Kasner-like description of spacelike singularities in spherically symmetric spacetimes with scalar matter

TL;DR

The paper establishes a rigorous Kasner-like description of spacelike singularities in spherically symmetric Einstein–Maxwell–scalar field spacetimes, proving precise inverse-polynomial blow-up rates for the Hawking mass, Kretschmann scalar, and matter fields near the r = 0 boundary. Central to the analysis is a double-null gauge framework that yields sharp asymptotics parameterized by Ψ, Ξ, and 𝔐, which encode a BKL-like expansion along the singular boundary and yield spatially varying Kasner exponents. The results generalize prior upper bounds to full leading-order asymptotics, incorporate electromagnetic fields to probe stability regimes, and relate the singularity structure to the BKL picture via a τ-foliation and Kasner maps. The work also outlines a stability program for spacelike singularities and provides illustrative examples including FLRW and scale-invariant collapsing spacetimes, highlighting the breadth and applicability of the Kasner-like description in gravitational collapse. Overall, the findings connect cosmological singularity heuristics with black-hole interior dynamics in a rigorous, quantitatively precise setting, and pave the way for further stability/scattering analyses beyond spherical symmetry.

Abstract

We study the properties of spacelike singularities in spherically symmetric spacetimes obeying the Einstein equations, in the presence of matter. We consider in particular matter described by a scalar field, both in the presence of an electromagnetic field and without. We prove that if a spacelike singularity obeying several reasonable assumptions is formed, then the Hawking mass, the Kretschmann scalar, and the matter fields have inverse polynomial blow-up rates near the singularity that may be described precisely. Furthermore, one may view the resulting spacetime in the context of the BKL heuristics regarding space-like singularities in relativistic cosmology. In particular, near any point $p$ on the singular boundary in our spherically symmetric spacetime, we obtain a leading order BKL-type expansion, including a description of Kasner exponents associated to $p$. This provides a rigorous description of a detailed correspondence between Kasner-like singularities most often associated to the cosmological setting, and the singularities observed in (spherically symmetric) gravitational collapse. Moreover, we outline a program concerning the study of the stability and instability of spacelike singularities in the latter picture, both outside of spherical symmetry and within (where the electromagnetic field acts as a proxy for angular momentum).

Figures (12)

• Figure 1: Penrose diagram representation of a dispersive solution to the Einstein-scalar field equations. This is a future causal geodesically complete spacetime with a complete future null infinity $\mathcal{I}^+$.
• Figure 2: Penrose diagram representation of a gravitational collapse solution to the Einstein-scalar field equations. The solution possesses a complete future null infinity $\mathcal{I}^+$ as well as a black hole region bounded to the past by the event horizon$\mathcal{H}^+$. The black hole interior contains an apparent horizon$\mathcal{A}$ at which $\partial_v r = 0$, whose future is a trapped region$\mathcal{T}$ in which $\partial_v r < 0$ and which culminates at a spacelike boundary $\mathcal{S} = \{ r = 0 \}$ connecting $i^+$ with the first singularity at the center, denoted $b_0$.
• Figure 3: A strongly singular spacetime to which Theorem \ref{['roughthm:asymp']} applies. The region $\mathcal{D}$ is foliated by trapped $2$-spheres, and possesses a future boundary $\mathcal{S}$ at which $r$ extends continuously to $0$. Examples of such strongly singular spacetimes exist both in the case $F_{\mu\nu} = 0$, and in the case that $F_{\mu \nu} \neq 0$ where one must additionally assume \ref{['eq:QTS']} and \ref{['eq:SKE']}.
• Figure 4: Penrose diagrams representing the Schwarzschild interior (left) and the Reissner-Nordström interior (right). The darker shaded region of the Schwarzschild interior represents a strongly singular spacetime in the sense of Theorem \ref{['roughthm:asymp']}, whose singular boundary $\mathcal{S}$ is destroyed upon perturbations adding a non-trivial charge $Q \neq 0$. This effect is due to the presence of a (global) Cauchy horizon $\mathcal{CH}^+$ in the Reissner-Nordström interior.
• Figure 5: The near-singularity region of a spacetime constructed in MeVdM. The region contains two Kasner-like regimes: $\mathcal{K}_1$ fails to satisfy an inequality of the form \ref{['eq:SKE']} and is thus unstable. Before reaching the singularity $\mathcal{S}$, there is an "inversion" to the subcritical Kasner-like region $\mathcal{K}_2$. The (nonlinear) inversion procedure occurs in the transition region $\mathcal{K}_{inv}$.

Theorems & Definitions (49)

• Theorem I: Asymptotics -- rough version
• Corollary II: Kasner-like behaviour, rough version
• Theorem III: Stability -- rough version
• Corollary IV: Instability
• Remark
• Lemma 2.1
• proof
• Remark
• Definition 1
• Definition 2