Gravitational collapse of thin shells of dust in asymptotically flat Shape Dynamics

Abstract

In a recent paper, one of us studied spherically symmetric, asymptotically flat solutions of Shape Dynamics, finding that the spatial metric has characteristics of a wormhole - two asymptotically flat ends and a minimal-area sphere, or `throat', in between. In this paper we investigate whether that solution can emerge as a result of gravitational collapse of matter. With this goal, we study the simplest kind of spherically-symmetric matter: an infinitely-thin shell of dust. Our system can be understood as a model of a star accreting a thin layer of matter. We solve the dynamics of the shell exactly and find that, indeed, as it collapses, the shell leaves in its wake the wormhole metric. In the maximal-slicing time we use for asymptotically flat solutions, the shell only approaches the throat asymptotically and does not cross it in a finite amount of time (as measured by a clock `at infinity'). This leaves open the possibility that a more realistic cosmological solution of Shape Dynamics might see this crossing happening in a finite amount of time (as measured by the change of relational/shape degrees of freedom).