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

TL;DR

The paper studies the interior of hairy black holes with a small charged scalar field, showing the terminal boundary is a crushing Kasner-like spacelike singularity. By reducing to spatially homogeneous Einstein–Maxwell–Klein–Gordon dynamics and comparing to RN interiors, it uncovers collapsed oscillations of the charged scalar and a Kasner-bounce mechanism that can switch between Kasner regimes, with the final exponents depending on an oscillatory phase $\alpha(\varepsilon)=|\sin(\omega_0\varepsilon^{-2}+O(\log\varepsilon^{-1}))|$. A detailed region-by-region analysis (redshift to Kasner) yields precise asymptotics: oscillations evolve into a proto-Kasner and then Kasner regime, with charge retention and bounded final charge $Q_{\infty}$, and in AdS cases this interior aligns with holographic superconductor models. The work connects to BKL cosmology heuristics, extends rigorous Kasner constructions to charged stiff matter in BH interiors, and provides a framework for understanding spacelike singularities inside more general (asymptotically flat or AdS) black holes. It further suggests potential applicability to non-hairy interiors via domain-of-dependence arguments and motivates open problems in extending these dynamics beyond spatial homogeneity.

Abstract

We study the interior of black holes in the presence of charged scalar hair of small amplitude $ε$ 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 $ε$ is via an expression of the form $|\sin\left(ω_0 \cdot ε^{-2}+ O(\log (ε^{-1}))\right)|$. - 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 $Λ\in \mathbb{R}$, in the $Λ<0$ 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].

Figures (8)

• Figure 1: Penrose diagram of a (non-hairy) black hole interior with a Cauchy horizon $\mathcal{CH}_{i^+}$ and a spacelike singularity $\mathcal{S}$. Left: one-ended black hole (gravitational collapse case). Right: two-ended black hole.
• Figure 2: Penrose diagram of the hairy black hole interior from Theorem \ref{['thm.intro']}. If $|\alpha| \geq 1 + \sigma > 1$, then the first Kasner regime matches the final Kasner regime and continues to $\{ r = 0 \}$. If $0 < \eta \leq |\alpha| \leq 1 - \sigma < 1$, then there is a Kasner bounce between the first and final Kasner regimes. A more detailed breakdown is given in Figure \ref{['Penrose_detailed']}.
• Figure 3: Penrose diagram of the spacetime corresponding to Theorem \ref{['CH.thm']}
• Figure 4: The proposed construction of a two-ended black hole with a spacelike singularity $\mathcal{S}$ via gluing argument. The union of the dark grey, orange, brown and yellow regions (including the green part of the spacelike singularity) is isometric to a subset of the hairy black hole of Figure \ref{['Penrose_simplified']}. $\mathcal{I}_L^+$ and $\mathcal{I}_R^+$ are the components of null infinity $\mathcal{I}^+$.
• Figure 5: The Penrose diagram of the hairy black hole interiors constructed in Theorem \ref{['violent.thm']}.

Theorems & Definitions (95)

• Conjecture A: Spacelike singularity conjecture, KerrStabKommemiJonathanICMr0
• Conjecture B
• Theorem I
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.1: Moi
• Remark 1.4
• Theorem 1.2: Black hole interior in gravitational collapse, r0Moi2Moi4
• Remark 1.5