785 research outputs found
Evolution systems for non-linear perturbations of background geometries
The formulation of the initial value problem for the Einstein equations is at
the heart of obtaining interesting new solutions using numerical relativity and
still very much under theoretical and applied scrutiny. We develop a
specialised background geometry approach, for systems where there is
non-trivial a priori knowledge about the spacetime under study. The background
three-geometry and associated connection are used to express the ADM evolution
equations in terms of physical non-linear deviations from that background.
Expressing the equations in first order form leads naturally to a system
closely linked to the Einstein-Christoffel system, introduced by Anderson and
York, and sharing its hyperbolicity properties. We illustrate the drastic
alteration of the source structure of the equations, and discuss why this is
likely to be numerically advantageous.Comment: 12 pages, 3 figures, Revtex v3.0. Revised version to appear in
Physical Review
Hyperboloidal evolution of test fields in three spatial dimensions
We present the numerical implementation of a clean solution to the outer
boundary and radiation extraction problems within the 3+1 formalism for
hyperbolic partial differential equations on a given background. Our approach
is based on compactification at null infinity in hyperboloidal scri fixing
coordinates. We report numerical tests for the particular example of a scalar
wave equation on Minkowski and Schwarzschild backgrounds. We address issues
related to the implementation of the hyperboloidal approach for the Einstein
equations, such as nonlinear source functions, matching, and evaluation of
formally singular terms at null infinity.Comment: 10 pages, 8 figure
On the determination of the last stable orbit for circular general relativistic binaries at the third post-Newtonian approximation
We discuss the analytical determination of the location of the Last Stable
Orbit (LSO) in circular general relativistic orbits of two point masses. We use
several different ``resummation methods'' (including new ones) based on the
consideration of gauge-invariant functions, and compare the results they give
at the third post-Newtonian (3PN) approximation of general relativity. Our
treatment is based on the 3PN Hamiltonian of Jaranowski and Sch\"afer. One of
the new methods we introduce is based on the consideration of the (invariant)
function linking the angular momentum and the angular frequency. We also
generalize the ``effective one-body'' approach of Buonanno and Damour by
introducing a non-minimal (i.e. ``non-geodesic'') effective dynamics at the 3PN
level. We find that the location of the LSO sensitively depends on the
(currently unknown) value of the dimensionless quantity \oms which
parametrizes a certain regularization ambiguity of the 3PN dynamics. We find,
however, that all the analytical methods we use numerically agree between
themselves if the value of this parameter is \oms\simeq-9. This suggests that
the correct value of \oms is near -9 (the precise value
\oms^*\equiv-{47/3}+{41/64}\pi^2=-9.3439... seems to play a special role). If
this is the case, we then show how to further improve the analytical
determination of various LSO quantities by using a ``Shanks'' transformation to
accelerate the convergence of the successive (already resummed) PN estimates.Comment: REVTeX, 25 pages, 3 figures, submitted to Phys. Rev.
Einstein boundary conditions for the 3+1 Einstein equations
In the 3+1 framework of the Einstein equations for the case of vanishing
shift vector and arbitrary lapse, we calculate explicitly the four boundary
equations arising from the vanishing of the projection of the Einstein tensor
along the normal to the boundary surface of the initial-boundary value problem.
Such conditions take the form of evolution equations along (as opposed to
across) the boundary for certain components of the extrinsic curvature and for
certain space-derivatives of the intrinsic metric. We argue that, in general,
such boundary conditions do not follow necessarily from the evolution equations
and the initial data, but need to be imposed on the boundary values of the
fundamental variables. Using the Einstein-Christoffel formulation, which is
strongly hyperbolic, we show how three of the boundary equations should be used
to prescribe the values of some incoming characteristic fields. Additionally,
we show that the fourth one imposes conditions on some outgoing fields.Comment: Revtex 4, 6 pages, text and references added, typos corrected, to
appear in Phys. Rev.
On the Performance Prediction of BLAS-based Tensor Contractions
Tensor operations are surging as the computational building blocks for a
variety of scientific simulations and the development of high-performance
kernels for such operations is known to be a challenging task. While for
operations on one- and two-dimensional tensors there exist standardized
interfaces and highly-optimized libraries (BLAS), for higher dimensional
tensors neither standards nor highly-tuned implementations exist yet. In this
paper, we consider contractions between two tensors of arbitrary dimensionality
and take on the challenge of generating high-performance implementations by
resorting to sequences of BLAS kernels. The approach consists in breaking the
contraction down into operations that only involve matrices or vectors. Since
in general there are many alternative ways of decomposing a contraction, we are
able to methodically derive a large family of algorithms. The main contribution
of this paper is a systematic methodology to accurately identify the fastest
algorithms in the bunch, without executing them. The goal is instead
accomplished with the help of a set of cache-aware micro-benchmarks for the
underlying BLAS kernels. The predictions we construct from such benchmarks
allow us to reliably single out the best-performing algorithms in a tiny
fraction of the time taken by the direct execution of the algorithms.Comment: Submitted to PMBS1
Spin effects in gravitational radiation backreaction II. Finite mass effects
A convenient formalism for averaging the losses produced by gravitational
radiation backreaction over one orbital period was developed in an earlier
paper. In the present paper we generalize this formalism to include the case of
a closed system composed from two bodies of comparable masses, one of them
having the spin S.
