Post-Newtonian approximation for isolated systems calculated by matched asymptotic expansions

Two long-standing problems with the post-Newtonian approximation for isolated slowly-moving systems in general relativity are: (i) the appearance at high post-Newtonian orders of divergent Poisson integrals, casting a doubt on the soundness of the post-Newtonian series; (ii) the domain of validity of the approximation which is limited to the near-zone of the source, and prevents one, a priori, from incorporating the condition of no-incoming radiation, to be imposed at past null infinity. In this article, we resolve the problem (i) by iterating the post-Newtonian hierarchy of equations by means of a new (Poisson-type) integral operator that is free of divergencies, and the problem (ii) by matching the post-Newtonian near-zone field to the exterior field of the source, known from previous work as a multipolar-post-Minkowskian expansion satisfying the relevant boundary conditions at infinity. As a result, we obtain an algorithm for iterating the post-Newtonian series up to any order, and we determine the terms, present in the post-Newtonian field, that are associated with the gravitational-radiation reaction onto an isolated slowly-moving matter system.Comment: 61 pages, to appear in Phys. Rev.

Gravitational-Wave Inspiral of Compact Binary Systems to 7/2 Post-Newtonian Order

The inspiral of compact binaries, driven by gravitational-radiation reaction, is investigated through 7/2 post-Newtonian (3.5PN) order beyond the quadrupole radiation. We outline the derivation of the 3.5PN-accurate binary's center-of-mass energy and emitted gravitational flux. The analysis consistently includes the relativistic effects in the binary's equations of motion and multipole moments, as well as the contributions of tails, and tails of tails, in the wave zone. However the result is not fully determined because of some physical incompleteness, present at the 3PN order, of the model of point-particle and the associated Hadamard-type self-field regularization. The orbital phase, whose prior knowledge is crucial for searching and analyzing the inspiral signal, is computed from the standard energy balance argument.Comment: 12 pages, version which includes the correction of an Erratum to be published in Phys. Rev. D (2005

Efficient Monte Carlo for high excursions of Gaussian random fields

Our focus is on the design and analysis of efficient Monte Carlo methods for computing tail probabilities for the suprema of Gaussian random fields, along with conditional expectations of functionals of the fields given the existence of excursions above high levels, b. Na\"{i}ve Monte Carlo takes an exponential, in b, computational cost to estimate these probabilities and conditional expectations for a prescribed relative accuracy. In contrast, our Monte Carlo procedures achieve, at worst, polynomial complexity in b, assuming only that the mean and covariance functions are H\"{o}lder continuous. We also explain how to fine tune the construction of our procedures in the presence of additional regularity, such as homogeneity and smoothness, in order to further improve the efficiency.Comment: Published in at http://dx.doi.org/10.1214/11-AAP792 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Statistical Mechanics of Horizons and Black Hole Thermodynamics

Although we know that black holes are characterized by a temperature and an entropy, we do not yet have a satisfactory microscopic ``statistical mechanical'' explanation for black hole thermodynamics. I describe a new approach that attributes the thermodynamic properties to ``would-be gauge'' degrees of freedom that become dynamical on the horizon. For the (2+1)-dimensional black hole, this approach gives the correct entropy. (Talk given at the Pacific Conference on Gravitation and Cosmology, Seoul, February 1996.)Comment: 11 pages, LaTe

Lorentzian regularization and the problem of point-like particles in general relativity

The two purposes of the paper are (1) to present a regularization of the self-field of point-like particles, based on Hadamard's concept of ``partie finie'', that permits in principle to maintain the Lorentz covariance of a relativistic field theory, (2) to use this regularization for defining a model of stress-energy tensor that describes point-particles in post-Newtonian expansions (e.g. 3PN) of general relativity. We consider specifically the case of a system of two point-particles. We first perform a Lorentz transformation of the system's variables which carries one of the particles to its rest frame, next implement the Hadamard regularization within that frame, and finally come back to the original variables with the help of the inverse Lorentz transformation. The Lorentzian regularization is defined in this way up to any order in the relativistic parameter 1/c^2. Following a previous work of ours, we then construct the delta-pseudo-functions associated with this regularization. Using an action principle, we derive the stress-energy tensor, made of delta-pseudo-functions, of point-like particles. The equations of motion take the same form as the geodesic equations of test particles on a fixed background, but the role of the background is now played by the regularized metric.Comment: 34 pages, to appear in J. Math. Phy

Gravitational wave forms for a three-body system in Lagrange's orbit: parameter determinations and a binary source test

Continuing work initiated in an earlier publication [Torigoe et al. Phys. Rev. Lett. {\bf 102}, 251101 (2009)], gravitational wave forms for a three-body system in Lagrange's orbit are considered especially in an analytic method. First, we derive an expression of the three-body wave forms at the mass quadrupole, octupole and current quadrupole orders. By using the expressions, we solve a gravitational-wave {\it inverse} problem of determining the source parameters to this particular configuration (three masses, a distance of the source to an observer, and the orbital inclination angle to the line of sight) through observations of the gravitational wave forms alone. For this purpose, the chirp mass to a three-body system in the particular configuration is expressed in terms of only the mass ratios by deleting initial angle positions. We discuss also whether and how a binary source can be distinguished from a three-body system in Lagrange's orbit or others.Comment: 21 pages, 3 figures, 1 table; text improved, typos corrected; accepted for publication in PR

Time-symmetric initial data for binary black holes in numerical relativity

We look for physically realistic initial data in numerical relativity which are in agreement with post-Newtonian approximations. We propose a particular solution of the time-symmetric constraint equation, appropriate to two momentarily static black holes, in the form of a conformal decomposition of the spatial metric. This solution is isometric to the post-Newtonian metric up to the 2PN order. It represents a non-linear deformation of the solution of Brill and Lindquist, i.e. an asymptotically flat region is connected to two asymptotically flat (in a certain weak sense) sheets, that are the images of the two singularities through appropriate inversion transformations. The total ADM mass M as well as the individual masses m_1 and m_2 (when they exist) are computed by surface integrals performed at infinity. Using second order perturbation theory on the Brill-Lindquist background, we prove that the binary's interacting mass-energy M-m_1-m_2 is well-defined at the 2PN order and in agreement with the known post-Newtonian result.Comment: 27 pages, to appear in Phys. Rev.

Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the post-Newtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many non-linear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail effects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary's orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.Comment: 109 pages, 1 figure; this version is an update of the Living Review article originally published in 2002; available on-line at http://www.livingreviews.org