post-Newtonian Approximation Of General Relativity Research Articles

Precession of planets' orbits between Newtonian gravity and Einstein's general relativity

The recent result that, contrary to a longstanding conviction older than 160 years, the precession of Mercury perihelion can be achieved in Newtonian gravity with a very high precision by correctly analyzing the situation without neglecting Mercury's mass is confirmed. Orbit's precession does not occur in an inertial (in Newtonian sense) frame of reference, but it occurs in the non-inertial frame of reference of the Sun because in Newtonian theory, the distance which is travelled by a body depends on the frame of reference in which the motion of the body is analyzed. After reviewing a previous solution, the precession is directly obtained from the orbit equation in the non-inertial frame of reference of the Sun. The Newtonian formula of the precession of planets' perihelion breaks down for the other planets because the predicted Newtonian result is indeed too large for Venus and Earth. In a previous paper, it was shown that corrections due to gravitational and rotational time dilation, in an intermediate framework (a particular post-Newtonian approximation of general relativity) which analyzes gravity between Newton and Einstein, solve the problem in accordance with some other results in the literature. By adding such corrections, a result consistent with the one of general relativity was indeed obtained. Those corrections also are re-analyzed in this paper by stressing the difference between general relativity and Newtonian physics by evoking the Einstein Equivalence Principle, and the third postulate of relativity. This framework of gravity between Newton and Einstein seems consistent with global covariance. Finally, it is important to stress that a better understanding of gravitational effects in an intermediate framework between Newtonian theory and general relativity, which is one of the goals of this paper, could, in principle, be crucial for a better understanding of the famous Dark Matter and Dark Energy problems.

Rotating neutron stars in the first-order post-Newtonian approximation

We study models of uniformly and differentially rotating neutron stars in the framework of the first-order post-Newtonian approximation in general relativity as established by Chandrasekhar. In particular, we adopt the polytropic equation of state in order to derive the appropriate hydrodynamic equations and a rotation law based on the generalized Clement’s model. To compute equilibrium configurations at the mass-shedding limit, i.e. at critical angular velocity (equivalently, Keplerian angular velocity), we develop an iterative numerical method, belonging to the category of the well-known “self-consistent field methods”, with two perturbation parameters: the “rotation parameter” ῡ and the “gravitation or relativity parameter” σ̄. These two parameters represent the effects of rotation and gravity on the configuration. We investigate the validity and the limits of our method by comparing our results with respective results of other computational methods and public domain codes. As it turns out, our method can derive satisfactory results for general-relativistic polytropic configurations at critical rotation.

Post-Keplerian waveform model for binary compact object as sources of space-based gravitational wave detector and its implications

Binary compact objects will be among the important sources for the future space-based gravitational wave detectors. Such binary compact objects include stellar massive binary black hole, binary neutron star, binary white dwarf and mixture of these compact objects. Regarding to the relatively low frequency, the gravitational interaction between the two objects of the binary is weak. Post-Newtonian approximation of general relativity is valid. Previous works about the waveform model for such binaries in the literature consider the dynamics for specific situations which involve detailed complicated matter dynamics between the two objects. We here take a different idea. We adopt the trick used in pulsar timing detection. For any gravity theories and any detailed complicated matter dynamics, the motion of the binary can always be described as a post-Keplerian expansion. And a post-Keplerian gravitational waveform model will be reduced. Instead of object masses, spins, matter’s equation of state parameters and dynamical parameters beyond general relativity, the involved parameters in our post-Keplerian waveform model are the Keplerian orbit elements and their adiabatic variations. Respect to current planning space-based gravitational wave detectors including LISA, Taiji and Tianqin, we find that the involved waveform model parameters can be well determined. And consequently the detail matter dynamics of the binary can be studied then. For binary with purely gravitational interactions, gravity theory can be constrained well.

Estimating the Parameters of Extended Gravity Theories with the Schwarzschild Precession of S2 Star

After giving a short overview of previous results on constraining of Extended Gravity by stellar orbits, we discuss the Schwarzschild orbital precession of S2 star assuming the congruence with predictions of General Relativity (GR). At the moment, the S2 star trajectory is remarkably fitted with the first post-Newtonian approximation of GR. In particular, both Keck and VLT (GRAVITY) teams declared that the gravitational redshift near its pericenter passage for the S2 star orbit corresponds to theoretical estimates found with the first post-Newtonian (pN) approximation. In 2020, the GRAVITY Collaboration detected the orbital precession of the S2 star around the supermassive black hole (SMBH) at the Galactic Center and showed that it is close to the GR prediction. Based on this observational fact, we evaluated parameters of the Extended Gravity theories with the Schwarzschild precession of the S2 star. Using the mentioned method, we estimate the orbital precession angles for some Extended Gravity models including power-law f(R), general Yukawa-like corrections, scalar–tensor gravity, and non-local gravity theories formulated in both metric and Palatini formalism. In this consideration, we assume that a gravitational field is spherically symmetric, therefore, alternative theories of gravity could be described only with a few parameters. Specifically, considering the orbital precession, we estimate the range of parameters of these Extended Gravity models for which the orbital precession is like in GR. Then we compare these results with our previous results, which were obtained by fitting the simulated orbits of S2 star to its observed astrometric positions. In case of power-law f(R), generic Yukawa-like correction, scalar–tensor gravity and non-local gravity theories, we were able to obtain a prograde orbital precession, like in GR. According to these results, the method is a useful tool to evaluate parameters of the gravitational potential at the Galactic Center.

