  We consider General Relativity (GR) on a space-time whose spatial slices are
compact manifolds $M$ with non-empty boundary $\partial M$. We argue that this
theory has a non-trivial space of `vacua', consisting of spatial metrics
obtained by an action on a reference flat metric by diffeomorpisms that are
non-trivial at the boundary. In an adiabatic limit the Einstein equations
reduce to geodesic motion on this space of vacua with respect to a particular
pseudo-Riemannian metric that we identify. We show how the momentum constraint
implies that this metric is fully determined by data on the boundary $\partial
M$ only, while the Hamiltonian constraint forces the geodesics to be null. We
comment on how the conserved momenta of the geodesic motion correspond to an
infinite set of conserved boundary charges of GR in this setup.
