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-Nordstr\"om-(dS/AdS) interior ($\epsilon = 0$). For all small $\epsilon \neq 0$, 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 one, 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.