The appearance of coordinate shocks in hyperbolic formalisms of General Relativity

I consider the appearance of shocks in hyperbolic formalisms of General Relativity. I study the particular case of the Bona-Massó 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-Massó slicing condition ∂tα = −α2f(α) trK to the case f = 1 + k/α2, with k an arbitrary constant. The other family of shocks can not be eliminated even in the case of harmonic slicing (f = 1). I also show the results of the numerical evolution of non-trivial initial slices 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 I concentrate in the Bona-Massó formalism, the phenomena of coordinate shocks should arise in any other hyperbolic formalism. In ∗Present address: Max Planck Institut für Gravitationsphysik, Albert Einstein Institut, Schlaatzweg 1, D-14473 Potsdam, Germany.