383 research outputs found
From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem
The notion of Nash equilibria plays a key role in the analysis of strategic
interactions in the framework of player games. Analysis of Nash equilibria
is however a complex issue when the number of players is large. In this article
we emphasize the role of optimal transport theory in: 1) the passage from Nash
to Cournot-Nash equilibria as the number of players tends to infinity, 2) the
analysis of Cournot-Nash equilibria
Remarks on existence and uniqueness of Cournot-Nash equilibria in the non-potential case
This article is devoted to various methods (optimal transport, fixed-point,
ordinary differential equations) to obtain existence and/or uniqueness of
Cournot-Nash equilibria for games with a continuum of players with both
attractive and repulsive effects. We mainly address separable situations but
for which the game does not have a potential. We also present several numerical
simulations which illustrate the applicability of our approach to compute
Cournot-Nash equilibria
Higher-order spin effects in the dynamics of compact binaries II. Radiation field
Motivated by the search for gravitational waves emitted by binary black
holes, we investigate the gravitational radiation field of point particles with
spins within the framework of the multipolar-post-Newtonian wave generation
formalism. We compute: (i) the spin-orbit (SO) coupling effects in the binary's
mass and current quadrupole moments one post-Newtonian (1PN) order beyond the
dominant effect, (ii) the SO contributions in the gravitational-wave energy
flux and (iii) the secular evolution of the binary's orbital phase up to 2.5PN
order. Crucial ingredients for obtaining the 2.5PN contribution in the orbital
phase are the binary's energy and the spin precession equations, derived in
paper I of this series. These results provide more accurate gravitational-wave
templates to be used in the data analysis of rapidly rotating Kerr-type
black-hole binaries with the ground-based detectors LIGO, Virgo, GEO 600 and
TAMA300, and the space-based detector LISA.Comment: includes the correction of an erratum to be published in Phys. Rev.
Gravitational-wave tail effects to quartic non-linear order
Gravitational-wave tails are due to the backscattering of linear waves onto
the space-time curvature generated by the total mass of the matter source. The
dominant tails correspond to quadratic non-linear interactions and arise at the
one-and-a-half post-Newtonian (1.5PN) order in the gravitational waveform. The
"tails-of-tails", which are cubic non-linear effects appearing at the 3PN order
in the waveform, are also known. We derive here higher non-linear tail effects,
namely those associated with quartic non-linear interactions or
"tails-of-tails-of-tails", which are shown to arise at the 4.5PN order. As an
application, we obtain at that order the complete coefficient in the total
gravitational-wave energy flux of compact binary systems moving on circular
orbits. Our result perfectly agrees with black-hole perturbation calculations
in the limit of extreme mass ratio of the two compact objects.Comment: 32 pages, no figure, matches with published versio
Third post-Newtonian spin-orbit effect in the gravitational radiation flux of compact binaries
Gravitational waves contain tail effects that are due to the backscattering
of linear waves in the curved space-time geometry around the source. The
knowledge as well as the accuracy of the two-body inspiraling post-Newtonian
(PN) dynamics and of the gravitational-wave signal has been recently improved,
notably by computing the spin-orbit (SO) terms induced by tail effects in the
gravitational-wave energy flux at the 3PN order. Here we sketch this
derivation, which yields the phasing formula including SO tail effects through
the same 3PN order. Those results can be employed to improve the accuracy of
analytical templates aimed at describing the whole process of inspiral, merger,
and ringdown.Comment: 6 pages; proceeding of the 9th LISA Symposium, Pari
On the structure of the post-Newtonian expansion in general relativity
In the continuation of a preceding work, we derive a new expression for the
metric in the near zone of an isolated matter system in post-Newtonian
approximations of general relativity. The post-Newtonian metric, a solution of
the field equations in harmonic coordinates, is formally valid up to any order,
and is cast in the form of a particular solution of the wave equation, plus a
specific homogeneous solution which ensures the asymptotic matching to the
multipolar expansion of the gravitational field in the exterior of the system.