Post-Newtonian satellite orbits

The first post-Newtonian approximation of general relativity is used to account for the motion of solar system bodies and near-Earth objects which are slow moving and produce weak gravitational fields. The $n$ -body relativistic equations of motion are given by the Einstein-Infeld-Hoffmann equations. For $n=2$ , we investigate the associated dynamics of two-body systems in the first post-Newtonian approximation. By direct integration of the associated planar equations of motion, we deduce a new expression that characterises the orbit of test particles in the first post-Newtonian regime generalising the well-known Binet equation for Newtonian mechanics. The expression so obtained does not appear to have been given in the literature and is consistent with classical orbiting theory in the Newtonian limit. Further, the accuracy of the post-Newtonian Binet equation is numerically verified by comparing secular variations of known expression with the full general relativistic orbit equation.

Modeling approaches for precise relativistic orbits: Analytical, Lie-series, and pN approximation

Accurate orbit modeling plays a key role in contemporary and future space missions such as GRACE and its successor GRACE-FO, GNSS, and altimetry missions. To fully exploit the technological capabilities and correctly interpret measurements, relativistic orbital effects need to be taken into account.Within the theory of General Relativity, equations of motion for freely falling test objects, such as satellites orbiting the Earth, are given by the geodesic equation. We analyze and compare different solution methods in a spherically symmetric background, i.e.for the Schwarzschild spacetime, as a test bed. We investigate satellite orbits and use direct numerical orbit integration as well as the semi-analytical Lie-series approach. The results are compared to the exact analytical reference solution in terms of elliptic functions. For a set of exemplary orbits, we determine the respective accuracy of the different methods.Within the post-Newtonian approximation of General Relativity, modified orbital equations are obtained by adding relativistic corrections to the Newtonian equations of motion. We analyze the accuracy of this approximation with respect to the general relativistic setting. Therefore, we solve the post-Newtonian equation of motion using the eXtended High Performance Satellite dynamics Simulator. For corresponding initial conditions, we compare orbits in the Schwarzschild spacetime to those in its post-Newtonian approximation. Moreover, we compare the magnitude of relativistic contributions to several typical perturbations of satellite orbits due to, e.g., solar radiation pressure, Earth’s albedo, and atmospheric drag. This comparison is done for our test scenarios and for a real GRACE orbit to highlight the importance of relativistic effects in geodetic space missions. For the considered orbits, first-order relativistic contributions give accelerations of about 20 nm/s2 and are dominant in the radial direction.

Conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity

The fourth post-Newtonian (4PN) two-body dynamics has been recently tackled by several different approaches: effective field theory, Arnowitt-Deser-Misner Hamiltonian, action-angle-Delaunay averaging, effective-one-body, gravitational self-force, first law of dynamics, and Fokker action. We review the achievements of these approaches and discuss the complementarity of their results. Our main conclusions are: (i) the results of the first complete derivation of the 4PN dynamics [T.Damour, P.Jaranowski, and G.Sch\"afer, Phys. Rev. D 89, 064058 (2014)] have been, piecewise, fully confirmed by several subsequent works; (ii) the results of the Delaunay-averaging technique [T.Damour, P.Jaranowski, and G.Sch\"afer, Phys. Rev. D 91, 084024 (2015)] have been confirmed by several independent works; and (iii) several claims in a recent Fokker-action computation [L.Bernard et al., arXiv:1512.02876v2 [gr-qc]] are incorrect, but can be corrected by the addition of a couple of ambiguity parameters linked to subtleties in the regularization of infrared and ultraviolet divergences.

