9 research outputs found
Comparing initial-data sets for binary black holes
We compare the results of constructing binary black hole initial data with
three different decompositions of the constraint equations of general
relativity. For each decomposition we compute the initial data using a
superposition of two Kerr-Schild black holes to fix the freely specifiable
data. We find that these initial-data sets differ significantly, with the ADM
energy varying by as much as 5% of the total mass. We find that all
initial-data sets currently used for evolutions might contain unphysical
gravitational radiation of the order of several percent of the total mass. This
is comparable to the amount of gravitational-wave energy observed during the
evolved collision. More astrophysically realistic initial data will require
more careful choices of the freely specifiable data and boundary conditions for
both the metric and extrinsic curvature. However, we find that the choice of
extrinsic curvature affects the resulting data sets more strongly than the
choice of conformal metric.Comment: 18 pages, 12 figures, accepted for publication in Phys. Rev.
Comparing Criteria for Circular Orbits in General Relativity
We study a simple analytic solution to Einstein's field equations describing
a thin spherical shell consisting of collisionless particles in circular orbit.
We then apply two independent criteria for the identification of circular
orbits, which have recently been used in the numerical construction of binary
black hole solutions, and find that both yield equivalent results. Our
calculation illustrates these two criteria in a particularly transparent
framework and provides further evidence that the deviations found in those
numerical binary black hole solutions are not caused by the different criteria
for circular orbits.Comment: 4 pages; to appear in PRD as a Brief Report; added and corrected
reference
Extrinsic Curvature and the Einstein Constraints
The Einstein initial-value equations in the extrinsic curvature (Hamiltonian)
representation and conformal thin sandwich (Lagrangian) representation are
brought into complete conformity by the use of a decomposition of symmetric
tensors which involves a weight function. In stationary spacetimes, there is a
natural choice of the weight function such that the transverse traceless part
of the extrinsic curvature (or canonical momentum) vanishes.Comment: 8 pages, no figures; added new section; significant polishing of tex
On geometric problems related to Brown-York and Liu-Yau quasilocal mass
We discuss some geometric problems related to the definitions of quasilocal
mass proposed by Brown-York \cite{BYmass1} \cite{BYmass2} and Liu-Yau
\cite{LY1} \cite{LY2}. Our discussion consists of three parts. In the first
part, we propose a new variational problem on compact manifolds with boundary,
which is motivated by the study of Brown-York mass. We prove that critical
points of this variation problem are exactly static metrics. In the second
part, we derive a derivative formula for the Brown-York mass of a smooth family
of closed 2 dimensional surfaces evolving in an ambient three dimensional
manifold. As an interesting by-product, we are able to write the ADM mass
\cite{ADM61} of an asymptotically flat 3-manifold as the sum of the Brown-York
mass of a coordinate sphere and an integral of the scalar curvature plus
a geometrically constructed function in the asymptotic region outside
. In the third part, we prove that for any closed, spacelike, 2-surface
in the Minkowski space for which the Liu-Yau mass is
defined, if bounds a compact spacelike hypersurface in ,
then the Liu-Yau mass of is strictly positive unless lies on
a hyperplane. We also show that the examples given by \'{O} Murchadha, Szabados
and Tod \cite{MST} are special cases of this result.Comment: 28 page
Improved numerical stability of stationary black hole evolution calculations
We experiment with modifications of the BSSN form of the Einstein field
equations (a reformulation of the ADM equations) and demonstrate how these
modifications affect the stability of numerical black hole evolution
calculations. We use excision to evolve both non-rotating and rotating
Kerr-Schild black holes in octant and equatorial symmetry, and without any
symmetry assumptions, and obtain accurate and stable simulations for specific
angular momenta J/M of up to about 0.9M.Comment: 13 pages, 11 figures, 1 typo in Eq. (20) correcte
Relativistic Hydrodynamic Evolutions with Black Hole Excision
We present a numerical code designed to study astrophysical phenomena
involving dynamical spacetimes containing black holes in the presence of
relativistic hydrodynamic matter. We present evolutions of the collapse of a
fluid star from the onset of collapse to the settling of the resulting black
hole to a final stationary state. In order to evolve stably after the black
hole forms, we excise a region inside the hole before a singularity is
encountered. This excision region is introduced after the appearance of an
apparent horizon, but while a significant amount of matter remains outside the
hole. We test our code by evolving accurately a vacuum Schwarzschild black
hole, a relativistic Bondi accretion flow onto a black hole, Oppenheimer-Snyder
dust collapse, and the collapse of nonrotating and rotating stars. These
systems are tracked reliably for hundreds of M following excision, where M is
the mass of the black hole. We perform these tests both in axisymmetry and in
full 3+1 dimensions. We then apply our code to study the effect of the stellar
spin parameter J/M^2 on the final outcome of gravitational collapse of rapidly
rotating n = 1 polytropes. We find that a black hole forms only if J/M^2<1, in
agreement with previous simulations. When J/M^2>1, the collapsing star forms a
torus which fragments into nonaxisymmetric clumps, capable of generating
appreciable ``splash'' gravitational radiation.Comment: 17 pages, 14 figures, submitted to PR