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arXiv · 1704.05790

Stability and instability of the sub-extremal Reissner-Nordstr\"om black hole interior for the Einstein-Maxwell-Klein-Gordon equations in spherical symmetry

Maxime Van de Moortel
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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.

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Maxime Van de Moortel. 2017-04-19. Stability and instability of the sub-extremal Reissner-Nordstr\"om black hole interior for the Einstein-Maxwell-Klein-Gordon equations in spherical symmetry. https://doi.org/10.1007/s00220-017-3079-3

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