Boosted Three-Dimensional Black-Hole Evolutions with Singularity Excision

Abstract

Binary black hole interactions provide potentially the strongest source of gravitational radiation for detectors currently under development. We present some results from the Binary Black Hole Grand Challenge Alliance three-dimensional Cauchy evolution module. These constitute essential steps towards modeling such interactions and predicting gravitational radiation waveforms. We report on single black hole evolutions and the first successful demonstration of a black hole moving freely through a three-dimensional computational grid via a Cauchy evolution: a hole moving ∼ 6M at 0.1c during a total evolution of duration ∼ 60M . The accurate computational modeling of black-hole interactions is essential to the confident detection of astrophysical gravitational radiation by future space-based detectors such as LISA and by the LIGO/VIRGO/GEO complex of ground-based detectors currently under construction. The sensitivity of these detectors will be significantly enhanced if accurate computer simulations of black-hole mergers can produce predictions of radiation waveforms [1]. The Binary Black Hole Grand Challenge Alliance [2] was funded in September 1993 to develop the computational infrastructure necessary accurately to simulate the coalescence of black-hole binaries. The primary objective of the resulting code will be the production of waveforms from binary black hole mergers. In this Letter we report on an important step towards achieving such simulations.A key difficulty in evolving black-hole spacetimes is handling the curvature singularity contained within each hole. The only viable means of accomplishing this over time scales required for binary coalescence appears to be black-hole excision: exclude all or part of the black-hole interior (and the singularity) from the computational domain and evolve only the exterior region [3,4

An explicit harmonic code for black-hole evolution using excision

“…The effects of these adjustments in suppressing long wavelength instabilities in standardized tests for periodic boundary conditions have been discussed by Babiuc et al [25]. In particular, the first and second terms in the adjustments (10) have been shown to be effective in suppressing constraint-violating nonlinear instabilities in shifted gauge-wave tests. The third term in (10), on the other hand, was first considered in [39] and leads to constraint damping in the linear regime.…”

A Fast Apparent-Horizon Finder for 3-Dimensional Cartesian Grids in Numerical Relativity

“…The AHFINDERDIRECT values are in excellent agreement with those of [1], and are dramatically more accurate. 15 Raising this to 50 terms changed the numerically computed critical µ by <10 −12 , and the horizon area by <10 −7 . 16 I used very high resolutions here, with grid spacings as small as 0.01 for the CACTUS 3D grid and 0.5 • for the AHFINDERDIRECT angular grid.…”

“…In cases where the distinction is important, I use a prefix (3) to denote quantities defined on a 3D (three-dimensional) neighbourhood of the apparent horizon surface. 1 For more recent work on this topic, see (for example) [2,3,11,15]. 2 For numerical purposes the usual approximate development to a nearly stationary state suffices [4,13,21,32].…”

Cauchy-perturbative matching and outer boundary conditions: Computational studies

“…(B) The values of multipole amplitudes and their time derivatives computed at the extraction 2-sphere on a given timeslice are imposed as inner boundary conditions on the 1D grid L and evolved [using the radial wave equations (4)-(6) for each (ℓ, m) mode] forward to the following timeslice 3 This provides the solution, for the new time level, on the whole perturbative region P. Since the outer boundary of L is located, by construction, well out in the wave zone, a simple radial outgoing wave Sommerfeld condition can be imposed there.…”

“…Present three-dimensional (3D) numerical relativity simulations face a number of fundamental and in some cases unsolved problems, including: coordinate choice, most suitable form of Einstein's equations, singularity avoiding techniques, gravitational wave extraction and outer boundary conditions. While a robust solution to the generic problem is still awaited, some interesting results have already been obtained, for instance, in the evolution of a generic 3D black hole [2], in the translation of a 3D black hole across a numerical grid [3], and in the extraction of gravitational wave information and imposition of outer boundary conditions [4].…”