Violent Nonlinear Collapse in the Interior of Charged Hairy Black Holes

Abstract

We construct a new one-parameter family, indexed by \(\epsilon \), of two-ended, spatially-homogeneous black hole interiors solving the Einstein–Maxwell–Klein–Gordon equations with a (possibly zero) cosmological constant \(\Lambda \) and bifurcating off a Reissner–Nordström-(dS/AdS) interior (\(\epsilon =0\)). For all small \(\epsilon \ne 0\), we prove that, although the black hole is charged, its terminal boundary is an everywhere-spacelike Kasner singularity foliated by spheres of zero radius r. Moreover, smaller perturbations (i.e. smaller \(|\epsilon |\)) are more singular than larger ones, in the sense that the Hawking mass and the curvature blow up following a power law of the form \(r^{-O(\epsilon ^{-2})}\) at the singularity \(\{r=0\}\). This unusual property originates from a dynamical phenomenon—violent nonlinear collapse—caused by the almost formation of a Cauchy horizon to the past of the spacelike singularity \(\{r=0\}\). This phenomenon was previously described numerically in the physics literature and referred to as “the collapse of the Einstein–Rosen bridge”. While we cover all values of \(\Lambda \in \mathbb {R}\), the case \(\Lambda <0\) is of particular significance to the AdS/CFT correspondence. Our result can also be viewed in general as a first step towards the understanding of the interior of hairy black holes.