Constant crunch coordinates for black hole simulations

Abstract

We reinvestigate the utility of time-independent constant mean curvature foliations for the numerical simulation of a single spherically-symmetric black hole. Each spacelike hypersurface of such a foliation is endowed with the same constant value of the trace of the extrinsic curvature tensor, K. Of the three families of K-constant surfaces possible (classified according to their asymptotic behaviors), we single out a sub-family of singularity-avoiding surfaces that may be particularly useful, and provide an analytic expression for the closest approach such surfaces make to the singularity. We then utilize a non-zero shift to yield families of K-constant surfaces which (1) avoid the black hole singularity, and thus the need to excise the singularity, (2) are asymptotically null, aiding in gravity wave extraction, (3) cover the physically relevant part of the spacetime, (4) are well behaved (regular) across the horizon, and (5) are static under evolution, and therefore have no "grid stretching/sucking" pathologies. Preliminary numerical runs demonstrate that we can stably evolve a single spherically-symmetric static black hole using this foliation. We wish to emphasize that this coordinatization produces K-constant surfaces for a single black hole spacetime that are regular, static and stable throughout their evolution. I. CONSTANT CRUNCH SURFACESIn this paper, we address a single question: Is there a numerically-viable coordinatization of a Schwarzschild black hole spacetime foliated by hypersurfaces of constant (not necessarily zero) mean extrinsic curvature? In other words, can we coordinatize the Schwarzschild spacetime with constant mean extrinsic curvature (T r(K) = constant) hypersurfaces so as to bound the growth of metric components and their gradients? We demonstrate here that the single shift freedom yields a spacetime metric that is static, and therefore bounds the growth in time of such gradients. A more complete analysis of the stability of our coordinatization, and a more thorough canvassing of the parameter space, will appear elsewhere.[1] Our foliation is consistent with that of Iriondo et al. [2], who provided a generic constant mean curvature (CMC) foliation of the Reissner-Nordström spacetime for the purpose of finding trapped surfaces. In this paper we focus on the utility of CMC slicings for the numerical simulation of black holes, in support of the emerging field of gravity-wave astrophysics.The trace of the extrinsic curvature tensor (T r(K) = K a a = K) at a point on a spacelike hypersurface measures the fractional rate of contraction of 3-volume along a unit normal to the surface. It represents the amount of "crunch" the 3-surface is experiencing at the point, at a given time. If all the observers throughout a spacelike hypersurface moving in time orthogonal to the surface experience the same amount of contraction per unit proper time, we say that the surface is a K-surface or a "constant crunch" surface. In this paper we examine foliations of a single sphericallysymmetric, static black h...

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