Continuing work initiated in earlier publications [Ichita, Yamada and Asada, Phys. Rev. D {\bf 83}, 084026 (2011); Yamada and Asada, Phys. Rev. D {\bf 86}, 124029 (2012)], we examine the post-Newtonian (PN) effects on the stability of the triangular solution in the relativistic three-body problem for general masses. For three finite masses, a condition for stability of the triangular solution is obtained at the first post-Newtonian (1PN) order, and it recovers previous results for the PN restricted three-body problem when one mass goes to zero. The stability regions still exist even at the 1PN order, though the PN triangular configuration for general masses is less stable than the PN restricted three-body case as well as the Newtonian one.
Continuing work initiated in an earlier publication [Ichita, Yamada and Asada, Phys. Rev. D 83, 084026 (2011)], we reexamine the post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem. For three finite masses, it is found that a triangular configuration satisfies the post-Newtonian equation of motion in general relativity, if and only if it has the relativistic corrections to each side length. This post-Newtonian configuration for three finite masses is not always equilateral and it recovers previous results for the restricted three-body problem when on…
Continuing work initiated in earlier publications [Yamada, Asada, Phys. Rev. D 82, 104019 (2010), 83, 024040 (2011)], we investigate the post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem. For three finite masses, it is found that the equilateral triangular configuration satisfies the post-Newtonian equation of motion in general relativity, if and only if all three masses are equal. When a test mass is included, the equilateral configuration is possible for two cases: (1) one mass is finite and the other two are zero, or (2) two of the masses are fi…
Continuing work initiated in an earlier publication [Yamada, Tsuchiya, and Asada, Phys. Rev. D 91, 124016 (2015)], we reexamine the linear stability of the triangular solution in the relativistic three-body problem for general masses by the standard linear algebraic analysis. In this paper, we start with the Einstein-Infeld-Hoffman form of equations of motion for $N$-body systems in the uniformly rotating frame. As an extension of the previous work, we consider general perturbations to the equilibrium, i.e. we take account of perturbations orthogonal to the orbital plane, as well as perturbati…
This paper investigates the coplanar and circular three-body problem in the parametrized post-Newtonian (PPN) formalism, for which we focus on a class of fully conservative theories characterized by the Eddington-Robertson parameters $β$ and $γ$. It is shown that there can still exist a collinear equilibrium configuration and a triangular one, each of which is a generalization of the post-Newtonian equilibrium configuration in general relativity. The collinear configuration can exist for arbitrary mass ratio, $β$, and $γ$. On the other hand, the PPN triangular configuration depends on the nonl…
Continuing work initiated in an earlier publication [H. Asada, Phys. Rev. D {\bf 80}, 064021 (2009)], the gravitational radiation reaction to Lagrange's equilateral triangular solution of the three-body problem is investigated in an analytic method. The previous work is based on the energy balance argument, which is sufficient for a two-body system because the number of degrees of freedom (the semimajor axis and the eccentricity in quasi-Keplerian cases, for instance) equals that of the constants of motion such as the total energy and the orbital angular momentum. In a system with three (or mo…
In a previous paper we derived a post-Newtonian approximation to cosmology which, in contrast to former Newtonian and post-Newtonian cosmological theories, has a well-posed initial value problem. In this paper, this new post-Newtonian theory is compared with the fully general relativistic theory, in the context of the k = 0 Friedmann Robertson Walker cosmologies. It is found that the post-Newtonian theory reproduces the results of its general relativistic counterpart, whilst the Newtonian theory does not.
We apply general relativity to construct the post-Newtonian background manifold that serves as a reference spacetime in relativistic geodesy for conducting relativistic calculation of the geoid's undulation and the deflection of the plumb line from the vertical. We chose an axisymmetric ellipsoidal body made up of perfect homogeneous fluid uniformly rotating around a fixed axis, as a source generating the reference geometry. We, then, reformulate and extend hydrodynamic calculations of rotating fluids done by previous researchers to the realm of relativistic geodesy to set up algebraic equatio…
We discuss in detail the weak field limit of f(R,G) gravity taking into account analytic functions of the Ricci scalar R and the Gauss-Bonnet invariant G. Specifically, we develop, in metric formalism, the Newtonian, Post Newtonian and Parameterized Post Newtonian limits starting from general f(R, G) Lagrangian. The special cases of f(R) and f(G) gravities are considered. In the case of the Newtonian limit of f(R, G) gravity, a general solution in terms of Green's functions is achieved.