Anisotropic models for globular clusters, Galactic bulges, and dark halos

Spherical systems with polytropic equations of state are of great interest in astrophysics. They are widely used to describe neutron stars, red giants, white dwarfs, brown dwarfs, main sequence stars, galactic halos, and globular clusters of diverse sizes. In this paper we construct analytically a family of self-gravitating spherical models in the post-Newtonian approximation of general relativity. These models present interesting cusps in their density profiles which are appropriate for the modeling of galaxies and dark matter halos. The systems described here are anisotropic in the sense that their equiprobability surfaces in velocity space are nonspherical, leading to an overabundance of radial or circular orbits, depending on the parameters of the model under consideration. Among the family of models, we find the post-Newtonian generalization of the Plummer and Hernquist models. A close inspection of their equation of state reveals that these solutions interpolate smoothly between a polytropic sphere in the asymptotic region and an inner core that resembles an isothermal sphere. Finally, we study the thermodynamics of these models and argue for their stability.

Is gravitational mass of a composite quantum body equivalent to its energy?

Abstract We define passive gravitational mass operator of a hydrogen atom in the post-Newtonian approximation of general relativity and show that it does not commute with energy operator, taken in the absence of gravitational field. Nevertheless, the equivalence between the expectation values of passive gravitational mass and energy is shown to survive for stationary quantum states. Inequivalence between passive gravitational mass and energy at a macroscopic level results in time dependent oscillations of the expectation values of passive gravitational mass for superpositions of stationary quantum states, where the equivalence restores after averaging over time. Inequivalence between gravitational mass and energy at a microscopic level reveals itself as unusual electromagnetic radiation, emitted by the atoms, supported and moved in the Earth gravitational field with constant velocity using spacecraft or satellite, which can be experimentally measured.

Non-relativistic gravitation: from Newton to Einstein and back

We present an improvement to the classical effective theory approach to the non-relativistic or post-Newtonian approximation of general relativity. The ‘potential metric field’ is decomposed through a temporal Kaluza–Klein ansatz into three non-relativistic gravity (NRG) fields: a scalar identified with the Newtonian potential, a 3-vector corresponding to the gravito-magnetic vector potential and a 3-tensor. The derivation of the Einstein–Infeld–Hoffmann Lagrangian simplifies such that each term corresponds to a single Feynman diagram providing a clear physical interpretation. Spin interactions are dominated by the exchange of the gravito-magnetic field. Leading correction diagrams corresponding to the 3PN correction to the spin–spin interaction and the 2.5PN correction to the spin–orbit interaction are presented.

Resurvey of order and chaos in spinning compact binaries

This paper is mainly devoted to applying the invariant, fast, Lyapunov indicator to clarify some doubt regarding the apparently conflicting results of chaos in spinning compact binaries at the second-order post-Newtonian approximation of general relativity from previous literatures. It is shown with a number of examples that no single physical parameter or initial condition can be described as responsible for causing chaos, but a complicated combination of all parameters and initial conditions is responsible. In other words, a universal rule for the dependence of chaos on each parameter or initial condition cannot be found in general. Chaos does not depend only on the mass ratio, and the maximal spins do not necessarily bring the strongest effect of chaos. Additionally, chaos does not always become drastic when the initial spin vectors are nearly perpendicular to the orbital plane, and the alignment of spins cannot trigger chaos by itself.

The Post-Newtonian Approximation for Relativistic Compact Binaries.

We discuss various aspects of the post-Newtonian approximation in general relativity. After presenting the foundation based on the Newtonian limit, we show a method to derive post-Newtonian equations of motion for relativistic compact binaries based on a surface integral approach and the strong field point particle limit. As an application we derive third post-Newtonian equations of motion for relativistic compact binaries which respect the Lorentz invariance in the post-Newtonian perturbative sense, admit a conserved energy, and are free from any ambiguity.

On the Stability of the Triangular Lagrangian Equilibrium Points in the Relativistic Restricted Three–Body Problem

The restricted three body problem is treated in the framework of the post-Newtonian approximation of general relativity. The equations of motion are linear zed around the libration points L4,5. Locations of the equilibrium points L4,5 are obtained. Existence and stability of these points are investigated. The relativistic correction to the classical mass ratio is determined.

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 is suitable for practical computations of the gravitational radiation reaction forces in harmonic coordinates. It also provides some insights on the structure of the post-Newtonian expansion in general relativity and the radiation reaction terms therein.

On the two-body problem in general relativity

We consider the two-body problem in post-Newtonian approximations of general relativity. We report the recent results concerning the equations of motion, and the associated Lagrangian formulation, of compact binary systems, at the third post-Newtonian order (∼1/ c 6 beyond the Newtonian acceleration). These equations are necessary when constructing the theoretical templates for searching and analyzing the gravitational-wave signals from inspiralling compact binaries in VIRGO and LISA type experiments.

