On the fate of Birkhoff's theorem in Shape Dynamics

TL;DR

This paper investigates the applicability of Birkhoff's theorem in Shape Dynamics, revealing that the system is underdetermined without additional matter coupling, challenging previous assumptions about asymptotic conditions.

Contribution

It demonstrates that standard falloff conditions are not valid in Shape Dynamics and identifies an undetermined function related to dilatational momentum, highlighting the need for matter coupling.

Findings

The system is underdetermined without matter coupling.

Standard asymptotic conditions are not valid in Shape Dynamics.

A new physical quantity related to dilatational momentum is identified.

Abstract

Spherically symmetric, asymptotically flat solutions of Shape Dynamics were previously studied assuming standard falloff conditions for the metric and the momenta. These ensure that the spacetime is asymptotically Minkowski, and that the falloff conditions are Poincar\'e-invariant. These requirements however are not legitimate in Shape Dynamics, which does not make assumptions on the structure or regularity of spacetime. Analyzing the same problem in full generality, I find that the system is underdetermined, as there is one function of time that is not fixed by any condition and appears to have physical relevance. This quantity can be fixed only by studying more realistic models coupled with matter, and it turns out to be related to the dilatational momentum of the matter surrounding the region under study.