 To see how best matching works in the case of GTR, first note that a certain subclass of solutions of that theory (namely, those which are globally hyperbolic) can be formulated in terms of the ‘3+1 formalism’, according to which the state of the universe at a particular time is represented by a determinate 3-manifold with associated Riemannian metric; dynamical equations then determine how such 3-geometries evolve in time. (For a summary of the 3+1 formalism, see e.g. Gourgoulhon (2012).) The Machian relationists working in the shape dynamics research program take this 3+1 approach to GTR as their starting point. They thus assume that instantaneous spaces which are the points in configuration space have the determinate topology of some closed 3-manifold without boundary. Configuration space is the space of Riemannian 3-metrics on that 3-manifold. The natural analogue of relative configuration space is, then, this space of Riemannian 3-metrics quotiented by diffeomorphisms, which are the generalisations of Leibniz shifts appropriate to GTR (see the entry on  the hole argument).  The analogue of shape space in this case is the space of Riemannian 3-metrics, but quotiented in addition by local dilatations (by ‘local’, we mean here a transformation which can vary from point to point).