24 research outputs found
Introductory lectures on the Effective One Body formalism
The Effective One Body (EOB) formalism is an analytical approach which aims
at providing an accurate description of the motion and radiation of coalescing
binary black holes. We present a brief review of the basic elements of this
approach.Comment: 22 pages, 3 figures, lectures given at the Second ICRANet
Stueckelberg Workshop on Relativistic Field Theories (Pescara, Italy,
September 3-8, 2007); to be published in the International Journal of Modern
Physics
Hadamard Regularization
Motivated by the problem of the dynamics of point-particles in high
post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a
certain class of functions which are smooth except at some isolated points
around which they admit a power-like singular expansion. We review the concepts
of (i) Hadamard ``partie finie'' of such functions at the location of singular
points, (ii) the partie finie of their divergent integral. We present and
investigate different expressions, useful in applications, for the latter
partie finie. To each singular function, we associate a partie-finie (Pf)
pseudo-function. The multiplication of pseudo-functions is defined by the
ordinary (pointwise) product. We construct a delta-pseudo-function on the class
of singular functions, which reduces to the usual notion of Dirac distribution
when applied on smooth functions with compact support. We introduce and analyse
a new derivative operator acting on pseudo-functions, and generalizing, in this
context, the Schwartz distributional derivative. This operator is uniquely
defined up to an arbitrary numerical constant. Time derivatives and partial
derivatives with respect to the singular points are also investigated. In the
course of the paper, all the formulas needed in the application to the physical
problem are derived.Comment: 50 pages, to appear in Journal of Mathematical Physic
Third post-Newtonian dynamics of compact binaries: Equations of motion in the center-of-mass frame
The equations of motion of compact binary systems and their associated
Lagrangian formulation have been derived in previous works at the third
post-Newtonian (3PN) approximation of general relativity in harmonic
coordinates. In the present work we investigate the binary's relative dynamics
in the center-of-mass frame (center of mass located at the origin of the
coordinates). We obtain the 3PN-accurate expressions of the center-of-mass
positions and equations of the relative binary motion. We show that the
equations derive from a Lagrangian (neglecting the radiation reaction), from
which we deduce the conserved center-of-mass energy and angular momentum at the
3PN order. The harmonic-coordinates center-of-mass Lagrangian is equivalent,
{\it via} a contact transformation of the particles' variables, to the
center-of-mass Hamiltonian in ADM coordinates that is known from the
post-Newtonian ADM-Hamiltonian formalism. As an application we investigate the
dynamical stability of circular binary orbits at the 3PN order.Comment: 31 pages, to appear in Classical and Quantum Gravit
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
General relativistic dynamics of compact binaries at the third post-Newtonian order
The general relativistic corrections in the equations of motion and
associated energy of a binary system of point-like masses are derived at the
third post-Newtonian (3PN) order. The derivation is based on a post-Newtonian
expansion of the metric in harmonic coordinates at the 3PN approximation. The
metric is parametrized by appropriate non-linear potentials, which are
evaluated in the case of two point-particles using a Lorentzian version of an
Hadamard regularization which has been defined in previous works.
Distributional forms and distributional derivatives constructed from this
regularization are employed systematically. The equations of motion of the
particles are geodesic-like with respect to the regularized metric. Crucial
contributions to the acceleration are associated with the non-distributivity of
the Hadamard regularization and the violation of the Leibniz rule by the
distributional derivative. The final equations of motion at the 3PN order are
invariant under global Lorentz transformations, and admit a conserved energy
(neglecting the radiation reaction force at the 2.5PN order). However, they are
not fully determined, as they depend on one arbitrary constant, which reflects
probably a physical incompleteness of the point-mass regularization. The
results of this paper should be useful when comparing theory to the
observations of gravitational waves from binary systems in future detectors
VIRGO and LIGO.Comment: 78 pages, submitted to Phys. Rev. D, with minor modification
Relativistic Celestial Mechanics with PPN Parameters
Starting from the global parametrized post-Newtonian (PPN) reference system
with two PPN parameters and we consider a space-bounded
subsystem of matter and construct a local reference system for that subsystem
in which the influence of external masses reduces to tidal effects. Both the
metric tensor of the local PPN reference system in the first post-Newtonian
approximation as well as the coordinate transformations between the global PPN
reference system and the local one are constructed in explicit form. The terms
proportional to reflecting a violation of the
equivalence principle are discussed in detail. We suggest an empirical
definition of multipole moments which are intended to play the same role in PPN
celestial mechanics as the Blanchet-Damour moments in General Relativity.
