Einstein boundary conditions in relation to constraint propagation for the initial-boundary value problem of the Einstein equations

Details Einstein boundary conditions in relation to constraint propagation for the initial-boundary value problem of the Einstein equations Einstein boundary conditions in relation to constraint propagation for the initial-boundary value problem of the Einstein equations

2003-10-13

arxiv.org/abs/gr-qc/0310064v4

Phys.Rev. D69 (2004) 124020

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

v1: 9 pages. v2: sign error corrected in eq.(8) and relevant discussion amended accordingly. v3: short section added, to appea

Scientific paper

10.1103/PhysRevD.69.124020

We show how the use of the normal projection of the Einstein tensor as a set of boundary conditions relates to the propagation of the constraints, for two representations of the Einstein equations with vanishing shift vector: the ADM formulation, which is ill posed, and the Einstein-Christoffel formulation, which is symmetric hyperbolic. Essentially, the components of the normal projection of the Einstein tensor that act as non-trivial boundary conditions are linear combinations of the evolution equations with the constraints that are not preserved at the boundary, in both cases. In the process, the relationship of the normal projection of the Einstein tensor to the recently introduced ``constraint-preserving'' boundary conditions becomes apparent.

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