Dynamical Bar Instability in Rotating Stars: Effect of General Relativity

We study the dynamical stability against bar-mode deformation of rapidly and differentially rotating stars in the first post-Newtonian approximation of general relativity. We vary the compaction of the star M/R (where M is the gravitational mass and $R$ the equatorial circumferential radius) between 0.01 and 0.05 to isolate the influence of relativistic gravitation on the instability. For compactions in this moderate range, the critical value of $\beta \equiv T/W$ for the onset of the dynamical instability (where T is the rotational kinetic energy and W the gravitational binding energy) slightly decreases from ~ 0.26 to ~ 0.25 with increasing compaction for our choice of the differential rotational law. Combined with our earlier findings based on simulations in full general relativity for stars with higher compaction, we conclude that relativistic gravitation {\em enhances} the dynamical bar-mode instability, i.e. the onset of instability sets in for smaller values of $\beta$ in relativistic gravity than in Newtonian gravity. We also find that once a triaxial structure forms after the bar-mode perturbation saturates in dynamically unstable stars, the triaxial shape is maintained, at least for several rotational periods. To check the reliability of our numerical integrations, we verify that the general relativistic Kelvin-Helmholtz circulation is well-conserved, in addition to rest-mass energy, total mass-energy, linear and angular momentum. Conservation of circulation indicates that our code is not seriously affected by numerical viscosity. We determine the amplitude and frequency of the quasi-periodic gravitational waves emitted during the bar formation process using the quadrupole formula.

ESTIMATION OF POST-NEWTONIAN APPROXIMATION OF GRAVITY THAT AFFECTS SATELLITES OF PLANET

The post-Newtonian approximation in general relativity will be used to estimate gravity for the jovian system and the comet shoemaker-Levy 9 that collided with Jupiter. We calculate and discuss the post-Newtonian corrections to the gravity of Jupiter and precessions of its satellites, which are compared with the Earth-Moon system.

Irrotational and Incompressible Ellipsoids in the First Post-Newtonian Approximation of General Relativity

Irrotational and Incompressible Ellipsoids in the First Post-Newtonian Approximation of General Relativity

Irrotational and Incompressible Binary Systems in the First Post-Newtonian Approximation of General Relativity

The first post-Newtonian (PN) hydrostatic equations for an irrotational fluid are solved for an incompressible binary system. The equilibrium configuration of the binary system is given by a small deformation from the irrotational Darwin-Riemann ellipsoid which is the solution at Newtonian order. It is found that the orbital separation at the innermost stable circular orbit (ISCO) decreases when one increases the compactness parameter $M_{\ast}/c^2 a_{\ast}$, in which $M_{\ast}$ and $a_{\ast}$ denote the mass and the radius of a star, respectively. If we compare the 1PN angular velocity of the binary system at the ISCO in units of $\sqrt{M_{\ast}/a_{\ast}^3}$ with that of Newtonian order, the angular velocity at the ISCO is almost the same value as that at Newtonian order when one increases the compactness parameter. Also, we do not find the instability point driven by the deformation at 1PN order, where a new sequence bifurcates throughout the equilibrium sequence of the binary system until the ISCO. We also investigate the validity of an ellipsoidal approximation, in which a 1PN solution is obtained assuming an ellipsoidal figure and neglecting the deformation. It is found that the ellipsoidal approximation gives a fairly accurate result for the total energy, total angular momentum and angular velocity. However, if we neglect the velocity potential of 1PN order, we tend to overestimate the angular velocity at the ISCO regardless of the shape of the star (ellipsoidal figure or deformed figure).

A post-Newtonian Lagrangian perturbation approach to large-scale structure formation

We formulate the Lagrangian perturbation theory to solve the non-linear dynamics of self-gravitating fluid within the framework of the post-Newtonian approximation in general relativity, using the (3+1) formalism. Our formulation coincides with Newtonian Lagrangian perturbation theory developed by Buchert for the scale much smaller than the horizon scale, and with the gauge invariant linearized theory in the longitudinal gauge conditions for the linear regime. These are achieved by using the gauge invariant quantities at the initial time when the linearized theory is valid enough. The post-Newtonian corrections in the solution of the trajectory field of fluid elements are calculated in the explicit forms. Thus our formulation allows us to investigate the evolution of the large-scale fluctuations involving relativistic corrections from the early regime such as the decoupling time of matter and radiation until today. As a result, we are able to show that naive Newtonian cosmology to the structure formation will be a good approximation even for the perturbations with scales not only inside but also beyond the present horizon scale in the longitudinal coordinates. Although the post-Newtonian corrections are small, it is shown that they have a growing transverse mode which is not present in Newtonian theory as well as in the gauge invariant linearized theory. Such post-Newtonian order effects might produce characteristic appearances of the large-scale structure formation, for example, through the observation of anisotropies of the cosmic microwave background radiation (CMB). Furthermore since our approach has a straightforward Newtonian limit, it will be also convenient for numerical implementation based on the presently available Newtonian simulation. ......