Relativistic shells: Dynamics, horizons, and shell crossing

Abstract

We consider the dynamics of timelike spherical thin matter shells in vacuum. A general formalism for thin shells matching two arbitrary spherical spacetimes is derived, and subsequently specialized to the vacuum case. We first examine the relative motion of two dust shells by focusing on the dynamics of the exterior shell, whereby the problem is reduced to that of a single shell with different active Schwarzschild masses on each side. We then examine the dynamics of shells with non-vanishing tangential pressure $p$, and show that there are no stable--stationary, or otherwise--solutions for configurations with a strictly linear barotropic equation of state, $p=ασ$, where $σ$ is the proper surface energy density and $α\in(-1,1)$. For {\em arbitrary} equations of state, we show that, provided the weak energy condition holds, the strong energy condition is necessary and sufficient for stability. We examine in detail the formation of trapped surfaces, and show explicitly that a thin boundary layer causes the apparent horizon to evolve discontinuously. Finally, we derive an analytical (necessary and sufficient) condition for neighboring shells to cross, and compare the discrete shell model with the well-known continuous Lema\^ıtre-Tolman-Bondi dust case.