With the post-Newtonian approxination, the energy of Schwarzschild spacetime in the Weinberg prescription is obtained. The energy for the first post-Newtonian approximation $E^{(1)} = m$ gives the Newtonian treatment of Schwarzschild spacetime. However, for the second post-Newtonian approximation, the erergy is shown that $E^{(1)}$ adds extra terms $E^{(2)}$ which consist of the energy stored in the configuration $E_{\rm config}$, in the gravitational field $E_{\rm field}$. and a term of surface integral. These extra terms gives post-Newtonian corrections to the Newtonian treatment.
The post-Newtonian hydrodynamic equations for a non-perfect fluid are developed within the framework of a post-Newtonian Boltzmann equation. The post-Newtonian components of the energy-momentum tensor are determined by considering the relativistic Eckart decomposition for a viscous and heat conducting fluid. From the relativistic Grad distribution function its post-Newtonian expression is derived. The hydrodynamic equations for the mass density, mass-energy density and momentum density are determined from a post-Newtonian transfer equation and Grad's distribution function. In the non-relativis…
The generation of gravitational waves from a post-Newtonian source endowed with a quantum spin, modeled by the Weyssenhoff fluid, is investigated in the context of Einstein-Cartan theory at the first post-Newtonian level by resorting to the Blanchet-Damour formalism. After having worked out the basic principles of the hydrodynamics in Einstein-Cartan framework, we study the Weyssenhoff fluid within the post-Newtonian approximation scheme. The complexity of the underlying dynamical equations suggests to employ a discrete description via the point-particle limit, a procedure which permits the an…
The post-Newtonian Jeans equation for stationary self-gravitating systems is derived from the post-Newtonian Boltzmann equation in spherical coordinates. The Jeans equation is coupled with the three Poisson equations from the post-Newtonian theory. The Poisson equations are functions of the energy-momentum tensor components which are determined from the post-Newtonian Maxwell--Jüttner distribution function. As an application, the effect of a central massive black hole on the velocity dispersion profile of the host galaxy is investigated and the influence of the post-Newtonian corrections are d…
Jeans instability within the framework of post-Newtonian Boltzmann and Poisson equations are analyzed. The components of the energy-momentum tensor are calculated from a post-Newtonian Maxwell-Jüttner distribution function. The perturbations of the distribution function and gravitational potentials from their background states with the representation of the perturbations as plane waves lead to a dispersion relation with post-Newtonian corrections. The influence of the post-Newtonian approximation on the Jeans mass is determined and it was shown that the mass necessary for an overdensity to beg…
We estimate virial masses of galaxy clusters using the parametrized post-Newtonian (PPN) virial theorem. Also, we show explicitly that post-Newtonian corrections can not address the mass discrepancy in the galaxy clusters.
Post-Newtonian expansions of the Brill-Lindquist and Misner-Lindquist solutions of the time-symmetric two-black-hole initial value problem are derived. The static Hamiltonians related to the expanded solutions, after identifying the bare masses in both solutions, are found to differ from each other at the third post-Newtonian approximation. By shifting the position variables of the black holes the post-Newtonian expansions of the three metrics can be made to coincide up to the fifth post-Newtonian order resulting in identical static Hamiltonians up the third post-Newtonian approximation. The c…
We present an elementary derivation of the planetary advance of the perihelion for a general spherically symmetric line element in the post- newtonian approximation.
The second post-Newtonian hydrodynamic equations are analyzed within the framework of a plane wave solution. The hydrodynamic equations for the mass and momentum density are coupled with six Poisson equations for the Newtonian and post-Newtonian gravitational potentials. Perturbations of the basic fields and gravitational potentials from a background state by assuming plane wave representations lead to a dispersion relation where the Jeans instability condition emerges. The influence of the first and second post-Newtonian approximations on the Jeans mass is determined. It was shown that the re…
Contents: I. Introduction; II. Summary of optimal signal filtering; III. Newtonian binary polarization waveforms; IV. Newtonian orbital phase evolution; V. Post-Newtonian wave-generation; VI. Inspiral binary waveform.
We discuss various aspects of the post-Newtonian approximation in general relativity. After presenting the foundation based on the Newtonian limit, we use the (3+1) formalism to formulate the post-Newtonian approximation for the perfect fluid. As an application we show the method for constructing the equilibrium configuration of nonaxisymmetric uniformly rotating fluid. We also discuss the gravitational waves including tail from post-Newtonian systems.
