Approximation Of General Relativity Research Articles

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.

On the Rotation Curve of Disk Galaxies in General Relativity

Recently, it has been suggested that the phenomenology of flat rotation curves observed at large radii in the equatorial plane of disk galaxies can be explained as a manifestation of general relativity (GR) instead of the effect of dark matter (DM) halos. In this paper, by using the well-known weak-field, low-velocity gravitomagnetic formulation of GR, the expected rotation curves in GR are rigorously obtained for purely baryonic disk models with realistic density profiles and compared with the predictions of Newtonian gravity for the same disks in absence of DM. As expected, the resulting rotation curves are indistinguishable, with GR corrections at all radii of the order v 2/c 2 ≈ 10−6. Next, the gravitomagnetic Jeans equations for two-integral stellar systems are derived, and then solved for the Miyamoto–Nagai disk model, showing that finite-thickness effects do not change the previous conclusions. Therefore, the observed phenomenology of galactic rotation curves at large radii requires DM in GR exactly as in Newtonian gravity, unless the cases here explored are reconsidered in the full GR framework with substantially different results (with the surprising consequence that the weak-field approximation of GR cannot be applied to the study of rotating systems in the weak-field regime). In this article, the mathematical framework is described in detail, so that the present study can be extended to other disk models, or to elliptical galaxies (where DM is also required in Newtonian gravity, but their rotational support can be much less than in disk galaxies).

The quadrupole moment of compact binaries to the fourth post-Newtonian order: II. Dimensional regularization and renormalization

The regularization and renormalization of the radiative mass-type quadrupole moment of inspiralling compact binaries (without spins) is investigated at the fourth post-Newtonian (4PN) approximation of general relativity. As clear from the conservative 4PN equations of motion, a dimensional regularization has to be implemented in order to properly treat the non-linear interactions experienced by gravitational waves during their propagation towards future null infinity. By implementing such procedure, we show that the poles coming from the source moment (computed in a companion paper) are exactly cancelled in the radiative moment, as expected for a physical quantity. We thus define and obtain a ‘renormalized’ source quadrupole, three-dimensional by nature, which is an important step towards the computation of the gravitational-wave flux with 4PN accuracy. Furthermore, we explicitly prove the equivalence between the dimensional regularization and the previously used Hadamard partie finie scheme up to the 3PN order.

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.

Using the quantity z(bq,m) in terms of general relativity

Albert Einstein’s general theory of relativity describes gravitation through a deformation of space-time. However, there is a high degree of parallelism between the linear approximation of general relativity and Maxwell’s theory of electromagnetism. The first of Maxwell’s laws allows one to derive Coulomb’s law and, therefore, the correct radius dependency of charge interaction. Both theories (general relativity and Maxwell’s theory) are absolutely equivalent to each other in the linear derivation. Therefore, mass and charge have analogous positions and functions. This is the starting point for considering that it should be possible to combine both theories into one. To do so, the general theory of relativity has to be formulated for the quantity z, which is a complex number that combines mass and charge. In this way, it becomes possible to combine general relativity and electromagnetism into one theory.

The mass quadrupole moment of compact binary systems at the fourth post-Newtonian order

The mass-type quadrupole moment of inspiralling compact binaries (without spins) is computed at the fourth post-Newtonian (4PN) approximation of general relativity. The multipole moments are defined by matching between the field in the exterior zone of the matter system and the PN field in the near zone, following the multipolar-post-Minkowskian (MPM)-PN formalism. The matching implies a specific regularization for handling infra-red (IR) divergences of the multipole moments at infinity, based on the Hadamard finite part procedure. On the other hand, the calculation entails ultra-violet (UV) divergences due to the modelling of compact objects by delta-functions, that are treated with dimensional regularization (DR). In future work we intend to systematically study the IR divergences by means of dimensional regularization as well. Our result constitutes an important step in the goal of obtaining the gravitational wave templates of inspiralling compact binary systems with 4PN/4.5PN accuracy.

Logarithmic tail contributions to the energy function of circular compact binaries

We combine different techniques to extract information about the logarithmic contributions to the two-body conservative dynamics within the post-Newtonian (PN) approximation of General Relativity. The logarithms come from the conservative part of non linear gravitational-wave tails and their iterations. Explicit, original expressions are found for conservative dynamics logarithmic tail terms up to 6PN order by adopting both traditional PN calculations and effective field theory (EFT) methods. We also determine all logarithmic terms at 7PN order, fixing a sub-leading logarithm from a tail-of-tail-of-tail process by comparison with self-force (SF) results. Moreover, we use renormalization group techniques to obtain the leading logarithmic terms to generic power $n$, appearing at $(3n+1)$PN order, and we resum the infinite series in a closed form. Half-integer PN orders enter the conservative dynamics starting at 5.5PN, but they do not generate logarithmic contributions up to next-to-next-to-leading order included. We nevertheless present their contribution at leading order in the small mass ratio limit.

Equations of motion of self-gravitatingN-body systems in the first post-Minkowskian approximation

We revisit the problem of the equations of motion of a system of $N$ self-interacting massive particles (without spins) in the first post-Minkowskian (1PM) approximation of general relativity. We write the equations of motion, gravitational field and associated conserved integrals of the motion in a form suitable for comparison with recently published post-Newtonian (PN) results at the 4PN order. We show that the Lagrangian associated with the equations of motion in harmonic coordinates is a generalized one, and compute all the terms linear in $G$ up to 5PN order. We discuss the Hamiltonian in the frame of the center of mass and exhibit a canonical transformation connecting it to previous results directly obtained with the Hamiltonian formalism of general relativity. Finally we recover the known result for the gravitational scattering angle of two particles at the 1PM order.

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.

