Time functions in numerical relativity: Marginally bound dust collapse

Abstract

We explore the problem of the existence of global maximal ($K=0$) and constant-mean-curvature ($K={K}_{0}$) time functions in general relativity. We attempt a rigorous definition of numerical relativity so as to bridge the gap between the field and mathematical relativity. We point out that numerical relativity can in principle construct any globally hyperbolic solution to Einstein's equations. This involves the construction of Cauchy time functions. Therefore we review what is known about the existence and uniqueness of such functions when their mean curvature is specified to be a constant on each time slice. We note that in strong-field solutions which contain singularities the question of existence is intimately connected to the nature of the singularity. Defining the class of "crushing singularities" we prove new theorems showing that $K=0$ or $K={K}_{0}$ time functions uniformly avoid such singularities (which include both Cauchy horizons and some curvature singularities). We then study the inhomogeneous generalizations of the Oppenheimer-Snyder spherical-dust-collapse spacetimes. These Tolman-Bondi solutions are classified as to their causal structure and found to contain naked singularities of a new type if the collapse is sufficiently inhomogeneous. We calculate the $K=0$ and $K={K}_{0}$ time slices for a variety of these spacetimes. We find that since some extreme dust collapses lead to noncrushing singularities, maximal time slicing can hit the singularity before covering the domain of outer communications of the resulting black hole. Furthermore, the use of $K={K}_{0}$ slices in the presence of a naked singularity is discussed.