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Initial-boundary value problems for Einstein's field equations

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We discuss well-posed initial-boundary value formulations
in general relativity. These formulations allow us to construct solutions of
Einstein's field equations inside a cylindrical region, given suitable initial
and boundary data. We analyze the restrictions on the boundary data that result
from the requirement of constraint propagation and the minimization of spurious
reflections, and choosing harmonic coordinates we show how to cast the problem
into well-posed form. Then, we consider the particular case where the boundary
represents null infinity of an asymptotically flat spacetime. Here, the rôle of
the boundary conditions is to provide adequate regularity and gauge conditions
at infinity.

As an application of our setup we mention ongoing work on
the computation of quasi-stationary scalar field configurations on a
non-rotating supermassive black hole background.