857 research outputs found
Implementation of higher-order absorbing boundary conditions for the Einstein equations
We present an implementation of absorbing boundary conditions for the
Einstein equations based on the recent work of Buchman and Sarbach. In this
paper, we assume that spacetime may be linearized about Minkowski space close
to the outer boundary, which is taken to be a coordinate sphere. We reformulate
the boundary conditions as conditions on the gauge-invariant
Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated
by rewriting the boundary conditions as a system of ODEs for a set of auxiliary
variables intrinsic to the boundary. From these we construct boundary data for
a set of well-posed constraint-preserving boundary conditions for the Einstein
equations in a first-order generalized harmonic formulation. This construction
has direct applications to outer boundary conditions in simulations of isolated
systems (e.g., binary black holes) as well as to the problem of
Cauchy-perturbative matching. As a test problem for our numerical
implementation, we consider linearized multipolar gravitational waves in TT
gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We
demonstrate that the perfectly absorbing boundary condition B_L of order L=l
yields no spurious reflections to linear order in perturbation theory. This is
in contrast to the lower-order absorbing boundary conditions B_L with L<l,
which include the widely used freezing-Psi_0 boundary condition that imposes
the vanishing of the Newman-Penrose scalar Psi_0.Comment: 25 pages, 9 figures. Minor clarifications. Final version to appear in
Class. Quantum Grav
Improved outer boundary conditions for Einstein's field equations
In a recent article, we constructed a hierarchy B_L of outer boundary
conditions for Einstein's field equations with the property that, for a
spherical outer boundary, it is perfectly absorbing for linearized
gravitational radiation up to a given angular momentum number L. In this
article, we generalize B_2 so that it can be applied to fairly general
foliations of spacetime by space-like hypersurfaces and general outer boundary
shapes and further, we improve B_2 in two steps: (i) we give a local boundary
condition C_2 which is perfectly absorbing including first order contributions
in 2M/R of curvature corrections for quadrupolar waves (where M is the mass of
the spacetime and R is a typical radius of the outer boundary) and which
significantly reduces spurious reflections due to backscatter, and (ii) we give
a non-local boundary condition D_2 which is exact when first order corrections
in 2M/R for both curvature and backscatter are considered, for quadrupolar
radiation.Comment: accepted Class. Quant. Grav. numerical relativity special issue; 17
pages and 1 figur
Explicit solution of the linearized Einstein equations in TT gauge for all multipoles
We write out the explicit form of the metric for a linearized gravitational
wave in the transverse-traceless gauge for any multipole, thus generalizing the
well-known quadrupole solution of Teukolsky. The solution is derived using the
generalized Regge-Wheeler-Zerilli formalism developed by Sarbach and Tiglio.Comment: 9 pages. Minor corrections, updated references. Final version to
appear in Class. Quantum Gra
Effective-one-body waveforms calibrated to numerical relativity simulations: coalescence of non-precessing, spinning, equal-mass black holes
We present the first attempt at calibrating the effective-one-body (EOB)
model to accurate numerical-relativity simulations of spinning, non-precessing
black-hole binaries. Aligning the EOB and numerical waveforms at low frequency
over a time interval of 1000M, we first estimate the phase and amplitude errors
in the numerical waveforms and then minimize the difference between numerical
and EOB waveforms by calibrating a handful of EOB-adjustable parameters. In the
equal-mass, spin aligned case, we find that phase and fractional amplitude
differences between the numerical and EOB (2,2) mode can be reduced to 0.01
radians and 1%, respectively, over the entire inspiral waveforms. In the
equal-mass, spin anti-aligned case, these differences can be reduced to 0.13
radians and 1% during inspiral and plunge, and to 0.4 radians and 10% during
merger and ringdown. The waveform agreement is within numerical errors in the
spin aligned case while slightly over numerical errors in the spin anti-aligned
case. Using Enhanced LIGO and Advanced LIGO noise curves, we find that the
overlap between the EOB and the numerical (2,2) mode, maximized over the
initial phase and time of arrival, is larger than 0.999 for binaries with total
mass 30-200Ms. In addition to the leading (2,2) mode, we compare four
subleading modes. We find good amplitude and frequency agreements between the
EOB and numerical modes for both spin configurations considered, except for the
(3,2) mode in the spin anti-aligned case. We believe that the larger difference
in the (3,2) mode is due to the lack of knowledge of post-Newtonian spin
effects in the higher modes.Comment: 15 pages, 8 figures, typos fixed in Eqs.(7-10
Schwarzschild Tests of the Wahlquist-Estabrook-Buchman-Bardeen Tetrad Formulation for Numerical Relativity
A first order symmetric hyperbolic tetrad formulation of the Einstein
equations developed by Estabrook and Wahlquist and put into a form suitable for
numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted
to explicit spherical symmetry and tested for accuracy and stability in the
evolution of spherically symmetric black holes (the Schwarzschild geometry).
