Kasner-Like Description of Spacelike Singularities in Spherically Symmetric Spacetimes with Scalar Matter

Abstract

We study the properties of spacelike singularities in spherically symmetric spacetimes obeying the Einstein equations, in the presence of matter. Building upon previous work of An–Zhang [4], we consider 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 spacelike 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 confirms heuristics of Buonanno–Damour–Veneziano [14]. As a result, we provide a rigorous description of a detailed, quantitative 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).