The link between general relativity and shape dynamics

We define the concept of a linking theory and show how two equivalent gauge theories possessing different gauge symmetries generically arise from a linking theory. We show that under special circumstances a linking theory can be constructed from a given gauge theory through ‘Kretchmannization’ of a given gauge theory, which becomes one of the two theories related by the linking theory. The other, so-called dual gauge theory, is then a gauge theory of the symmetry underlying the ‘Kretschmannization’. We then prove the equivalence of general relativity and Shape Dynamics, a theory with fixed foliation but spatial conformal invariance. This streamlines the rather complicated construction of this equivalence performed in Gomes et al (2011 Class. Quantum Grav. 28 045005). We use this streamlined argument to extend the result to general relativity with asymptotically flat boundary conditions. The improved understanding of linking theories naturally leads to the Lagrangian formulation of shape dynamics, which allows us to partially relate the degrees of freedom.