Abstract
We consider General Relativity (GR) on a space-time whose spatial slices are compact manifolds M with non-empty boundary ∂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 ∂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.
Highlights
Symmetries are equivalences that are quotiented out
We review how in this gauge the Einstein equations split into two constraints and a dynamic equation that can be obtained from a Lagrangian in natural form
This is all the better since we show that the Hamiltonian constraint — in addition to removing the remaining local diffeomorphisms — implies a constraint on the adiabatic motion that is equivalent to requiring the geodesics on the space of vacua to be null
Summary
Given a theory with time-independent — i.e. equilibrium — solutions it is natural to investigate if one can obtain new solutions by introducing a slow time dependence. One area where this has proven highly successful and interesting is that of topological solitons where the approximation method goes under the name of the Manton or moduli-space approximation [7]. There one starts with an infinite dimensional field space and reduces the problem to that of motion on a finite dimensional subspace, that is often referred to as the moduli space. The Hamiltonian constraint — a consequence of the time reparametrization invariance — restricts the dynamics to motion along null curves
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