In the present paper, using the first-order approximation of the $n$-body Lagrangian (derived on the basis of the post-Newtonian gravitational theory of Einstein, Infeld, and Hoffman), we explicitly write down the equations of motion for the planar circular restricted three-body problem. Additionally, with some simplified assumptions, we obtain two formulas for estimating the values of the mass/distance and velocity/speed of light ratios appropriate for a given post-Newtonian approximation. We show that the formulas derived in the present study, lead to a good numerical conservation of the Jac…
It is shown that the recently found non-uniqueness of the third post-Newtonian binary point-mass ADM-Hamiltonian is related to the non-uniqueness at the third post-Newtonian approximation of the applied ADM-coordinate conditions.
In this paper we explore how far the post-Newtonian theory goes in overcoming the difficulties associated with anisotropic homogeneous cosmologies in the Newtonian approximation. It will be shown that, unlike in the Newtonian case, the cosmological equations of the post-Newtonian approximation are much more in the spirit of general relativity with regard to the nine Bianchi types and issues of singularities. The situations of vanishing rotation and vanishing shear are treated separately. The homogeneous Bianchi I model is considered as an example of a rotation-free cosmology with anisotropy. I…
Newtonian cosmological perturbation equations valid to full nonlinear order are well known in the literature. Assuming the absence of the transverse-tracefree part of the metric, we present the general relativistic counterpart valid to full nonlinear order. The relativistic equations are presented without taking the slicing (temporal gauge) condition. The equations do have the proper Newtonian and first post-Newtonian limits. We also present the relativistic pressure correction terms in the Newtonian hydrodynamic equations.
We study a solution of the field equations for dilatonic gravity and obtain its post-post-newtonian limit. It turns out that terms to this and higher orders in the expansion may become important in strong gravitational fields, even though the post-newtonian limit coincides with that of General Relativity. This suggests that strong gravitational fields can only be studied by exact solutions of the field equations.
In 2002, Poujade and Blanchet succeeded in matching the post-Newtonian solution to the Einstein field equation to the post-Minkowskian field up to any arbitrary order as well as reproducing, in a different way, the results of the 1998 paper by Blanchet in which he showed how to match the post-Minkowskian series to the post-Newtonian expansion. Comparing these two papers, it might be asked whether it is possible to match the post-Newtonian field to the post-Minkowskian one by means of a method similar to the one used in the 1998 paper. The answer is affirmative, and in the present paper we prov…
The set up of matched filters for the detection of gravitational waves from in-spiraling compact binaries is usually carried out using the restricted post-Newtonian approximation: the filter phase is modelled including post-Newtonian corrections, whereas the amplitude is retained at the Newtonian order. Here we investigate the effects of the introduction of post-Newtonian corrections also to the amplitude and we discuss some of the implications for signal detection and parameter estimation.
We derive the equations of motion for binary systems of compact bodies in the post-Newtonian (PN) approximation to general relativity. Results are given through 2PN order (order (v/c)^4 beyond Newtonian theory), and for gravitational radiation reaction effects at 2.5PN and 3.5PN orders. The method is based on a framework for direct integration of the relaxed Einstein equations (DIRE) developed earlier, in which the equations of motion through 3.5PN order can be expressed in terms of Poisson-like potentials that are generalizations of the instantaneous Newtonian gravitational potential, and in …
In this paper, we derive the post-Newtonian equations of the ideal Magnetohydrodynamics. To do so, we use the modern approach to post-Newtonian theory, where the harmonic gauge is used instead of the standard post-Newtonian gauge, and find the post-Newtonian metric in the presence of the electromagnetic fields. We show that although the electric field does not contribute in the metric and curvature of the spacetime, the magnetic field appears in the time-time component of the metric. The appearance of the magnetic field, in principle, leads to new relativistic contributions to the magnetohydro…
Using post-Newtonian equations of motion for fluid bodies valid to the second post-Newtonian order, we derive the equations of motion for binary systems with finite-sized, non-spinning but arbitrarily shaped bodies. In particular we study the contributions of the internal structure of the bodies (such as self-gravity) that would diverge if the size of the bodies were to shrink to zero. Using a set of virial relations accurate to the first post-Newtonian order that reflect the stationarity of each body, and redefining the masses to include 1PN and 2PN self-gravity terms, we demonstrate the comp…
In this paper, second post-Newtonian approximation of Einstein-aether theory is obtained by Chandrasekhar's approach. Five parameterized post-Newtonian parameters in first post-Newtonian approximation are presented after a time transformation and they are identical with previous works, in which $γ=1$, $β=1$ and two preferred-frame parameters remain. Meanwhile, in second post-Newtonian approximation, a parameter, which represents third order nonlinearity for gravity, is zero the same as in general relativity. For an application for future deep space laser ranging missions, we reduce the metric …
The post-Newtonian motion around an oblate spheroid: the mixed orbital effects due to the Newtonian oblateness and the post-Newtonian mass monopole accelerations
The increasing sophistication and accuracy of numerical simulations of compact binaries (especially binary black holes) presents the opportunity to test the regime in which post-Newtonian (PN) predictions for the emitted gravitational waves are accurate. In order to confront numerical results with those of post-Newtonian theory, it is convenient to compare multipolar decompositions of the two waveforms. It is pointed out here that the individual modes can be computed to higher post-Newtonian order by examining the radiative multipole moments of the system, rather than by decomposing the 2.5PN …
The post-Newtonian approximation is useful to discuss gravitational waveforms for binary inspirals. In this note, for retrograde circular orbits in the equatorial plane, we discuss the region of validity of the post-Newtonian approximation by using the gravitational energy flux derived numerically and analytically in the black hole perturbation approach. It is found that the edge of the allowable region behaves quite intricately, including prograde orbits even if we treat higher post-Newtonian orders.
