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We formulate an initial boundary value problem (IBVP) for the vacuum Einsteinequations by describing the boundary conditions of a spacetime metric in itsassociated gauge. This gauge is determined, equivariantly with respect todiffeomorphisms, by the spacetime metric. The vacuum spacetime metric $g$ andits associated gauge $\phi_g$ are solved simultaneously in local harmoniccoordinates. Further we show that vacuum spacetimes satisfying fixedinitial-boundary conditions and corner conditions are geometrically unique nearthe initial surface. Finally, in analogy to the solution of the Cauchy problem,we also construct a unique maximal globally hyperbolic solution of the IBVP.

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