The lapse and shift which specify the evolution of the coordinates relative to
the tetrad congruence are reset at frequent time intervals to keep the
constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the
spatial coordinate satisfying a kind of minimal rate of strain condition. By
arranging through initial conditions that the constant-time hypersurfaces are
asymptotically hyperbolic, we simplify the boundary value problem and improve
stability of the evolution. Results are obtained for both tetrad gauges
(``Nester'' and ``Lorentz'') of the WEBB formalism using finite difference
numerical methods. We are able to obtain stable unconstrained evolution with
the Nester gauge for certain initial conditions, but not with the Lorentz
gauge.Comment: (accepted by Phys. Rev. D) minor changes; typos correcte
The NINJA-2 catalog of hybrid post-Newtonian/numerical-relativity waveforms for non-precessing black-hole binaries
The Numerical INJection Analysis (NINJA) project is a collaborative effort between members of the numerical relativity and gravitational wave data analysis communities. The purpose of NINJA is to study the sensitivity of existing gravitational-wave search and parameter-estimation algorithms using numerically generated waveforms, and to foster closer collaboration between the numerical relativity and data analysis communities. The first NINJA project used only a small number of injections of short numerical-relativity waveforms, which limited its ability to draw quantitative conclusions. The goal of the NINJA-2 project is to overcome these limitations with long post-Newtonian - numerical relativity hybrid waveforms, large numbers of injections, and the use of real detector data. We report on the submission requirements for the NINJA-2 project and the construction of the waveform catalog. Eight numerical relativity groups have contributed 63 hybrid waveforms consisting of a numerical portion modelling the late inspiral, merger, and ringdown stitched to a post-Newtonian portion modelling the early inspiral. We summarize the techniques used by each group in constructing their submissions. We also report on the procedures used to validate these submissions, including examination in the time and frequency domains and comparisons of waveforms from different groups against each other. These procedures have so far considered only the mode. Based on these studies we judge that the hybrid waveforms are suitable for NINJA-2 studies. We note some of the plans for these investigations
Testing outer boundary treatments for the Einstein equations
Various methods of treating outer boundaries in numerical relativity are
compared using a simple test problem: a Schwarzschild black hole with an
outgoing gravitational wave perturbation. Numerical solutions computed using
different boundary treatments are compared to a `reference' numerical solution
obtained by placing the outer boundary at a very large radius. For each
boundary treatment, the full solutions including constraint violations and
extracted gravitational waves are compared to those of the reference solution,
thereby assessing the reflections caused by the artificial boundary. These
tests use a first-order generalized harmonic formulation of the Einstein
equations. Constraint-preserving boundary conditions for this system are
reviewed, and an improved boundary condition on the gauge degrees of freedom is
presented. Alternate boundary conditions evaluated here include freezing the
incoming characteristic fields, Sommerfeld boundary conditions, and the
constraint-preserving boundary conditions of Kreiss and Winicour. Rather
different approaches to boundary treatments, such as sponge layers and spatial
compactification, are also tested. Overall the best treatment found here
combines boundary conditions that preserve the constraints, freeze the
Newman-Penrose scalar Psi_0, and control gauge reflections.Comment: Modified to agree with version accepted for publication in Class.
Quantum Gra
An axisymmetric evolution code for the Einstein equations on hyperboloidal slices
We present the first stable dynamical numerical evolutions of the Einstein
equations in terms of a conformally rescaled metric on hyperboloidal
hypersurfaces extending to future null infinity. Axisymmetry is imposed in
order to reduce the computational cost. The formulation is based on an earlier
axisymmetric evolution scheme, adapted to time slices of constant mean
curvature. Ideas from a previous study by Moncrief and the author are applied
in order to regularize the formally singular evolution equations at future null
infinity. Long-term stable and convergent evolutions of Schwarzschild spacetime
are obtained, including a gravitational perturbation. The Bondi news function
is evaluated at future null infinity.Comment: 21 pages, 4 figures. Minor additions, updated to agree with journal
versio
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