Kasner bounces and fluctuating collapse inside hairy black holes with charged matter

Abstract: We study the interior of black holes in the presence of charged scalar hair of small amplitude $\epsilon$ on the event horizon and show their terminal boundary is a crushing Kasner-like singularity. These spacetimes are spatially homogeneous and they differ significantly from the hairy black holes with uncharged matter previously studied in [M. Van de Moortel, Violent nonlinear collapse inside charged hairy black holes, Arch. Rational. Mech. Anal., 248, 89, 2024] in that the electric field is dynamical and subject to the backreaction of charged matter. This charged backreaction causes drastically different dynamics compared to the uncharged case that impact the formation of the spacelike singularity, exhibiting novel phenomena such as - Collapsed oscillations: oscillatory growth of the scalar hair, nonlinearly induced by the collapse. - A fluctuating collapse: The final Kasner exponents' dependency in $\epsilon$ is via an expression of the form $|\sin\left(\omega_0 \cdot \epsilon^{-2}+ O(\log (\epsilon^{-1}))\right)|$. - A Kasner bounce: a transition from an unstable Kasner metric to a different stable Kasner metric. The Kasner bounce occurring in our spacetime is reminiscent of the celebrated BKL scenario in cosmology. We additionally propose a construction indicating the relevance of the above phenomena -- including Kasner bounces -- to spacelike singularities inside more general (asymptotically flat) black holes, beyond the hairy case. While our result applies to all values of $\Lambda \in \mathbb{R}$, in the $\Lambda<0$ case, our spacetime corresponds to the interior region of a charged asymptotically Anti-de-Sitter stationary black hole, also known as a holographic superconductor, and whose exterior region was rigorously constructed in the recent mathematical work [W. Zheng, Asymptotically Anti-de Sitter Spherically Symmetric Hairy Black Holes, arXiv.2410.04758].