On the structure of singularities in self-gravitating relativistic dust spacetimes

Abstract

This dissertation presents several results on the dynamics of self-gravitating fluid models in general relativity. The broad focus is on understanding and classifying singular structures arising in gravitational collapse, using techniques from the theory of partial differential equations to study the existence of these structures and their characteristic features. In the first locus of study, we examine caustics—envelopes formed by the trajectories of fluid particles—which arise in proposed dynamical extensions for shell-crossing singularities occurring in the Einstein-dust system. A local existence result is established, describing the dynamics in a neighbourhood of such caustics. Specifically, we obtain spacetimes (M, g) containing a caustic C, which, in the spherically symmetric quotient, 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. The metric has limited C^{1,1/2} regularity 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. To complement this local construction, a novel family of static, spherically symmetric spacetimes is identified. Each spacetime contains an eternal annular 2-dust region bounded by a pair of caustics. The second locus of study is the interaction of charged null fluids in general relativity, specifically in the context of the bouncing continuation proposed by Ori. In this model, charged massless particles may instantaneously change direction (bounce) after losing all their 4-momentum due to electrostatic repulsion. We investigate timelike bounce hypersurfaces in spherical symmetry: scenarios in which an incoming beam of charged null dust changes direction along a timelike surface, which is the (free) boundary of an interacting 2-dust region. It is shown that every timelike curve in the (spherically symmetric quotient of) Minkowski or Reissner-Nordstrom spacetimes arises as the bounce hypersurface B of a charged null dust beam incident from past null infinity. We construct a spacetime (M, g) describing the full trajectory of the beam, which includes gluing to Reissner-Nordstrom and Vaidya regions. Here, we encounter a much milder singularity than before: despite the energy density diverging at B, the metric has C^{2,1} regularity and satisfies Einstein’s equation classically. We also obtain examples of timelike bounce hypersurfaces terminating in a null point. Finally, since these constructions are teleological, we consider a given charged incoming beam from past null infinity. We formulate and solve a free boundary problem which represents the formation of a timelike bounce hypersurface.