Compact spherically symmetric solutions and gravitational collapse in SD

Abstract

I will review the current status of our understanding of spherically symmetric compact solutions of Shape Dynamics, which have nontrivial degrees of freedom when matter is present. I will show some new solutions of GR in a CMC foliation: a single thin spherical shell of matter in equilibrium in a compact foliation of de Sitter, and the simplest possible model of a black hole or compact star. This is provided by a universe with the topology of a 3-sphere with two thin spherical shells of dust. One of the shells models the `fixed stars’, or the `rest of the universe’, while the other shell models collapsing matter. Both are needed for a truly relational description of gravitational collapse. It turns out that such a solution of GR cannot be evolved past a point at which the foliationceases to be admissible, but it still makes sense past that point as a solution of Shape Dynamics, because the shape degrees of freedom seem to be unaffected. My conjecture is that we have found another example of departure between GR and SD, and this departure happens whenever ordinary matter undergoes gravitational collapse.