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 Σ≠? and boundary conditions on the manifold T= Re+ 0×Σ. We assume the boundaries Σ and { 0 }×, Σ 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 Σ, 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∪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.
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Similar content being viewed by others Explore related subjectsDiscover the latest articles and news from researchers in related subjects, suggested using machine learning. Author information Authors and AffiliationsAlbert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, Schlaatzweg 1, 14473 Potsdam,¶Germany, , , , , , DE
Helmut Friedrich & Gabriel Nagy
Received: Received: 11 June 1998 / Accepted: 15 September 1998
About this article Cite this articleFriedrich, H., Nagy, G. The Initial Boundary Value Problem for Einstein's Vacuum Field Equation. Comm Math Phys 201, 619–655 (1999). https://doi.org/10.1007/s002200050571
Issue Date: April 1999
DOI: https://doi.org/10.1007/s002200050571
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