The new form provides some insights on the structure of the post-Newtonian
expansion in general relativity and the gravitational radiation reaction terms
therein.Comment: 22 pages, to appear in Phys. Rev.
High-order half-integral conservative post-Newtonian coefficients in the redshift factor of black hole binaries
The post-Newtonian approximation is still the most widely used approach to
obtaining explicit solutions in general relativity, especially for the
relativistic two-body problem with arbitrary mass ratio. Within many of its
applications, it is often required to use a regularization procedure. Though
frequently misunderstood, the regularization is essential for waveform
generation without reference to the internal structure of orbiting bodies. In
recent years, direct comparison with the self-force approach, constructed
specifically for highly relativistic particles in the extreme mass ratio limit,
has enabled preliminary confirmation of the foundations of both computational
methods, including their very independent regularization procedures, with high
numerical precision. In this paper, we build upon earlier work to carry this
comparison still further, by examining next-to-next-to-leading order
contributions beyond the half integral 5.5PN conservative effect, which arise
from terms to cubic and higher orders in the metric and its multipole moments,
thus extending scrutiny of the post-Newtonian methods to one of the highest
orders yet achieved. We do this by explicitly constructing tail-of-tail terms
at 6.5PN and 7.5PN order, computing the redshift factor for compact binaries in
the small mass ratio limit, and comparing directly with numerically and
analytically computed terms in the self-force approach, obtained using
solutions for metric perturbations in the Schwarzschild space-time, and a
combination of exact series representations possibly with more typical PN
expansions. While self-force results may be relativistic but with restricted
mass ratio, our methods, valid primarily in the weak-field slowly-moving
regime, are nevertheless in principle applicable for arbitrary mass ratios.Comment: 33 pages, no figure; minor correction
Half-integral conservative post-Newtonian approximations in the redshift factor of black hole binaries
Recent perturbative self-force computations (Shah, Friedman & Whiting,
submitted to Phys. Rev. {\bf D}, arXiv:1312.1952 [gr-qc]), both numerical and
analytical, have determined that half-integral post-Newtonian terms arise in
the conservative dynamics of black-hole binaries moving on exactly circular
orbits. We look at the possible origin of these terms within the post-Newtonian
approximation, find that they essentially originate from non-linear
"tail-of-tail" integrals and show that, as demonstrated in the previous paper,
their first occurrence is at the 5.5PN order. The post-Newtonian method we use
is based on a multipolar-post-Minkowskian treatment of the field outside a
general matter source, which is re-expanded in the near zone and extended
inside the source thanks to a matching argument. Applying the formula obtained
for generic sources to compact binaries, we obtain the redshift factor of
circular black hole binaries (without spins) at 5.5PN order in the extreme mass
ratio limit. Our result fully agrees with the determination of the 5.5PN
coefficient by means of perturbative self-force computations reported in the
previously cited paper.Comment: 18 pages, no figures, references updated and minor corrections
include
On the equations of motion of point-particle binaries at the third post-Newtonian order
We investigate the dynamics of two point-like particles through the third
post-Newtonian (3PN) approximation of general relativity. The infinite
self-field of each point-mass is regularized by means of Hadamard's concept of
``partie finie''. Distributional forms associated with the regularization are
used systematically in the computation. We determine the stress-energy tensor
of point-like particles compatible with the previous regularization. The
Einstein field equations in harmonic coordinates are iterated to the 3PN order.
The 3PN equations of motion are Lorentz-invariant and admit a conserved energy
(neglecting the 2.5PN radiation reaction). They depend on an undetermined
coefficient, in agreement with an earlier result of Jaranowski and Schaefer.
This suggests an incompleteness of the formalism (in this stage of development)
at the 3PN order. In this paper we present the equations of motion in the
center-of-mass frame and in the case of circular orbits.Comment: 12 pages, to appear in Physics Letters A, minor changes include
- âŠ