Stability and Instability of the Sub-extremal Reissner–Nordström Black Hole Interior for the Einstein–Maxwell–Klein–Gordon Equations in Spherical Symmetry
Authors
Van De Moortel, Maxime
Publication Date
2018-05Journal Title
Communications in Mathematical Physics
ISSN
0010-3616
Publisher
Springer Nature
Volume
360
Issue
1
Pages
103-168
Language
en
Type
Article
This Version
VoR
Metadata
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Van De Moortel, M. (2018). Stability and Instability of the Sub-extremal Reissner–Nordström Black Hole Interior for the Einstein–Maxwell–Klein–Gordon Equations in Spherical Symmetry. Communications in Mathematical Physics, 360 (1), 103-168. https://doi.org/10.1007/s00220-017-3079-3
Abstract
We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole -approaching a sub-extremal Reissner-Nordstr\"om background fast enough at infinity- in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting :
1. Stability : We prove that spherically symmetric characteristic initial data to the Einstein-Maxwell- Klein-Gordon equations approaching a Reissner-Nordstr\"om background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space-time possessing a Cauchy horizon in a neighbourhood of time-like infinity. Moreover if the decay is even stronger, we prove that the spacetime metric admits a continuous extension to the Cauchy horizon. This generalizes the celebrated stability result of Dafermos for Einstein-Maxwell-real-scalar-field in spherical symmetry.
2. Instability : We prove that for the class of space-times considered in the stability part, whose scalar field in addition obeys a polynomial averaged-L^2 (consistent) lower bound on the event horizon, the scalar field obeys an integrated lower bound transversally to the Cauchy horizon. As a consequence we prove that the non-degenerate energy is infinite on any null surface crossing the Cauchy horizon and the curvature of a geodesic vector field blows up at the Cauchy horizon near time-like infinity. This generalizes an instability result due to Luk and Oh for Einstein-Maxwell-real-scalar-field in spherical symmetry.
This instability of the black hole interior can also be viewed as a step towards the resolution of the C^2 strong cosmic censorship conjecture for one-ended asymptotically initial data.
Sponsorship
EPSRC (1652828)
Embargo Lift Date
2100-01-01
Identifiers
External DOI: https://doi.org/10.1007/s00220-017-3079-3
This record's URL: https://www.repository.cam.ac.uk/handle/1810/273785
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