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Geometrical Well Posed Systems for the Einstein Equations

Y. Choquet-Bruhat·J. W. York, Jr
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Abstract

We show that, given an arbitrary shift, the lapse $N$ can be chosen so that the extrinsic curvature $K$ of the space slices with metric $\overline g$ in arbitrary coordinates of a solution of Einstein's equations satisfies a quasi-linear wave equation. We give a geometric first order symmetric hyperbolic system verified in vacuum by $\overline g$, $K$ and $N$. We show that one can also obtain a quasi-linear wave equation for $K$ by requiring $N$ to satisfy at each time an elliptic equation which fixes the value of the mean extrinsic curvature of the space slices.

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Y. Choquet-Bruhat, J. W. York, Jr. 1995-06-28. Geometrical Well Posed Systems for the Einstein Equations. https://arxiv.org/abs/gr-qc/9506071

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