Timelike bounce hypersurfaces in charged null dust collapse

Abstract

We establish results on the dynamics of interacting charged null fluids in general relativity, specifically in the context of the bouncing continuation proposed in [Ori91]. In this model - the setting for a number of prominent case studies on black hole formation - charged massless particles may instantaneously change direction (bounce) after losing all their 4-momentum due to electrostatic repulsion. We initiate the study of timelike bounce hypersurfaces in spherical symmetry: scenarios in which an incoming beam of charged null dust changes direction along a timelike surface $\mathcal{B}$, which is the (free) boundary of an interacting 2-dust region. We identify a novel decoupling of the equations of motion in this region. First, it is shown that every timelike curve segment $\gamma$ in the spherically symmetric quotient of Minkowski or Reissner-Nordstr\"om spacetimes arises as the bounce hypersurface $\mathcal{B}$ of a charged null dust beam incident from past null infinity $\mathcal{I}^-$. We construct a spacetime $(\mathcal{M},g_{\mu\nu})$ describing the full trajectory of the beam, which includes gluing to Reissner-Nordstr\"om and Vaidya regions. Across $\mathcal{B}$ the metric has regularity $g_{\mu\nu}\in C^{2,1}$ and satisfies Einstein's equation classically, while $C^\infty$ gluing may be achieved across all other interfaces. We also obtain examples of timelike bounce hypersurfaces terminating in a null point. Since these constructions are teleological, we secondly 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. The result is conditional, applying only in the exterior region of a Reissner-Nordstr\"om spacetime, and subject to a technical regularity condition.