The initial boundary value problem for Einstein's vacuum field equation

We study the initial boundary value problem for Einstein's vacuum field equation. We prescribe initial data on an orientable, compact, 3-dimensional manifold S with boundary Sigma not equal 0 and boundary conditions on the manifold T = R-0(+) x Sigma. We assume the boundaries Sigma and (0) x Sigma of S and T to be identified in the natural way. Furthermore, we prescribe certain gauge source functions which determine the evolution of the fields. Provided that all data are smooth and certain consistency conditions are met on Sigma, we show that there exists a smooth solution to Einstein's equation Ric[g] = 0 on a manifold which has (after an identification) a neighbourhood of S in T boolean OR S as a boundary. The solution is such that S is space-like, the initial data are induced by the solution on S, and, in the region of T where the solution is defined, T is time-like and the boundary conditions are satisfied.