Gauge structure of the Einstein field equations in Bondi-like coordinates

TL;DR

This paper analyzes the gauge structure of Einstein's equations in Bondi-like coordinates, revealing that such gauges often lead to weak hyperbolicity and ill-posedness in the characteristic initial value problem of GR.

Contribution

It identifies the root cause of weak hyperbolicity in Bondi-like gauges and demonstrates its implications through linear analysis and numerical tests.

Findings

Bondi-like gauges often cause weak hyperbolicity in Einstein equations.

Weak hyperbolicity leads to ill-posedness in standard norms.

Numerical tests confirm practical effects of weak hyperbolicity.

Abstract

The characteristic initial (boundary) value problem has numerous applications in general relativity (GR) involving numerical studies, and is often formulated using Bondi-like coordinates. Recently it was shown that several prototype formulations of this type are only weakly hyperbolic. Presently we examine the root cause of this result. In a linear analysis we identify the gauge, constraint and physical blocks in the principal part of the Einstein field equations in such a gauge, and show that the subsystem related to the gauge variables is only weakly hyperbolic. Weak hyperbolicity of the full system follows as a consequence in many cases. We demonstrate this explicitly in specific examples, and thus argue that Bondi-like gauges result in weakly hyperbolic free evolution systems under quite general conditions. Consequently the characteristic initial (boundary) value problem of GR in these gauges is rendered ill-posed in the simplest norms one would like to employ. The possibility of finding good alternative norms, in which well-posedness is achieved, is discussed. So motivated, we present numerical convergence tests with an implementation of full GR which demonstrate the effect of weak hyperbolicity in practice.