Covariant equations of motion beyond the spin-dipole particle approximation (original) (raw)

Covariant equations of motion of extended bodies with arbitrary mass and spin multipoles

Physical Review D

Gravitational wave detectors allow to test general relativity and to study the internal structure and orbital dynamics of neutron stars and black holes in inspiraling binary systems with a potentially unlimited rigor. Currently, analytic calculations of gravitational wave signal emitted by inspiralling compact binaries are based on the numerical integration of the asymptotic post-Newtonian expansions of the equations of motion in a pole-dipole approximation that includes masses and spins of the bodies composing the binary. Further progress in the accurate construction of gravitational-wave templates of the compact binaries strictly depends on our ability to significantly improve theoretical description of gravitational dynamics of extended bodies by taking into account the higher-order (quadrupole, octupole, etc.) multipoles in equations of motion of the bodies both in the radiative and conservative approximations of general relativity and other viable alternative theories of gravity. This paper employs the post-Newtonian approximations of a scalar-tensor theory of gravity along with the mathematical apparatus of the Cartesian symmetric trace-free tensors and the Blanchet-Damour multipole formalism to derive translational and rotational equations of motion of N extended bodies having arbitrary distribution of mass and velocity of matter. We assume that spacetime manifold can be covered globally by a single coordinate chart which asymptotically goes over to the Minkowskian coordinate chart at spatial infinity. We also introduce N local coordinate charts adapted to each body and covering a finite domain of space around the body. Gravitational field in the neighborhood of each body is parametrized by an infinite set of mass and spin multipoles of the body as well as by the set of tidal gravitoelectric and gravitomagnetic multipoles of external N − 1 bodies. The origin of the local coordinates is set moving along the accelerated worldline of the center of mass of the corresponding body by an appropriate choice of the internal and external dipole moments of the gravitational field. Translational equations of motion of the body's center of mass and rotational equations of motion for its spin are derived by integrating microscopic equations of motion of body's matter and applying the method of asymptotic matching technique to splice together the post-Newtonian solutions of the field equations of the scalar-tensor theory of gravity for the metric tensor and scalar field obtained in the global and local coordinate charts. The asymptotic matching is also used for separating the post-Newtonian self-field effects from the external gravitational environment and constructing the effective background spacetime manifold. It allows us to present the equations of translational and rotational motion of each body in covariant form by making use of the Einstein principle of equivalence. This relaxes the slowmotion approximation and makes the covariant post-Newtonian equations of motion of extended bodies with weak self-gravity applicable for the case of relativistic speeds. Though the covariant equations of the first post-Newtonian order are still missing terms from the second post-Newtonian approximation they may be instrumental to get a glimpse of the last several orbital revolutions of stars in ultra-compact binary system just before merging. Our approach significantly generalizes the Mathisson-Papapetrou-Dixon covariant equations of motion with regard to the number of body's multipoles and the post-Newtonian terms having been taken into account. The equations of translational and rotational motion derived in the present paper include the entire infinite set of covariantly-defined mass and spin multipoles of the bodies. Thus, they can be used for much more accurate prediction of orbital dynamics of tidally deformed stars in inspiraling binary systems and construction of templates of gravitational waves at the merger stage of coalescing binary when the strong tidal distortions and gravitational coupling of higher-order mass and spin multipoles of the stars play a dominant role in the last few seconds of the binary life.

General-relativistic celestial mechanics. III. Rotational equations of motion

Physical Review D, 1993

The rotational laws of motion for arbitrarily shaped, weakly self-gravitating bodies, members of gravitationally interacting N-body systems, are obtained at the first post-Newtonian approximation of general relativity. The derivation uses our previously introduced framework, characterized by the combined use of N local (body-attached) reference systems with one global reference system, and by the introduction of new sets of relativistic multipole moments, and relativistic tidal moments. We show how to associate with each body (considered in its corresponding local frame) a first-post-Newtonian-accurate spin vector, whose local-time evolution is entirely determined by the coupling between the multipole moments of that body and the tidal moments it experiences. The leading relativistic effects in the spin motion are discussed: gravitational Larmor theorem (de Sitter-Fokker-Eddington precession) and post-Newtonian contributions to the torque associated with the quadrupole moment and the quadrupole tidal tensor.

