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We show that one can construct two equivalent gauge theories from a linkingtheory and give a general construction principle for linking theories which weuse to construct a linking theory that proves the equivalence of GeneralRelativity and Shape Dynamics, a theory with fixed foliation but spatialconformal invariance. This streamlines the rather complicated construction ofthis equivalence performed previously. We use this streamlined argument toextend the result to General Relativity with asymptotically flat boundaryconditions. The improved understanding of linking theories naturally leads tothe Lagrangian formulation of Shape Dynamics, which allows us to partiallyrelate the degrees of freedom.

The Link between General Relativity and Shape Dynamics