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.

14 pages, 9 figures. Formatted using Revtex4. To appear Phys. Rev. D 2001, Added numerical results, updated references and revised figures

References in corpus (1)

• Towards a Stable Numerical Evolution of Strongly Gravitating Systems in General Relativity: The Conformal Treatments

Cited by in corpus (20)

• Numerical Relativity: A review
• Hyperboloidal foliations and scri-fixing
• Null infinity waveforms from extreme-mass-ratio inspirals in Kerr spacetime
• Binary black hole coalescence in the large-mass-ratio limit: the hyperboloidal layer method and waveforms at null infinity
• Constant mean curvature slices in the extended Schwarzschild solution and collapse of the lapse: Part I
• A hyperboloidal study of tail decay rates for scalar and Yang-Mills fields
• Hyperboloidal Approach to Quasinormal Modes
• Numerical stability of a new conformal-traceless 3+1 formulation of the Einstein equation
• Maximal Slicing for Puncture Evolutions of Schwarzschild and Reissner-Nordström Black Holes
• Stationary hyperboloidal slicings with evolved gauge conditions
• Late Time Analysis for Maximal Slicing of Reissner-Nordström Puncture Evolutions
• Spherically symmetric black hole spacetimes on hyperboloidal slices
• Axisymmetric constant mean curvature slices in the Kerr space-time
• Free evolution of the hyperboloidal initial value problem in spherical symmetry
• Height-function-based 4D reference metrics for hyperboloidal evolution
• Conformal diagrams for stationary and dynamical strong-field hyperboloidal slices
• The constraints as evolution equations for numerical relativity
• Free Hyperboloidal Evolution of the Einstein-Maxwell-Klein-Gordon System
• Slice Stretching Effects for Maximal Slicing of a Schwarzschild Black Hole
• Hyperboloidal neutron star and black hole initial data in spherical symmetry