Elliptic-hyperbolic systems and the Einstein equations

Details Elliptic-hyperbolic systems and the Einstein equations Elliptic-hyperbolic systems and the Einstein equations

2001-10-25

arxiv.org/abs/gr-qc/0110111v3

Annales Henri Poincare 4 (2003) 1-34

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

29 pages, several minor corrections, submitted to Ann. H. Poincare

Scientific paper

The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasilinear elliptic--hyperbolic system of evolution equations. We prove that the Cauchy problem is locally strongly well--posed and that a continuation principle holds. For initial data satisfying the Einstein constraint and gauge conditions, the solutions to the elliptic-hyperbolic system defined by the gauge fixed Einstein evolution equations are shown to give vacuum spacetimes.

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