The appearance of coordinate shocks in hyperbolic formalisms of General Realtivity

TL;DR

This paper investigates the formation of coordinate shocks in hyperbolic formulations of General Relativity, particularly in the Bona-Masso formalism, showing how they develop and can be mitigated or persist depending on gauge choices.

Contribution

It demonstrates the conditions under which coordinate shocks form in the Bona-Masso formalism and shows their inevitability in certain gauge choices, with numerical validation.

Findings

Coordinate shocks develop in the Bona-Masso formalism.

One family of shocks can be eliminated by restricting slicing conditions.

Numerical simulations confirm shock formation in various spacetime models.

Abstract

I consider the appearance of shocks in hyperbolic formalisms of General Relativity. I study the particular case of the Bona-Masso formalism with zero shift vector and show how shocks associated with two families of characteristic fields can develop. These shocks do not represent discontinuities in the geometry of spacetime, but rather regions where the coordinate system becomes pathological. For this reason I call them coordinate shocks. I show how one family of shocks can be eliminated by restricting the Bona-Masso slicing condition to a special case. The other family of shocks, however, can not be eliminated even in the case of harmonic slicing. I also show the results of numerical simulations in the special cases of a flat two-dimensional spacetime, a flat four-dimensional spacetime with a spherically symmetric slicing, and a spherically symmetric black hole spacetime. In all three cases coordinate shocks readily develop, confirming the predictions of the mathematical analysis. Although here I concentrate in the Bona-Masso formalism, the phenomena of coordinate shocks should arise in any other hyperbolic formalism. In particular, since the appearance of the shocks is determined by the choice of gauge, the results presented here imply that in any formalism the use of a harmonic slicing can generate shocks.