Well-posed forms of the 3+1 conformally-decomposed Einstein equations

TL;DR

This paper develops well-posed conformally-decomposed formulations of the 3+1 Einstein equations, ensuring stable evolution by analyzing their characteristic structure and identifying parameter families with causal properties.

Contribution

It introduces new well-posed formulations by densitizing the lapse and combining constraints, revealing a three-parameter family with causal characteristics.

Findings

Existence of a 3-parameter family of well-posed formulations.

Identification of formulations with causal characteristics within the lightcone.

Explicit characterization of the free functions and parameters in the system.

Abstract

We show that well-posed, conformally-decomposed formulations of the 3+1 Einstein equations can be obtained by densitizing the lapse and by combining the constraints with the evolution equations. We compute the characteristics structure and verify the constraint propagation of these new well-posed formulations. In these formulations, the trace of the extrinsic curvature and the determinant of the 3-metric are singled out from the rest of the dynamical variables, but are evolved as part of the well-posed evolution system. The only free functions are the lapse density and the shift vector. We find that there is a 3-parameter freedom in formulating these equations in a well-posed manner, and that part of the parameter space found consists of formulations with causal characteristics, namely, characteristics that lie only within the lightcone. In particular there is a 1-parameter family of systems whose characteristics are either normal to the slicing or lie along the lightcone of the evolving metric.