A Birkhoff theorem for Shape Dynamics

TL;DR

This paper derives an analogue of the Birkhoff theorem within Shape Dynamics, revealing a unique, non-singular isotropic wormhole solution that differs from Schwarzschild black holes, highlighting differences between Shape Dynamics and General Relativity.

Contribution

It provides the first derivation of a Birkhoff-like theorem in Shape Dynamics using a spherically symmetric ansatz, demonstrating a unique wormhole solution distinct from GR.

Findings

The solution is an isotropic wormhole without singularities.

The derivation is simpler, not requiring Christoffel symbols.

The solution differs from Schwarzschild, breaking the duality with GR.

Abstract

Shape Dynamics is a theory of gravity that replaces refoliation invariance for spatial Weyl invariance. Those solutions of the Einstein equations that have global, constant mean curvature slicings, are mirrored by solutions in Shape Dynamics. However, there are solutions of Shape Dynamics that have no counterpart in General relativity, just as there are solutions of GR that are not completely foliable by global constant mean curvature slicings (such as the Schwarzschild spacetime). It is therefore interesting to analyze directly the equations of motion of Shape Dynamics in order to find its own solutions, irrespective of properties of known solutions of GR. Here I perform a first study in this direction by utilizing the equations of motion of Shape Dynamics in a spherically symmetric, asymptotically flat ansatz to derive an analogue of the Birkhoff theorem. There are two significant differences with respect to the usual Birkhoff theorem in GR. The first regards the construction of the solution: the spatial Weyl gauge freedom of shape dynamics is used to simplify the problem, and boundary conditions are required. In fact the derivation is simpler than the usual Birkhof theorem as no Christoffel symbols are needed. The second, and most important difference is that the solution obtained is uniquely the isotropic wormhole solution, in which no singularity is present, as opposed to maximally extended Schwarzschild. This provides an explicit example of the breaking of the duality between General relativity and Shape Dynamics, and exhibits some of its consequences.