11 research outputs found
Approximate Analytical Solutions to the Initial Data Problem of Black Hole Binary Systems
We present approximate analytical solutions to the Hamiltonian and momentum
constraint equations, corresponding to systems composed of two black holes with
arbitrary linear and angular momentum. The analytical nature of these initial
data solutions makes them easier to implement in numerical evolutions than the
traditional numerical approach of solving the elliptic equations derived from
the Einstein constraints. Although in general the problem of setting up initial
conditions for black hole binary simulations is complicated by the presence of
singularities, we show that the methods presented in this work provide initial
data with and norms of violation of the constraint equations
falling below those of the truncation error (residual error due to
discretization) present in finite difference codes for the range of grid
resolutions currently used. Thus, these data sets are suitable for use in
evolution codes. Detailed results are presented for the case of a head-on
collision of two equal-mass M black holes with specific angular momentum 0.5M
at an initial separation of 10M. A straightforward superposition method yields
data adequate for resolutions of , and an "attenuated" superposition
yields data usable to resolutions at least as fine as . In addition, the
attenuated approximate data may be more tractable in a full (computational)
exact solution to the initial value problem.Comment: 6 pages, 5 postscript figures. Minor changes and some points
clarified. Accepted for publication in Phys. Rev.
Locating Boosted Kerr and Schwarzschild Apparent Horizons
We describe a finite-difference method for locating apparent horizons and
illustrate its capabilities on boosted Kerr and Schwarzschild black holes. Our
model spacetime is given by the Kerr-Schild metric. We apply a Lorentz boost to
this spacetime metric and then carry out a 3+1 decomposition. The result is a
slicing of Kerr/Schwarzschild in which the black hole is propagated and Lorentz
contracted. We show that our method can locate distorted apparent horizons
efficiently and accurately.Comment: Submitted to Physical Review D. 12 pages and 22 figure
Extended Lifetime in Computational Evolution of Isolated Black Holes
Solving the 4-d Einstein equations as evolution in time requires solving
equations of two types: the four elliptic initial data (constraint) equations,
followed by the six second order evolution equations. Analytically the
constraint equations remain solved under the action of the evolution, and one
approach is to simply monitor them ({\it unconstrained} evolution).
The problem of the 3-d computational simulation of even a single isolated
vacuum black hole has proven to be remarkably difficult. Recently, we have
become aware of two publications that describe very long term evolution, at
least for single isolated black holes. An essential feature in each of these
results is {\it constraint subtraction}. Additionally, each of these approaches
is based on what we call "modern," hyperbolic formulations of the Einstein
equations. It is generally assumed, based on computational experience, that the
use of such modern formulations is essential for long-term black hole
stability. We report here on comparable lifetime results based on the much
simpler ("traditional") - formulation.
We have also carried out a series of {\it constrained} 3-d evolutions of
single isolated black holes. We find that constraint solution can produce
substantially stabilized long-term single hole evolutions. However, we have
found that for large domains, neither constraint-subtracted nor constrained
- evolutions carried out in Cartesian coordinates admit
arbitrarily long-lived simulations. The failure appears to arise from features
at the inner excision boundary; the behavior does generally improve with
resolution.Comment: 20 pages, 6 figure
Analysis of ``Gauge Modes'' in Linearized Relativity
By writing the complete set of (ADM) equations for linearized waves,
we are able to demonstrate the properties of the initial data and of the
evolution of a wave problem set by Alcubierre and Schutz. We show that the
gauge modes and constraint error modes arise in a straightforward way in the
analysis, and are of a form which will be controlled in any well specified
convergent computational discretization of the differential equations.Comment: 11pages LaTe
Initial Data and Coordinates for Multiple Black Hole Systems
We present here an alternative approach to data setting for spacetimes with
multiple moving black holes generalizing the Kerr-Schild form for rotating or
non-rotating single black holes to multiple moving holes. Because this scheme
preserves the Kerr-Schild form near the holes, it selects out the behaviour of
null rays near the holes, may simplify horizon tracking, and may prove useful
in computational applications. For computational evolution, a discussion of
coordinates (lapse function and shift vector) is given which preserves some of
the properties of the single-hole Kerr-Schild form
Generic Tracking of Multiple Apparent Horizons with Level Flow
We report the development of the first apparent horizon locator capable of
finding multiple apparent horizons in a ``generic'' numerical black hole
spacetime. We use a level-flow method which, starting from a single arbitrary
initial trial surface, can undergo topology changes as it flows towards
disjoint apparent horizons if they are present. The level flow method has two
advantages: 1) The solution is independent of changes in the initial guess and
2) The solution can have multiple components. We illustrate our method of
locating apparent horizons by tracking horizon components in a short
Kerr-Schild binary black hole grazing collision.Comment: 13 pages including figures, submitted to Phys Rev
Three-dimensional adaptive evolution of gravitational waves in numerical relativity
Adaptive techniques are crucial for successful numerical modeling of
gravitational waves from astrophysical sources such as coalescing compact
binaries, since the radiation typically has wavelengths much larger than the
scale of the sources. We have carried out an important step toward this goal,
the evolution of weak gravitational waves using adaptive mesh refinement in the
Einstein equations. The 2-level adaptive simulation is compared with unigrid
runs at coarse and fine resolution, and is shown to track closely the features
of the fine grid run.Comment: REVTeX, 7 pages, including three figures; submitted to Physical
Review
Grazing Collisions of Black Holes via the Excision of Singularities
We present the first simulations of non-headon (grazing) collisions of binary
black holes in which the black hole singularities have been excised from the
computational domain. Initially two equal mass black holes are separated a
distance and with impact parameter . Initial data are
based on superposed, boosted (velocity ) solutions of single black
holes in Kerr-Schild coordinates. Both rotating and non-rotating black holes
are considered. The excised regions containing the singularities are specified
by following the dynamics of apparent horizons. Evolutions of up to are obtained in which two initially separate apparent horizons are present
for . At that time a single enveloping apparent horizon forms,
indicating that the holes have merged. Apparent horizon area estimates suggest
gravitational radiation of about 2.6% of the total mass. The evolutions end
after a moderate amount of time because of instabilities.Comment: 2 References corrected, reference to figure update
Approximate analytical solutions to the initial data problem of black hole binary systems
We present approximate analytical solutions to the Hamiltonian and momentum constraint equations, corresponding to systems composed of two black holes with arbitrary linear and angular momentum. The analytical nature of these initial data solutions makes them easier to implement in numerical evolutions than the traditional numerical approach of solving the elliptic equations derived from the Einstein constraints. Although in general the problem of setting up initial conditions for black hole binary simulations is complicated by the presence of singularities, we show that the methods presented in this work provide initial data with l 1 and l ϱ norms of violation of the constraint equations falling below those of the truncation error Í‘residual error due to discretizationÍ’ present in finite difference codes for the range of grid resolutions currently used. Thus, these data sets are suitable for use in evolution codes. Detailed results are presented for the case of a head-on collision of two equal-mass M black holes with specific angular momentum 0.5M at an initial separation of 10M . A straightforward superposition method yields data adequate for resolutions of hÏM /4, and an ''attenuated'' superposition yields data usable to resolutions at least as fine as hÏM /8. In addition, the attenuated approximate data may be more tractable in a full Í‘computationalÍ’ exact solution to the initial value problem