We investigated hydrodynamical post-Newtonian models of selfgravitating stationary black-hole-disk systems. The post-Newtonian scheme presented here and also in our recent paper is a continuation of previous, purely Newtonian studies of selfgravitating hydrodynamical disks rotating according to the Keplerian rotation law. The post-Newtonian relativistic corrections are significant even at the 1PN level. The 1PN correction to the angular velocity can be of the order of 10% of its Newtonian value. It can be expressed as a combination of geometric and hydrodynamical terms. Moreover, in contrast t…
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…
We construct a set of post-Newtonian reference frames for a comprehensive study of the orbital dynamics and rotational motion of the Moon and Earth by means of lunar laser ranging (LLR) with the precision of one millimeter. We also derive the post-Newtonian coordinate transformations between the frames and analyze the residual gauge freedom, which is used for removing spurious post-Newtonian effects from the equations of motion of the solar system bodies.
Newtonian Cosmology is commonly used in astrophysical problems, because of its obvious simplicity when compared with general relativity. However it has inherent difficulties, the most obvious of which is the non-existence of a well-posed initial value problem. In this paper we investigate how far these problems are met by using the post-Newtonian approximation in cosmology.
We study the Jeans gravitational instability for a mixture of baryonic and dark matter particles, in the post-Newtonian approximation. We adopt a kinetic model consisting of a coupled system of post-Newtonian collisionless Boltzmann equations, for each species, coupled to the post-Newtonian Poisson equations. We derive the stability criterion, accounting for both post-Newtonian corrections and the presence of dark matter. It is shown that both effects give rise to smaller Jeans masses, in comparison with the standard Jeans criterion, meaning that a smaller mass is needed to begin the gravitati…
We discuss an effective theory for the quantum static gravitational potential in spherical symmetry up to the first post-Newtonian correction. We build a suitable Lagrangian from the weak field limit of the Einstein-Hilbert action coupled to pressureless matter. Classical solutions of the field equation lead to the correct post-Newtonian expansion. Furthermore, we portray the Newtonian results in a quantum framework by means of a coherent quantum state, which is properly corrected to accomodate post-Newtonian corrections. These considerations provide a link between the corpuscular model of Dva…
We point out the existence of a new type of growing transverse mode in the gravitational instability. This appears as a post-Newtonian effect to Newtonian dynamics. We demonstrate this existence by formulating the Lagrangian perturbation theory in the framework of the cosmological post-Newtonian approximation in general relativity. Such post-Newtonian order effects might produce characteristic appearances of large-scale structure formation, for example, through the observation of anisotropy of the cosmic microwave background radiation (CMB).
We use surface adapted ellipsoidal coordinates to obtain a first order post-Newtonian (PN) approximation to Dedekind ellipsoids with the intention of proceeding to higher orders.
The accuracy of the post-Newtonian waveforms, both in standard and Pade form, is determined by computing their matched-filtering overlap integral with a reference waveform obtained from black-hole perturbation theory.
The second-order post-Newtonian solution for the light propagation in the field of Kerr-Newman black hole is achieved via an iterative method. Based on this result, we further obtain the second-order post-Newtonian light deflection in Kerr-Newman spacetime, which is formulated in an united form for any arbitrarily incident directions. All results are exhibited in the coordinate system constituted by the initial light-direction vector, the impact vector, and their cross-product.
