Black hole nonmodal linear stability under odd perturbations: The Reissner-Nordström case

Abstract

Following a program on black hole nonmodal linear stability initiated by one of the authors [Phys. Rev. Lett. 112, 191101 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.191101], we study odd linear perturbations of the Einstein-Maxwell equations around a Reissner-Nordström anti-de Sitter black hole. We show that all the gauge invariant information in the metric and Maxwell field perturbations is encoded in the spacetime scalars F=δ(Fαβ∗Fαβ) and Q=δ(148Cαβγδ∗Cαβγδ), where Cαβγδ is the Weyl tensor, Fαβ is the Maxwell field, a star denotes Hodge dual, and δ means first order variation, and that the linearized Einstein-Maxwell equations are equivalent to a coupled system of wave equations for F and Q. For a non-negative cosmological constant we prove that F and Q are pointwise bounded on the outer static region. The fields are shown to diverge as the Cauchy horizon is approached from the inner dynamical region, providing evidence supporting strong cosmic censorship. In the asymptotically anti-de Sitter case the dynamics depends on the boundary condition at the conformal timelike boundary, and there are instabilities if Robin boundary conditions are chosen.