Black hole nonmodal linear stability: odd perturbations of Reissner-Nordstr\"om

Abstract

Following a program on black hole nonmodal linear stability initiated in Phys.\ Rev.\ Lett.\ {\bf 112} (2014) 191101, we study odd linear perturbations of the Einstein-Maxwell equations around a Reissner-Nordstr\"om (A)dS black hole. We show that all the gauge invariant information in the metric and Maxwell field perturbations is encoded in the spacetime scalars $\mathcal{F} =\delta (F^*_{\alpha \beta} F^{\alpha \beta})$ and $\mathcal{Q} =\delta (\tfrac{1}{48} C^*_{\alpha \beta \gamma \delta} C^{\alpha \beta \gamma \delta})$, where $C_{\alpha \beta \gamma \delta}$ is the Weyl tensor, $F_{\alpha \beta}$ the Maxwell field, a star denotes Hodge dual and $\delta$ means first order variation, and that the linearized Einstein-Maxwell equations are equivalent to a coupled system of wave equations for $\mathcal{F}$ and $\mathcal{Q}$. For nonnegative cosmological constant we prove that $\mathcal{F}$ and $\mathcal{Q}$ 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 AdS case the dynamics depends on the boundary condition at the conformal timelike boundary and there are instabilities if Robin boundary conditions are chosen.