On the initial boundary value problem for the vacuum Einstein equations and geometric uniqueness

Abstract: We formulate an initial boundary value problem (IBVP) for the vacuum Einstein equations by describing the boundary conditions of a spacetime metric in its associated gauge. This gauge is determined, equivariantly with respect to diffeomorphisms, by the spacetime metric. The vacuum spacetime metric $g$ and its associated gauge $\phi_g$ are solved simultaneously in local harmonic coordinates. Further we show that vacuum spacetimes satisfying fixed initial-boundary conditions and corner conditions are geometrically unique near the initial surface. Finally, in analogy to the solution of the Cauchy problem, we also construct a unique maximal globally hyperbolic solution of the IBVP.