Center-of-mass equations of motion and conserved integrals of compact binary systems at the fourth post-Newtonian order

The dynamics of compact binary systems at the fourth post-Newtonian (4PN) approximation of general relativity has been recently completed in a self-consistent way. In this paper, we compute the ten Poincar\'e constants of the motion and present the equations of motion in the frame of the center of mass (CM), together with the corresponding CM Lagrangian, conserved energy and conserved angular momentum. Next, we investigate the reduction of the CM dynamics to the case of quasi-circular orbits. The non local (in time) tail effect at the 4PN order is consistently included, as well as the relevant radiation-reaction dissipative contributions to the energy and angular momentum.

Dimensional regularization of the IR divergences in the Fokker action of point-particle binaries at the fourth post-Newtonian order

The Fokker action of point-particle binaries at the fourth post-Newtonian (4PN) approximation of general relativity has been determined previously. However two ambiguity parameters associated with infra-red (IR) divergencies of spatial integrals had to be introduced. These two parameters were fixed by comparison with gravitational self-force (GSF) calculations of the conserved energy and periastron advance for circular orbits in the test-mass limit. In the present paper together with a companion paper, we determine both these ambiguities from first principle, by means of dimensional regularization. Our computation is thus entirely defined within the dimensional regularization scheme, for treating at once the IR and ultra-violet (UV) divergencies. In particular, we obtain crucial contributions coming from the Einstein-Hilbert part of the action and from the non-local tail term in arbitrary dimensions, which resolve the ambiguities.

The ether theory as implying that electromagnetism is the Newtonian approximation of general relativity

The ether theory as implying that electromagnetism is the Newtonian approximation of general relativity

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

AbstractWe 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.

The third and a half-post-Newtonian gravitational wave quadrupole mode for quasi-circular inspiralling compact binaries

We compute the quadrupole mode of the gravitational waveform of inspiralling compact binaries at the third and half-post-Newtonian (3.5PN) approximation of general relativity. The computation is performed using the multipolar post-Newtonian formalism, and restricted to binaries without spins moving on quasi-circular orbits. The new inputs mainly include the 3.5PN terms in the mass quadrupole moment of the source, and the control of required subdominant corrections to the contributions of hereditary integrals (tails and nonlinear memory effect). The result is given in the form of the quadrupolar mode (2, 2) in a spin-weighted spherical harmonic decomposition of the waveform, and may be used for comparison with the counterpart quantity computed in numerical relativity. It is a step towards the computation of the full 3.5PN waveform, whose knowledge is expected to reduce the errors on the location parameters of the source.

Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space–time

Gravitational radiation of binary systems can be studied by using the adiabatic approximation in General Relativity. In this approach a small astrophysical object follows a trajectory consisting of a chained series of bounded geodesics (orbits) in the outer region of a Kerr Black Hole, representing the space time created by a bigger object. In our paper, we study the entire class of orbits, both of constant radius (spherical orbits), as well as non-null eccentricity orbits, showing a number of properties on the physical parameters and trajectories. The main result is the determination of the geometrical locus of all the orbits in the space of physical parameters in Kerr space–time. This becomes a powerful tool to know if different orbits can be connected by a continuous change of their physical parameters. A discussion on the influence of different values of the angular momentum of the hole is given. Main results have been obtained by analytical methods.

Gauge and constraint degrees of freedom: from analytical to numerical approximations in General Relativity

The harmonic formulation of Einstein's field equations is considered, where the gauge conditions are introduced as dynamical constraints. The difference between the fully constrained approach (used in analytical approximations) and the free evolution one (used in most numerical approximations) is pointed out. As a generalization, quasi-stationary gauge conditions are also discussed, including numerical experiments with the gauge-waves testbed. The complementary 3+1 approach is also considered, where constraints are related instead with energy and momentum first integrals and the gauge must be provided separately. The relationship between the two formalisms is discussed in a more general framework (Z4 formalism). Different strategies in black hole simulations follow when introducing singularity avoidance as a requirement. More flexible quasi-stationary gauge conditions are proposed in this context, which can be seen as generalizations of the current “freezing shift” prescriptions.

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.

General relativistic dynamics of compact binary systems

The equations of motion of compact binary systems have been derived in the post-Newtonian (PN) approximation of general relativity. The current level of accuracy is 3.5PN order. The conservative part of the equations of motion (neglecting the radiation reaction damping terms) is deducible from a generalized Lagrangian in harmonic coordinates, or equivalently from an ordinary Hamiltonian in ADM coordinates. As an application, we investigate the problem of the dynamical stability of circular binary orbits against gravitational perturbations up to the 3PN order. We find that there is no innermost stable circular orbit or ISCO at the 3PN order for equal masses. To cite this article: L. Blanchet, C. R. Physique 8 (2007).

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.

Erratum: Energy losses by gravitational radiation in inspiraling compact binaries to5/2post-Newtonian order [Phys. Rev. D54, 001417 (1996)

This paper derives the total power or energy loss rate generated in the form of gravitational waves by an inspiralling compact binary system to the five halves post-Newtonian (2.5PN) approximation of general relativity. Extending a recently developed gravitational-wave generation formalism valid for arbitrary (slowly-moving) systems, we compute the mass multipole moments of the system and the relevant tails present in the wave zone to 2.5PN order. In the case of two point-masses moving on a quasi-circular orbit, we find that the 2.5PN contribution in the energy loss rate is entirely due to tails. Relying on an energy balance argument we derive the laws of variation of the instantaneous frequency and phase of the binary. The 2.5PN order in the accumulated phase is significantly large, being grossly of the same order of magnitude as the previous 2PN order, but opposite in sign. However finite mass effects at 2.5PN order are small. The results of this paper should be useful when analyzing the data from inspiralling compact binaries in future gravitational-wave detectors like VIRGO and LIGO.