Caustics in the spherically symmetric Einstein-dust system

TL;DR

This paper investigates the formation and properties of caustics in spherically symmetric Einstein-dust spacetimes, establishing local existence results, analyzing regularity near caustics, and identifying new static solutions with dust regions bounded by caustics.

Contribution

It provides a local existence theorem for spacetimes with caustics, analyzes metric regularity and Einstein's equations near caustics, and introduces new static solutions with dust regions bounded by caustics.

Findings

Curvature invariants diverge at caustics.

The metric has limited regularity $C^{1,1/2}$ near caustics.

A new family of static spacetimes with dust bounded by caustics is identified.

Abstract

Caustics-envelopes formed by the trajectories of fluid particles-arise in proposed dynamical extensions for shell-crossing singularities occurring in the Einstein-dust system. In this study, a local existence result is established, describing the dynamics in a neighbourhood of such caustics. Specifically, we obtain spherically symmetric spacetimes $(M,g_{\mu\nu})$ containing a caustic $\mathcal{C}$, which, in the quotient $M/SO(3)$, is a timelike curve forming a singular boundary between a 2-dust region and a vacuum region. The spacetimes are constructed from solutions to a PDE problem posed with a spacelike direction of evolution. Curvature invariants and energy densities diverge as the caustic is approached. Consequently the metric has limited regularity $g\in C^{1,1/2}$ and is shown to satisfy Einstein's equation weakly. On the complement of the caustic, the metric is smooth and satisfies Einstein's equation classically. A (degenerate) coordinate system is identified in which the dynamical variables are smooth with extension to the caustic. Finally, a novel family of static, spherically symmetric spacetimes is identified, complementing the local construction above. Each spacetime contains an eternal annular 2-dust region bounded by a pair of caustics.