Abstract
Motivated by the Strong Cosmic Censorship Conjecture for asymptotically Anti-de Sitter (AdS) spacetimes, we initiate the study of massive scalar waves satisfying Box _g psi - mu psi =0 on the interior of AdS black holes. We prescribe initial data on a spacelike hypersurface of a Reissner–Nordström–AdS black hole and impose Dirichlet (reflecting) boundary conditions at infinity. It was known previously that such waves only decay at a sharp logarithmic rate (in contrast to a polynomial rate as in the asymptotically flat regime) in the black hole exterior. In view of this slow decay, the question of uniform boundedness in the black hole interior and continuity at the Cauchy horizon has remained up to now open. We answer this question in the affirmative.
Highlights
We initiate the study of linear waves satisfying gψ − μψ = 0 (1.1)on the interior of asymptotically Anti-de Sitter (AdS) black holes (M, g)
The logarithmic decay is too slow to adapt the mechanism exploited in previous studies of black hole interiors [14,17,26]
Before we address the question of stability of the Cauchy horizon, it is essential to understand the behavior in the exterior region of Kerr–AdS or Reissner–Nordström–AdS
Summary
On the interior of asymptotically Anti-de Sitter (AdS) black holes (M, g). In the context of asymptotically AdS spacetimes it is natural to consider (possibly negative) mass parameters μ satisfying the Breitenlohner–Freedman [6]. The present paper is an attempt to shed some first light on SCC in the asymptotically AdS case: We will show (Theorem 1) that, despite the slow decay on the exterior, boundedness in the interior and continuous extendibility to the Cauchy horizon still holds for solutions of (1.1) on Reissner–Nordström–AdS black holes. For this part we use integrated energy decay estimates for bounded angular momenta established in [40] This superpolynomial decay in the exterior is sufficient so as to follow the method of [26] with vector fields of the form (1.2) to show boundedness and continuity at the Cauchy horizon, up to the additional difficulty caused by the fact that we allow a possibly negative Klein–Gordon mass parameter. Parts of the proof require a separate analysis which are treated in Sects. 4 and 5
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