FIRST-ORDER HYPERBOLIC FORMALISM FOR NUMERICAL RELATIVITY

The causal structure of Einstein's evolution equations is considered. We show that in general they can be written as a first-order system of balance laws for any choice of slicing or shift. We also show how cer tain terms in the evolution equations, which can lead to numerical ina ccuracies, can be eliminated by using the Hamiltonian constraint. Furt hermore, we show that the entire system is hyperbolic when the time co ordinate is chosen in an invariant algebraic way, and for any fixed ch oice of the shift. This is achieved by using the momentum constraints in such a way that no additional space or time derivatives of the equa tions need to be computed. The slicings that allow hyperbolicity in th is formulation belong to a large class, including harmonic, maximal, a nd many others that have been commonly used in numerical relativity. W e provide details of some of the advanced numerical methods that this formulation of the equations allows, and we also discuss certain advan tages that a hyperbolic formulation provides when treating boundary co nditions.