy until the upper bound is saturated. Meanwhile, in the AdS d+1 bulk geometry the dual volume of the interior lowtemperature hyperbolic black hole linearly increases, as dictated by the CV correspondence. Assuming the validity of Conjecture 1, and following the discussion in Section IV, CH + is a spacelike singular boundary, making timelike geodesics future incomplete. 11 That is, the spacetime terminates at CH + and cannot be extended across, making the region beyond it, effectively, non-existent. In light of (22), however, recall that most of the interior volume of low-temperature hyperbolic black holes is sourced by the region between CH + and the singularity at r s , where (22) maximizes. Hence, as time goes on the bulk volume will saturate much earlier than the complexity of the dual CFT, yielding a discrepancy in the bulk/boundary description which will only accumulate at late times. Therefore, we conjecture that in order to preserve the complexity-theoretic gauge/gravity black hole framework, the interior metric must be extendable to a larger Lorentzian manifold across CH + (i. towards a positive resolution of the C 0 SCC for a Kerr-AdS black hole, given the parameters admit Baire genericity. Interestingly, this instability was shown not to result from mere superradiance or blueshift effects but instead from a novel resonance phenomenon (i.e. relation between Diophantine approximation and black hole interiors) found in axisymmetric solutions.

V. CONCLUSIONS
In this work we presented a novel quantum treatment of an old problem concerning metric extendability in black hole interiors with inner horizons. This is reminiscent of the early quantum approach to black holes which led to completely redefining the subject in the 70s. We employed a quantum information framework which has been recently shown to play a key role in gravity and spacetime emergence, and to our knowledge, has never been considered in this context. In particular, we revisited, from a quantum complexitytheoretic perspective, the C 0 formulation of the SCC in the low-temperature limit of a hyperbolic AdS d+1 black hole dual to a CFT living on a (d − 1)-dimensional hyperboloid H d−1 . Our analysis showed that inextendability of the metric across CH + (validity of Conjecture 1) would lead to late-time violation of the CV-duality, as most of the interior black hole volume is sourced from the region beyond CH + . To retain the CV-duality and save the bulk/boundary correspondence, we conjectured imposing a lower bound on metric extendability of order the classical recurrence time.