Asymptotically flat black holes with a singular Cauchy horizon and a spacelike singularity

TL;DR

The paper delivers the first explicit constructions of asymptotically flat, spherically symmetric black holes within the Einstein–Maxwell–charged-scalar-field framework that host coexisting null CHs and spacelike singularities, with the interior Kasner-type regime degenerating to $(1,0,0)$ at the CH–S interface. Central to the method is a novel spacelike–characteristic gluing technique that joins regular uncharged data spheres to dynamical horizons and RN-trapped surfaces, enabling global one-ended and two-ended spacetimes with controlled interior dynamics and late-time tails. The results illuminate the interior structure of gravitational collapse in a concrete, non-perturbative setting and provide concrete examples compatible with, and extending, Strong Cosmic Censorship expectations for charged scalar fields. By coupling local interior analysis to global geometric constructions, the work offers a robust framework for exploring generic features of black hole interiors beyond exact solutions and opens avenues for analyzing robustness under perturbations and non-spherical symmetry.

Abstract

In our recent work [Van de Moortel, The coexistence of null and spacelike singularities inside spherically symmetric black holes], we analyzed the transition between null and spacelike singularities in spherically symmetric dynamical black holes and demonstrated that the spacelike portion is described by a Kasner metric with positive varying exponents that degenerate to $(1,0,0)$ near the null-spacelike transition. In the present paper, we provide examples of global spacetimes satisfying the assumptions of this previous result and apply its analysis to obtain a large class of asymptotically flat (spherically symmetric) black hole spacetimes that exhibit coexisting null and spacelike singularities. Our main results include: _The construction of one-ended asymptotically flat black hole spacetimes solving the Einstein-Maxwell-charged-scalar-field equations. The proof relies on a new spacelike-characteristic gluing method between any uncharged spherically symmetric solution and the event horizon of a charged dynamical black hole. _The construction of a large class of two-ended asymptotically flat black hole spacetimes solving the Einstein-Maxwell-(uncharged)-scalar-field equations. In both cases, we show that the terminal boundary in the black hole interior only has two distinct components: a weakly singular (null) Cauchy horizon $\mathcal{CH}_{i^+}$ where curvature blows up and a strong singularity $\mathcal{S}=\{r=0\}$. Our construction provides the first examples of black holes with coexisting null and spacelike singularities. These examples hold particular significance in the one-ended case as a model of gravitational collapse, where this phenomenon is conjecturally generic for the Einstein-scalar-field model, even beyond spherical symmetry.

Figures (18)

• Figure 1: Penrose diagram of the gravitational collapse (=one-ended) spacetimes obtained in Theorem \ref{['thm.II']}. $\mathcal{CH}_{i^+}$ is a weakly singular Cauchy horizon and $\mathcal{S}=\{r=0\}$ is a strong singularity, spacelike near $\mathcal{CH}_{i^+}\cap\mathcal{S}$ and $b_{\Gamma}$.
• Figure 2: Penrose diagram of the solution obtained in Theorem \ref{['thm.I']} with bifurcate initial data $\underline{C}_{in} \cup C_{out}$.
• Figure 3: Left: Two-ended black hole with a Cauchy horizon $\mathcal{CH}_{i^+}$ and no spacelike singularity. Right: Two-ended black hole with coexisting Cauchy horizon $\mathcal{CH}_{i^+}$ and singularity $\mathcal{S}=\{r=0\}$ (Theorem \ref{['thm.III']}).
• Figure 4: Penrose diagram of the Oppenheimer--Snyder solution of the Einstein-dust equations.
• Figure 5: Penrose diagram of the Oppenheimer--Snyder like uncharged solution of \ref{['1.1']}-\ref{['5.1']} in Theorem \ref{['OS.thm.intro']} for $q=0$. $\mathcal{S}$ is exactly Schwarzschild's singularity near $i^+$, and exactly the FLRW singularity near $b_{\Gamma}$. Note the absence of a Cauchy horizon ($\mathcal{CH}_{i^+}=\emptyset$), in contrast to the examples of Theorem \ref{['thm.II']} (depicted in Figure \ref{['fig:spacelikeconj']}).

Theorems & Definitions (109)

• Conjecture 1.1: Dafermos:2004jpMihalisICMKommemibreakdown
• Theorem I
• Theorem II
• Remark 1.1
• Theorem 1.2: Black hole construction by gluing
• Remark 1.2
• Theorem 1.3
• Theorem III
• Theorem IV
• Remark 1.3