Steven Brandt

A Simple Construction of Initial Data for Multiple Black Holes

Binary black hole spacetimes are one of the great challenges for numerical general relativity, even if no matter sources are present. Here we consider the problem of finding initial data for several black holes in vacuum with arbitrary momenta and spins. In general relativity, initial data on a hypersurface cannot be specified freely, because the Einstein equations give rise to four equations, three momentum constraints, and the Hamiltonian constraint that the initial data has to satisfy. The purpose of this Letter is to introduce a novel approach which is significantly simpler than the conventional method based on throats and conformal imaging. In all that follows we will assume vacuum, that the metric is conformally flat, and that the extrinsic curvature is tracefree. A convenient form of the constraints of general relativity can be obtained by rescaling the physical three-metric g ph ab and its extrinsic curvature K ph ab by a conformal factor c,

SYMMETRY WITHOUT SYMMETRY: NUMERICAL SIMULATION OF AXISYMMETRIC SYSTEMS USING CARTESIAN GRIDS

We present a new technique for the numerical simulation of axisymmetric systems. This technique avoids the coordinate singularities which often arise when cylindrical or polar-spherical coordinate finite difference grids are used, particularly in simulating tensor partial differential equations like those of 3+1 numerical relativity. For a system axisymmetric about the z axis, the basic idea is to use a three-dimensional Cartesian(x,y,z) coordinate grid which covers (say) the y=0 plane, but is only one finite-difference-molecule–width thick in the y direction. The field variables in the central y=0 grid plane can be updated using normal (x,y,z)-coordinate finite differencing, while those in the y≠ 0 grid planes can be computed from those in the central plane by using the axisymmetry assumption and interpolation. We demonstrate the effectiveness of the approach on a set of fully nonlinear test computations in 3+1 numerical general relativity, involving both black holes and collapsing gravitational waves.

Boosted three-dimensional black-hole evolutions with singularity excision

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 near 6M at 0.1c during a total evolution of duration near 60M. [S0031-9007(98)05652-X]

GRAVITATIONAL WAVE EXTRACTION AND OUTER BOUNDARY CONDITIONS BY PERTURBATIVE MATCHING

We present a method for extracting gravitational radiation from a three-dimensional numerical relativity simulation and, using the extracted data, to provide outer boundary conditions. The method treats dynamical gravitational variables as nonspherical perturbations of Schwarzschild geometry. We discuss a code which implements this method and present results of tests which have been performed with a three-dimensional numerical relativity code.

Stable Characteristic Evolution of Generic Three-Dimensional Single-Black-Hole Spacetimes

We report new results which establish that the accurate 3dimensional numerical simulation of generic single-black-hole spacetimes has been achieved by characteristic evolution with unlimited long term stability. Our results cover a selection of distorted, moving and spinning single black holes, with evolution times up to 60,000M. 04.25.Dm,04.30.Db

Dynamics of Apparent and Event Horizons.

The dynamics of apparent and event horizons of various black hole spacetimes, including those containing distorted, rotating and colliding black holes, are studied. We have developed a powerful and efficient new method for locating the event horizon, making possible the study of both types of horizons in numerical relativity. We show that both the event and apparent horizons, in all dynamical black hole spacetimes studied, oscillate with the quasinormal frequency.

Test-beds and applications for apparent horizon finders in numerical relativity

We present a series of test-beds for numerical codes designed to find apparent horizons. We compare three apparent horizon finders that use different numerical methods: one of them in axisymmetry and two which are fully three dimensional. We concentrate first on a toy model that has a simple horizon structure, and then go on to study single and multiple black hole datasets. We use our finders to look for apparent horizons in Brill wave initial data where we discover that some results published previously are not correct. For pure wave and multiple black hole spacetimes, we apply our finders to survey parameter space.

Evolution of distorted rotating black holes. I. Methods and tests

We have developed a new numerical code to study the evolution of distorted, rotating black holes. We discuss the numerical methods and gauge conditions we developed to evolve such spacetimes. The code has been put through a series of tests and we report on (a) results of comparisons with codes designed to evolve nonrotating holes, (b) the evolution of Kerr spacetimes for which analytic properties are known, and (c) the evolution of distorted rotating holes. The code accurately reproduces results of the previous NCSA nonrotating code and passes convergence tests. New features of the evolution of rotating black holes not seen in nonrotating holes are identified. With this code we can evolve rotating black holes up to about t=100M, depending on the resolution and angular momentum. We also describe a new family of black hole initial data sets which represent rotating holes with a wide range of distortion parameters, and distorted nonrotating black holes with odd-parity radiation. Finally, we study the limiting slices for a maximally sliced rotating black hole and find good agreement with theoretical predictions.

Three-dimensional distorted black holes

We present three-dimensional, non-axisymmetric distorted black-hole initial data which generalize the axisymmetric, distorted, non-rotating [1] and rotating [2] single black-hole data developed by Bernstein, Brandt and Seidel. These initial data should be useful for studying the dynamics of fully 3D, distorted black holes, such as those created by the spiralling coalescence of two black holes. We describe the mathematical construction of several families of such datasets, and show how to construct numerical solutions. We survey quantities associated with the numerically constructed solutions, such as ADM masses, apparent horizons, measurements of the horizon distortion and the maximum possible radiation loss.

Head--on Collision of Two Unequal Mass Black Holes

We present results from the first fully nonlinear numerical calculations of the head-on collision of two unequal mass black holes. Selected waveforms of the most dominant