Starting with the metric tensor in the local PPN reference system we derive
translational equations of motion of a test particle in that system. The
translational and rotational equations of motion for center of mass and spin of
each of extended massive bodies possessing arbitrary multipole structure
are derived. As an application of the general equations of motion a
monopole-spin dipole model is considered and the known PPN equations of motion
of mass monopoles with spins are rederived.Comment: 71 page
Third post-Newtonian dynamics of compact binaries: Noetherian conserved quantities and equivalence between the harmonic-coordinate and ADM-Hamiltonian formalisms
A Lagrangian from which derive the third post-Newtonian (3PN) equations of
motion of compact binaries (neglecting the radiation reaction damping) is
obtained. The 3PN equations of motion were computed previously by Blanchet and
Faye in harmonic coordinates. The Lagrangian depends on the harmonic-coordinate
positions, velocities and accelerations of the two bodies. At the 3PN order,
the appearance of one undetermined physical parameter \lambda reflects an
incompleteness of the point-mass regularization used when deriving the
equations of motion. In addition the Lagrangian involves two unphysical
(gauge-dependent) constants r'_1 and r'_2 parametrizing some logarithmic terms.
The expressions of the ten Noetherian conserved quantities, associated with the
invariance of the Lagrangian under the Poincar\'e group, are computed. By
performing an infinitesimal ``contact'' transformation of the motion, we prove
that the 3PN harmonic-coordinate Lagrangian is physically equivalent to the 3PN
Arnowitt-Deser-Misner Hamiltonian obtained recently by Damour, Jaranowski and
Sch\"afer.Comment: 30 pages, to appear in Classical and Quantum Gravit
Gravitational field and equations of motion of compact binaries to 5/2 post-Newtonian order
We derive the gravitational field and equations of motion of compact binary
systems up to the 5/2 post-Newtonian approximation of general relativity (where
radiation-reaction effects first appear). The approximate post-Newtonian
gravitational field might be used in the problem of initial conditions for the
numerical evolution of binary black-hole space-times. On the other hand we
recover the Damour-Deruelle 2.5PN equations of motion of compact binary
systems. Our method is based on an expression of the post-Newtonian metric
valid for general (continuous) fluids. We substitute into the fluid metric the
standard stress-energy tensor appropriate for a system of two point-like
particles. We remove systematically the infinite self-field of each particle by
means of the Hadamard partie finie regularization.Comment: 41 pages to appear in Physical Review
Equation of motion for relativistic compact binaries with the strong field point particle limit : the second and half post-Newtonian order
We study the equation of motion appropriate to an inspiralling binary star
system whose constituent stars have strong internal gravity. We use the
post-Newtonian approximation with the strong field point particle limit by
which we can introduce into general relativity a notion of a point-like
particle with strong internal gravity without using Dirac delta distribution.
Besides this limit, to deal with strong internal gravity we express the
equation of motion in surface integral forms and calculate these integrals
explicitly. As a result we obtain the equation of motion for a binary of
compact bodies accurate through the second and half post-Newtonian (2.5 PN)
order. This equation is derived in the harmonic coordinate. Our resulting
equation perfectly agrees with Damour and Deruelle 2.5 PN equation of motion.
Hence it is found that the 2.5 PN equation of motion is applicable to a
relativistic compact binary.Comment: 48 pages, revtex, accepted for publication in Phys. Rev.
Gravitational field and equations of motion of spinning compact binaries to 2.5 post-Newtonian order
We derive spin-orbit coupling effects on the gravitational field and
equations of motion of compact binaries in the 2.5 post-Newtonian approximation
to general relativity, one PN order beyond where spin effects first appear. Our
method is based on that of Blanchet, Faye, and Ponsot, who use a post-Newtonian
metric valid for general (continuous) fluids and represent pointlike compact
objects with a delta-function stress-energy tensor, regularizing divergent
terms by taking the Hadamard finite part. To obtain post-Newtonian spin
effects, we use a different delta-function stress-energy tensor introduced by
Bailey and Israel. In a future paper we will use the 2.5PN equations of motion
for spinning bodies to derive the gravitational-wave luminosity and phase
evolution of binary inspirals, which will be useful in constructing matched
filters for signal analysis. The gravitational field derived here may help in
posing initial data for numerical evolutions of binary black hole mergers.Comment: 18 pages, no figur