Violent Nonlinear Collapse in the Interior of Charged Hairy Black Holes

We construct a new one-parameter family, indexed by \(\epsilon \) $ϵ$ , of two-ended, spatially-homogeneous black hole interiors solving the Einstein–Maxwell–Klein–Gordon equations with a (possibly zero) cosmological constant \(\Lambda \) $\mathrm{\Lambda }$ and bifurcating off a Reissner–Nordström-(dS/AdS) interior ( \(\epsilon =0\) $ϵ=0$ ). For all small \(\epsilon \ne 0\) $ϵ\ne 0$ , 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 \(|\epsilon |\) $|ϵ|$ ) 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 \(r^{-O(\epsilon ^{-2})}\) $r^{-O({ϵ}^{-2})}$ at the singularity \(\{r=0\}\) $\{r=0\}$ . 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 \(\{r=0\}\) $\{r=0\}$ . 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 \(\Lambda \in \mathbb {R}\) $\mathrm{\Lambda }\in \mathbb{R}$ , the case \(\Lambda <0\) $\mathrm{\Lambda }<0$ 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.