Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I. The interior of the black hole region

Abstract

This is the first and main paper of a two-part series, in which we prove the C 2 -formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it is known through the works of Dafermos and Dafermos-Rodnianski that the maximal globally hyperbolic future development of any admissible twoended asymptotically flat Cauchy initial data set possesses a non-empty Cauchy horizon, across which the spacetime is C 0 -future-extendible (in particular, the C 0 -formulation of the strong cosmic censorship conjecture is false). Nevertheless, the main conclusion of the present series of papers is that for a generic (in the sense of being open and dense relative to appropriate topologies) class of such data, the spacetime is future-inextendible with a Lorentzian metric of higher regularity (specifically, C 2 ).In this paper, we prove that the solution is C 2 -future-inextendible under the condition that the scalar field obeys an L 2 -averaged polynomial lower bound along each of the event horizons. This, in particular, improves upon a previous result of Dafermos, which required instead a pointwise lower bound. Key to the proof are appropriate stability and instability results in the interior of the black hole region, whose proofs are in turn based on ideas from the work of Dafermos-Luk on the stability of the Kerr Cauchy horizon (without symmetry) and from our previous paper on linear instability of Reissner-Nordström Cauchy horizon. In the second paper of the series [36], which concerns analysis in the exterior of the black hole region, we show that the L 2 -averaged polynomial lower bound needed for the instability result indeed holds for a generic class of admissible two-ended asymptotically flat Cauchy initial data.

The BTZ black hole violates strong cosmic censorship

“…Alternatively, from (3.15b) it is equivalent toT (ω, m) = ∞. 8 It follows from (3.16) that quasinormal frequencies are given by a = −p or a−c+1 = −p for p = 0, 1, 2, · · · . Using the definitions (3.4) for (a, b, c), this yields two sectors of exterior quasinormal mode frequencies [23]:…”

“…If the initial data is perturbed then the resulting spacetime can still be continuously extended beyond a Cauchy horizon [6,5,7]. But, generically, this extension is not C 2 [4,8] and it is believed that, generically, no extension can satisfy Einstein's equation, even in a weak sense, at the Cauchy horizon (see [9] for a discussion). Thus, in the perturbed spacetime, there is strong evidence that the Cauchy horizon is replaced by a curvature singularity and determinism is restored.…”

Strong cosmic censorship for a scalar field in a Born-Infeld–de Sitter black hole

“…Nevertheless, there are some subtleties of describing the extension of the perturbation at the Cauchy horizon, and hence several formulation versions of SCC have been proposed. For example, the perturbation can not be C r (r ∈ N ) smooth at the Cauchy horizon in the C r version of SCC [4,5]. However, since weak solutions can have many important physical applications in which C r smooth solutions are not available, a more appropriate way to characterize the extendibility is whether the perturbation being inextendible as a weak solution of the equations of motion.…”

Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

“…Theorem 6.6. (C 2 stability result for Einstein-Maxwell-(real)-scalar field system, Luk and Oh, [45]). The C 2 -formulation of the Strong Cosmic Censorship Conjecture for the Einstein-Maxwell-(real)-scalar field system in spherical symmetry with 2-ended asymptotically at initial data on R × S 2 is true.…”