Caustics in the spherically symmetric Einstein-dust system

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.