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In this article we summarize what is known about the initial-boundary value problem for general relativity and discuss present problems related to it. [ABSTRACT FROM AUTHOR]

In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In particular, they should be compatible with the constraints, yield a well posed initial-boundary value formulation and incorporate some physically desirable properties like, for instance, minimizing reflections of gravitational radiation. Motivated by the problem in General Relativity, we analyze a model problem, consisting of a formulation of Maxwell's equations on a spatially compact region of space–time with timelike boundaries. The form in which the equations are written is such that their structure is very similar to the Einstein–Christoffel symmetric hyperbolic formulations of Einstein's field equations. For this model problem, we specify a family of Sommerfeld-type constraint-preserving boundary conditions and show that the resulting initial-boundary value formulations are well posed. We expect that these results can be generalized to the Einstein–Christoffel formulations of General Relativity, at least in the case of linearizations about a stationary background. [ABSTRACT FROM AUTHOR]

A persistent challenge in numerical relativity is the correct specification of boundary conditions. In this work we consider a many-parameter family of symmetric hyperbolic initial-boundary value formulations for the linearized Einstein equations and analyze its well posedness using the Laplace–Fourier technique. By using this technique ill posed modes can be detected and thus a necessary condition for well posedness is provided. We focus on the following types of boundary conditions: (i) boundary conditions that have been shown to preserve the constraints, (ii) boundary conditions that result from setting the ingoing constraint characteristic fields to zero, and (iii) boundary conditions that result from considering the projection of Einstein’s equations along the normal to the boundary surface. While we show that in case (i) there are no ill posed modes, our analysis reveals that, unless the parameters in the formulation are chosen with care, there exist ill posed constraint violating modes in the remaining cases. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In this paper we address the problem of specifying boundary conditions for Einstein's equations when linearized around Minkowski space using the generalized Einstein-Christoffel symmetric hyperbolic system of evolution equations. The boundary conditions we work out guarantee that the constraints are satisfied provided they are satisfied on the initial slice and ensures a well posed initial-boundary value formulation. We consider the case of a manifold with a non-smooth boundary, as is the usual case of the cubic boxes commonly used in numerical relativity. The techniques discussed should be applicable to more general cases, as linearizations around more complicated backgrounds, and may be used to establish well posedness in the full non-linear case. [ABSTRACT FROM AUTHOR]