We employ the equations of motion given by Barker and O'Connell, where terms
up to linear order in the spin (the spin-orbit interaction terms) are kept. To
obtain the radiative losses up to terms linear in the spin, the equations of
motion are taken to the same order. Then the magnitude L of the angular
momentum L, the angle kappa subtended by S and L and the energy E are
conserved. The analysis of the radial motion leads to a new parametrization of
the orbit.
From the instantaneous gravitational radiation losses computed by Kidder the
leading terms and the spin-orbit terms are taken. Following Apostolatos,
Cutler, Sussman and Thorne, the evolution of the vectors S and L in the
momentary plane spanned by these vectors is separated from the evolution of the
plane in space. The radiation-induced change in the spin is smaller than the
leading-order spin terms in the momentary angular momentum loss. This enables
us to compute the averaged losses in the constants of motion E, L and L_S=L cos
kappa. In the latter, the radiative spin loss terms average to zero. An
alternative description using the orbital elements a,e and kappa is given.
The finite mass effects contribute terms, comparable in magnitude, to the
basic, test-particle spin terms in the averaged losses.Comment: 12 pages, 1 figure, Phys.Rev.D15, March, 199
Effective one-body approach to general relativistic two-body dynamics
We map the general relativistic two-body problem onto that of a test particle
moving in an effective external metric. This effective-one-body approach
defines, in a non-perturbative manner, the late dynamical evolution of a
coalescing binary system of compact objects. The transition from the adiabatic
inspiral, driven by gravitational radiation damping, to an unstable plunge,
induced by strong spacetime curvature, is predicted to occur for orbits more
tightly bound than the innermost stable circular orbit in a Schwarzschild
metric of mass M = m1 + m2. The binding energy, angular momentum and orbital
frequency of the innermost stable circular orbit for the time-symmetric
two-body problem are determined as a function of the mass ratio.Comment: 52 pages, RevTex, epsfig, 8 figure
Coalescence of Two Spinning Black Holes: An Effective One-Body Approach
We generalize to the case of spinning black holes a recently introduced
``effective one-body'' approach to the general relativistic dynamics of binary
systems. The combination of the effective one-body approach, and of a Pad\'e
definition of some crucial effective radial functions, is shown to define a
dynamics with much improved post-Newtonian convergence properties, even for
black hole separations of the order of . We discuss the approximate
existence of a two-parameter family of ``spherical orbits'' (with constant
radius), and, of a corresponding one-parameter family of ``last stable
spherical orbits'' (LSSO). These orbits are of special interest for forthcoming
LIGO/VIRGO/GEO gravitational wave observations. It is argued that for most (but
not all) of the parameter space of two spinning holes the effective one-body
approach gives a reliable analytical tool for describing the dynamics of the
last orbits before coalescence. This tool predicts, in a quantitative way, how
certain spin orientations increase the binding energy of the LSSO. This leads
to a detection bias, in LIGO/VIRGO/GEO observations, favouring spinning black
hole systems, and makes it urgent to complete the conservative effective
one-body dynamics given here by adding (resummed) radiation reaction effects,
and by constructing gravitational waveform templates that include spin effects.
Finally, our approach predicts that the spin of the final hole formed by the
coalescence of two arbitrarily spinning holes never approaches extremality.Comment: 26 pages, two eps figures, accepted in Phys. Rev. D, minor updating
of the text, clarifications added and inclusion of a few new reference
Improved filters for gravitational waves from inspiralling compact binaries
The order of the post-Newtonian expansion needed, to extract in a reliable
and accurate manner the fully general relativistic gravitational wave signal
from inspiralling compact binaries, is explored. A class of approximate wave
forms, called P-approximants, is constructed based on the following two inputs:
(a) The introduction of two new energy-type and flux-type functions e(v) and
f(v), respectively, (b) the systematic use of Pade approximation for
constructing successive approximants of e(v) and f(v). The new P-approximants
are not only more effectual (larger overlaps) and more faithful (smaller
biases) than the standard Taylor approximants, but also converge faster and
monotonically. The presently available O(v/c)^5-accurate post-Newtonian results
can be used to construct P-approximate wave forms that provide overlaps with
the exact wave form larger than 96.5% implying that more than 90% of potential
events can be detected with the aid of P-approximants as opposed to a mere
10-15 % that would be detectable using standard post-Newtonian approximants.Comment: Latex ([prd,aps,eqsecnum,epsf]{revtex}), 40 pages including 12
encapsulated figures. (The paper, together with all the figures and tables is
available from ftp://carina.astro.cf.ac.uk/pub/incoming/sathya/dis97.uu
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