Potential for ill-posedness in several 2nd-order formulations of the Einstein equations

TL;DR

This paper investigates the well-posedness of several second-order formulations of the Einstein equations, revealing potential ill-posedness and challenging assumptions about their stability based on first-order reductions.

Contribution

It demonstrates that common second-order formulations like ADM, BSSN, and Einstein-Christoffel are not strongly hyperbolic, questioning their well-posedness assumptions.

Findings

None of the formulations are strongly hyperbolic.

Some formulations are weakly hyperbolic, possibly well-posed under restrictive conditions.

Well-posedness cannot be inferred from first-order reductions.

Abstract

Second-order formulations of the 3+1 Einstein equations obtained by eliminating the extrinsic curvature in terms of the time derivative of the metric are examined with the aim of establishing whether they are well posed, in cases of somewhat wide interest, such as ADM, BSSN and generalized Einstein-Christoffel. The criterion for well-posedness of second-order systems employed is due to Kreiss and Ortiz. By this criterion, none of the three cases are strongly hyperbolic, but some of them are weakly hyperbolic, which means that they may yet be well posed but only under very restrictive conditions for the terms of order lower than second in the equations (which are not studied here). As a result, intuitive transferences of the property of well-posedness from first-order reductions of the Einstein equations to their originating second-order versions are unwarranted if not false.