Discrete differential forms in general relativity

Abstract

A major obstacle in the numerical simulation of general relativistic spacetimes is the fact that coordinates have to be specified in order to obtain a well-defined numerical evolution. While the choice of coordinates has no impact on the geometry, it does influence the evolution in ways which are rather difficult to predict. For this reason it might be useful to devise numerical procedures which are manifestly coordinate invariant. In this paper we present an alternative way to obtain discrete versions of the Einstein equations. We formulate the Einstein equations as an exterior system in terms of differential forms. Such a formulation can be used to interpret the variables and equations in a discrete setting as discrete differential forms. We describe the basic equations and discuss the accuracy of the discrete equations.