Relativistic celestial mechanics with PPN parameters (original) (raw)
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.
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General-relativistic celestial mechanics. IV. Theory of satellite motion
Physical Review D, 1994
The basic equations needed for developing a complete relativistic theory of artificial Earth satellites are explicitly written down. These equations are given both in a local, geocentric frame and in the global, barycentric one. They are derived within our recently introduced general-relativistic celestial mechanics framework. Our approach is more satisfactory than previous ones, especially with regard to its consistency, completeness, and flexibility. In particular, the problem of representing the relativistic gravitational effects associated with the quadrupole and higher multipole moments of the moving Earth, which caused difficulties in several other approaches, is easily dealt with in our approach thanks to the use of previously developed tools: the definition of relativistic multipole moments and transformation theory between reference frames. With this last paper in a series we hope to indicate the way of using our formalism in specific problems in applied celestial mechanics and astrometry.
Relativistic equations of motion of celestial bodies
Symposium - International Astronomical Union, 1996
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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.
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We give a surface integral derivation of post-1-Newtonian translational equations of motion for a system of arbitrarily structured bodies, including the coupling to all the bodies' mass and current multipole moments. The derivation requires only that the post-1-Newtonian vacuum field equations are satisfied in weak-field regions between the bodies; the bodies' internal gravity can be arbitrarily strong. In particular black holes are not excluded. The derivation extends previous results due to Damour, Soffel and Xu (DSX) for weakly self-gravitating bodies in which the post-1-Newtonian field equations are satisfied everywhere. The derivation consists of a number of steps: (i) The definition of each body's current and mass multipole moments and center-of-mass worldline in terms of the behavior of the metric in a weak-field region surrounding the body. (ii) The definition for each body of a set of gravitoelectric and gravitomagnetic tidal moments that act on that body, again in terms of the behavior of the metric in a weak-field region surrounding the body. For the special case of weakly self-gravitating bodies, our definitions of these multipole and tidal moments agree with definitions given previously by DSX. (iii) The derivation of a formula, for any given body, of the second time derivative of its mass dipole moment in terms of its other multipole and tidal moments and their time derivatives. This formula was obtained previously by DSX for weakly self-gravitating bodies. (iv) A derivation of the relation between the tidal moments acting on each body and the multipole moments and center-of-mass worldlines of all the other bodies. A formalism to compute this relation was developed by DSX; we simplify their formalism and compute the relation explicitly. (v) The deduction from the previous steps of the explicit translational equations of motion, whose form has not been previously derived.
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.
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