Ill-posedness in the Einstein equations

TL;DR

The paper demonstrates that the commonly used ADM formulation of Einstein equations is ill-posed, with unbounded solution norms relative to initial data, but practical numerical simulations may still remain stable.

Contribution

It identifies and analyzes the ill-posedness of the standard ADM and conformally-decomposed Einstein equations formulations used in numerical relativity.

Findings

Standard ADM form is ill-posed with unbounded solution norms.

Numerical experiments suggest practical stability despite theoretical ill-posedness.

Ill-posedness may not critically affect certain numerical simulations.

Abstract

It is shown that the formulation of the Einstein equations widely in use in numerical relativity, namely, the standard ADM form, as well as some of its variations (including the most recent conformally-decomposed version), suffers from a certain but standard type of ill-posedness. Specifically, the norm of the solution is not bounded by the norm of the initial data irrespective of the data. A long-running numerical experiment is performed as well, showing that the type of ill-posedness observed may not be serious in specific practical applications, as is known from many numerical simulations.