We present a method of post-Newtonian expansion to solve the homogeneous Regge-Wheeler equation which describes gravitational waves on the Schwarzschild spacetime. The advantage of our method is that it allows a systematic iterative analysis of the solution. Then we obtain the Regge-Wheeler function which is purely ingoing at the horizon in closed analytic form, with accuracy required to determine the gravitational wave luminosity to (post)$^{4}$-Newtonian order (i.e., order $v^8$ beyond Newtonian) from a particle orbiting around a Schwarzschild black hole. Our result, valid in the small-mass …
The objetive of this work is to investigate the influence of the corrections to the spherical symmetrical accretion of an infinity gas cloud characterized by a polytropic equation into a massive object due to the post-Newtonian approximation.The post-Newtonian corrections to the critical values of the flow velocity, sound velocity and radial distance are obtained from the system of hydrodynamics equations in spherical coordinates. The critical point in the post-Newtonian approximation accretion does not correspond to the transonic point like in Newtonian theory. An equation for the Mach number…
We present two of our efforts directed toward the numerical analysis of neutron star mergers, which are the most plausible sources for gravitational wave detectors that should begin operating in the near future. First we present Newtonian 3D simulations including radiation reaction (2.5PN) effects. We discuss the gravitational wave signals and luminosity from the merger with/without radiation reaction effects. Second we present the matching problem between post-Newtonian formulations and general relativity in numerical treatments. We prepare a spherical, static neutron star in a post-Newtonian…
We present a recent result on the parameterized post-Newtonian (PPN) limit of Horndeski's gravity theory and its consistency with solar system observations.
Continuing our analytic computation of the first-order self-force contribution to Detweiler's redshift variable we provide the exact expressions of the ninth and ninth-and-a-half post-Newtonian terms.
We derive the post-Newtonian limit of a general class of teleparallel gravity theories, whose action is given by a free function of three scalar quantities obtained from the torsion of the teleparallel connection. This class of theories is chosen to be sufficiently generic in order to include the $f(T)$ class of theories as well as new general relativity as subclasses. To derive its post-Newtonian limit, we first impose the Weitzenböck gauge, and then introduce a post-Newtonian approximation of the tetrad field around a Minkowski background solution. Our results show that the class of theories…
The Parametrized Post-Newtonian expansion of gravitational theories with a scalar field coupled to the Gauss-Bonnet invariant is performed and confrontation of such theories with Solar system experiments is discussed.
In this paper, we develop an iterative scheme to enable the explicit calculation of an arbitrary post-Newtonian order for a relativistic body that reduces to the Maclaurin spheroid in the appropriate limit. This scheme allows for an analysis of the structure of the solution in the vicinity of bifurcation points along the Maclaurin sequence. The post-Newtonian expansion is solved explicitly to the fourth order and its accuracy and convergence are studied by comparing it to highly accurate numerical results.
In this work the influence of the post-Newtonian corrections to the equations of stellar structure is analysed. The post-Newtonian Lane-Emden equation follows from the corresponding momentum density balance equation. From a polytropic equation of state the solutions of the Lane-Endem equations in the Newtonian and post-Newtonian theories are determined and the physical quantities for the \textit{Sun}, for the white dwarf \textit{Sirius B} and for neutron stars with masses $M\simeq1.4M_\odot, 1.8M_\odot$ and $2.0M_\odot$ are calculated. It is shown that the post-Newtonian corrections to the fie…
Expression for second post-Newtonian level gravitational deflection angle of massive particles is obtained in a model independent framework. Several of its important implications including the possibility of testing gravitational theories at that level are discussed.
We derive a coordinate-independent formulation of the post-1-Newtonian approximation to general relativity. This formulation is a generalization of the Newton-Cartan geometric formulation of Newtonian gravity. It involves several fields and a connection, but no spacetime metric at the fundamental level. We show that the usual coordinate-dependent equations of post-Newtonian gravity are recovered when one specializes to asymptotically flat spacetimes and to appropriate classes of coordinates.
Regular and chaotic test particle motion in gravitational fields due to stellar bodies with quadrupolar and octupolar deformation are studied using Poincare surfaces of section. In first instance, we analyze the purely Newtonian case and we find that the octupolar term induces a distortion in the KAM curves corresponding to regular trajectories as well as an increase in chaoticity, even in the case corresponding to oblate deformation. Then we examine the effect of the first general relativistic corrections, provided by the post Newtonian approach. For typical values of the post Newtonian multi…
