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

Following a program on black hole nonmodal linear stability initiated by one of the authors [Phys. Rev. Lett. 112, 191101 (2014)], we study odd linear perturbations of the Einstein-Maxwell equations around a Reissner-Nordstrm 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 = delta(F*F-alpha beta(alpha beta)) and Q = delta (1/48C*C-alpha beta gamma delta(alpha beta gamma delta), where C-alpha beta gamma delta is the Weyl tensor, F-alpha beta is the Maxwell field, a star denotes Hodge dual, and d 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.