Relativistic equations of motion of celestial bodies

Symposium - International Astronomical Union, 1996

The problem of relativistic equations of motion for extended celestial bodies in the first post-Newtonian approximation is reviewed. It is argued that the problems dealing with kinematical aspects have been solved in a satisfactory way, but more work has to be done on the dynamical side. Concepts like angular velocity, moments of inertia, Tisserand axes etc. still have to be introduced in a rigorous manner at the 1PN level.

Relativistic celestial mechanics with PPN parameters

Physical Review D, 2000

Starting from the global parametrized post-Newtonian (PPN) reference system with two PPN parameters γ and β 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 η = 4β − γ − 3 reflecting a violation of the equivalence principle are discussed in detail.

Post-Newtonian celestial dynamics in cosmology: Field equations

Physical Review D, 2013

Post-Newtonian celestial dynamics is a relativistic theory of motion of massive bodies and test particles under the influence of relatively weak gravitational forces. Standard approach for development of this theory relies upon the key concept of the isolated astronomical system supplemented by the assumption that the background space-time is flat. The standard post-Newtonian theory of motion was instrumental in explanation of the existing experimental data on binary pulsars, satellite and lunar laser ranging, and in building precise ephemerides of planets in the solar system. Recent studies of the formation of large-scale structure in our universe indicate that the standard post-Newtonian mechanics fails to describe more subtle dynamical effects in motion of the bodies comprising the astronomical systems of larger size -galaxies and clusters of galaxies -where the Riemann curvature of the expanding FLRW universe interacts with the local gravitational field of the astronomical system and, as such, can not be ignored.

New perturbative method for solving the gravitational N-body problem in general relativity

2013

We present a new approach to describe the dynamics of an isolated, gravitationally bound astronomical N -body system in the weak field and slow-motion approximation of general relativity. Celestial bodies are described using an arbitrary energy-momentum tensor and assumed to possess any number of internal multipole moments. The solution of the gravitational field equations in any reference frame is presented as a sum of three terms: i) the inertial flat spacetime in that frame, ii) unperturbed solutions for each body in the system boosted to the coordinates of this frame, and iii) the gravitational interaction term. Such an ansatz allows us to reconstruct all features of the gravitational field and to develop a theory of relativistic reference frames. We use the harmonic gauge conditions to impose a significant constraint on the structure of the post-Galilean coordinate transformation functions that relate global coordinates in the inertial reference frame to the local coordinates of the non-inertial frame associated with a particular body. The remaining parts of these functions are constrained using dynamical conditions, which are obtained by constructing the relativistic proper reference frame associated with a particular body. In this frame, the effect of external forces acting on the body is balanced by the fictitious frame-reaction force that is needed to keep the body at rest with respect to the frame, conserving its relativistic linear momentum. We find that this is sufficient to determine explicitly all the terms of the coordinate transformation. The same method is then used to develop the inverse transformations. The resulting post-Galilean coordinate transformations have an approximate group structure that extends the Poincaré group of global transformations to the case of a gravitational N -body system. We present and discuss the structure of the metric tensors corresponding to the reference frames involved, the rules for transforming relativistic gravitational potentials, the coordinate transformations between frames and the resulting relativistic equations of motion. PACS numbers: 03.30.+p, 04.25.Nx, 04.80.-y, 06.30.Gv, 95.10.Eg, 95.10.Jk, 95.55.Pe

Motion and rotation of celestial bodies in the post-Newtonian approximation

Celestial Mechanics, 1987

Consistent post-Newtonian description of motion and precession in a system of N extended slowly rotating bodies is developed in the framework of the post-Newtonian approximation scheme (PNA). The solution of Einstein equations for the metric in the local reference system related to a body of the system is obtained. This metric is used to derive the equations of motion and precession of the considered body on the basis of some relativistic generalization of the model of rigid body. These equations are solved in order to find the first order corrections to nutation theory and to the osculating orbital elements of the body. Another important application of such local metric, concerning the motion of a test particle (e.g., artificial satellite) orbiting the body, is also investigated in this paper.