On the foundations of general relativistic celestial mechanics

Battista, Emmanuele; Esposito, Giampiero; Dell’Agnello, S.

doi:10.1142/s0217751x17300228

Cited by 6 publications

(10 citation statements)

References 61 publications

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“…The body-adapted local coordinates replicate the inertial Lorentzian coordinates only in a limited domain of spacetime manifold M inside a world tube around the body under consideration. Thus, a complete coordinate-based solution of the external and internal problems of celestial mechanics requires introduction of N + 1 coordinate charts -one global and N local ones [54,55]. It agrees with the topological structure of spacetime defined by a set of the overlapping coordinate charts making the atlas of spacetime manifold M [56].…”

Section: Introductionmentioning

confidence: 88%

“…This should be compared with the law of transformation (42) applied to the full metric g αβ on spacetime manifold M which includes besides the external part also the internal and internal-external coupling components of the metric tensor perturbations but they are mutually canceled out in (42) leaving only the external terms, thus, converting (42) to (84) without making any additional assumptions about the structure of the effective background manifold M . The cancellation of the internal and internal-external components of the metric tensor perturbations in (42) is a manifestation of the effacing principle [95] that excludes the internal structure of body B from the definition of the effective background manifoldM used for description of motion of the body [55]. Compatibility of equations (42) and (84) confirms that the internal and external problems of the relativistic celestial mechanics in N -body system are completely decoupled regardless of the structure of the extended bodies and can be extrapolated to compact astrophysical objects like neutron stars and black holes.…”

Section: The Effective Background Manifoldmentioning

confidence: 99%

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Covariant equations of motion beyond the spin-dipole particle approximation

Kopeikin

2019

Eur. Phys. J. Plus

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The present paper studies the post-Newtonian dynamics of N -body problem in general relativity. We derive covariant equations of translational and rotational motion of N extended bodies having arbitrary distribution of mass and velocity of matter by employing the set of global and local coordinate charts on curved spacetime manifold M of N -body system along with the mathematical apparatus of the Cartesian symmetric trace-free tensors and Blanchet-Damour multipole formalism. We separate the self-field effects of the bodies from the external gravitational environment and construct the effective background spacetime manifold by making use of the asymptotic matching technique. We make worldline of the center of mass of each body identical with that of the origin of the body-adapted local coordinates by the appropriate choice of the dipole moments. The covariant equations of motion are obtained on the background manifoldM by applying the Einstein principle of equivalence and the Fermi-Walker law of transportation of the linear momentum and spin of each body. Our approach significantly extends the Mathisson-Papapetrou-Dixon covariant equations of motion beyond the spin-dipole particle approximation by accounting for the entire infinite set of the internal multipoles of the bodies which are gravitationally coupled with the curvature tensor of the background manifoldM and its covariant derivatives. The results of our study can be used for much more accurate prediction of orbital dynamics of extended bodies in inspiraling binary systems and construction of templates of gravitational waves at the merger stage when the strong gravitational interaction between the higher-order multipoles of the bodies play a dominant role. The covariant theory of the post-Newtonian equations of motion beyond the spin-dipole approximation is a solid foundation for future improvements in long-term accuracy of relativistic celestial ephemerides of the solar system bodies.PACS. 04.20.Cv fundamental problems and general formalism -04.25.-g approximation methods; equations of motion -04.25.Nx post-Newtonian approximation; perturbation theory -95.10.Ce N-body problem;celestial mechanics

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Section: Introductionmentioning

confidence: 88%

Section: The Effective Background Manifoldmentioning

confidence: 99%

Covariant equations of motion beyond the spin-dipole particle approximation

Kopeikin

2019

Eur. Phys. J. Plus

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“…In GR, neither analytical nor numerical solutions of 3-body problem of the full theory have been found, and most of the studies were restricted to PN approximations, see, for example, Refs. [25,[51][52][53][54][55] and references therein. In particular, the 1PN collinear solution was found in [56] and proved that it is unique in [57].…”

Section: Introductionmentioning

confidence: 99%

Gravitational waveforms, polarizations, response functions, and energy losses of triple systems in Einstein-aether theory

Lin¹,

Zhao²,

Zhang³

et al. 2019

Phys. Rev. D

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Gravitationally bound hierarchies containing three or more components are very common in our Universe. In this paper we study periodic gravitational wave (GW) form, their polarizations, response function, its Fourier transform, and energy loss rate of a triple system through three different channels of radiation, the scalar, vector and tensor modes, in Einstein-aether theory of gravity. The theory violates locally the Lorentz symmetry, and yet satisfies all the theoretical and observational constraints by properly choosing its four coupling constants ci's. In particular, in the weak-field approximations and with the recently obtained constraints of the theory, we first analyze the energy loss rate of a binary system, and find that the dipole contributions from the scalar and vector modes could be of the order of O (c14) O (GN m/d) 2 , where c14 (≡ c1 + c4) is constrained to c14 O 10 −5 by current observations, and GN , m and d are, respectively, the Newtonian constant, mass and size of the source. On the other hand, the "strong-field" effects for a binary system of neutron stars are about six orders lower than that of GR. So, in this paper we ignore these "strong-field" effects and first develop the general formulas to the lowest post-Newtonian order, by taking the coupling of the aether field with matter into account. Within this approximation, we find that the scalar breather mode and the scalar longitudinal mode are all suppressed by a factor of O (c14) with respect to the transverse-traceless modes (h+ and h×), while the vectorial modes (hX and hY ) are suppressed by a factor of c13 O 10 −15 . Applying the general formulas to a triple system with periodic orbits, we find that the corresponding GW form, response function, and its Fourier transform depend sensitively on the configuration of the triple system, their orientation with respect to the detectors, and the binding energies of the three compact bodies.

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“…Mathematical properties of the manifold M are fully determined in general relativity by the metric tensor g αβ which is found by solving Einstein's field equations. General-relativistic celestial mechanics admits a minimal number of fundamental constants characterizing geometry of curved spacetime -the universal gravitational constant G and the fundamental speed of gravity c which is assumed to be equal the speed of light in vacuum [98,99]. For experimental purposes Will [88] denotes the fundamental speed in gravity sector as c g to distinguish it from the fundamental speed c in matter sector of theory but he understands it in a rather narrow sense as the speed of weak gravitational waves propagating in radiative zone of an isolated gravitating system.…”

Section: Introductionmentioning

confidence: 99%

“…The presence of additional (hypothetical) long-range fields coupled to gravity brings about other fundamental parameters of the scalartensor theory like β and γ which are well-known in PPN formalism [88]. The basic principles of the parameterized relativistic celestial mechanics of extended bodies in scalar-tensor theory of gravity remain basically the same as in general relativity [17,99].…”

Section: Introductionmentioning

confidence: 99%

Covariant Equations of Motion of Extended Bodies with Arbitrary Mass and Spin Multipoles

Kopeikin

2018

Preprint

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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 applicab...

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On the foundations of general relativistic celestial mechanics

Cited by 6 publications

References 61 publications

Covariant equations of motion beyond the spin-dipole particle approximation

Covariant equations of motion beyond the spin-dipole particle approximation

Gravitational waveforms, polarizations, response functions, and energy losses of triple systems in Einstein-aether theory

Covariant Equations of Motion of Extended Bodies with Arbitrary Mass and Spin Multipoles

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