Gravitational instability of the inner static region of a Reissner–Nordström black hole

Abstract

Reissner–Nordström black holes have two static regions: r > ro and 0 < r < ri, where ri and ro are the inner and outer horizon radii, respectively. The stability of the exterior static region was established a long time ago. In this work we prove that the interior static region is unstable under linear gravitational perturbations, by showing that field perturbations compactly supported within this region will generically excite a mode that grows exponentially in time. This result gives an alternative reason to mass inflation to consider the spacetime extension beyond the Cauchy horizon as physically irrelevant, and thus provides support to the strong cosmic censorship conjecture, which is also backed by recent evidence of a linear gravitational instability in the interior region of Kerr black holes found by the authors. The use of intertwiners to solve the evolution of initial data plays a key role, and adapts without a change to the case of super-extremal Reissner–Nordström black holes, allowing us to complete the proof of the linear instability of this naked singularity. A particular intertwiner is found such that the intertwined Zerilli field has a geometrical meaning—it is the first-order variation of a particular Riemann tensor invariant. Using this, calculations can be carried out explicitly for every harmonic number.