Oppenheimer-Snyder collapse in polar time slicing

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Abstract
Oppenheimer-Snyder collapse provides a good test-bed calculation for general-relativistic numerical codes that evolve matter in strong gravitational fields. Such codes are often based on the Arnowitt-Deser-Misner 3+1 formalism, for which an appropriate choice of coordinates (lapse and shift functions) is the key to a successful spacetime evolution. Polar time slicing has been proposed as a good choice of time coordinate because of its singularity-avoidance features. We express the entire Oppenheimer-Snyder solution analytically in polar time slicing to facilitate calibrations of numerical codes written in this gauge. We examine two possible choices of radial coordinate: isotropic and Schwarzschild. We find that, while polar slicing does indeed hold back collapse near the center of a black hole, it is not alone sufficient to allow a singularity-free numerical evolution. For example, with the Schwarzschild radial coordinate the radial metric coefficient develops an ever-increasing spike at the surface of the black hole. This does not occur in isotropic coordinates, where the metric coefficient reaches a finite maximum value at the center. By contrast, the isotropic radial metric coefficient in maximal time slicing diverges exponentially at the center at late times. These generic features may be important in constructing a numerical code.