Constant Crunch Coordinates for Black Hole Simulations

TL;DR

This paper explores the use of time-independent constant mean curvature foliations in black hole simulations, providing analytic expressions and demonstrating stable evolution without singularity excision.

Contribution

It introduces a new class of $K$-constant surfaces with desirable properties for black hole simulations, including singularity avoidance and stability.

Findings

Stable evolution of a static black hole using the proposed foliation.

Analytic expressions for the closest approach of $K$-constant surfaces to the singularity.

Surfaces that are regular, static, and well-behaved across the horizon.

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.