Stability and Instability of the Sub-extremal Reissner–Nordström Black Hole Interior for the Einstein–Maxwell–Klein–Gordon Equations in Spherical Symmetry

• Published in: Communications in mathematical physics 2018-05, Vol.360 (1), p.103-168
• Main Author: Van de Moortel, Maxime
• Format: Article
• Language: English
• Subjects: Black holes; Censorship; Classical and Quantum Gravitation; Complex Systems; Curvature; Decay rate; Event horizon; Infinity; Lower bounds; Mathematical analysis; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Polynomials; Quantum Physics; Relativity Theory; Stability; Symmetry; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-017-3079-3

We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole—approaching a sub-extremal Reissner–Nordström background fast enough—in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting: Stability We prove that spherically symmetric characteristic initial data to the Einstein–Maxwell–Klein–Gordon equations approaching a Reissner–Nordström 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 space–time 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. 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 flat initial data.