Sergei M. Kopeikin

The IAU 2000 Resolutions for Astrometry, Celestial Mechanics, and Metrology in the Relativistic Framework: Explanatory Supplement

We discuss the IAU resolutions B1.3, B1.4, B1.5, and B1.9 that were adopted during the 24th General Assembly in Manchester, 2000, and provides details on and explanations for these resolutions. It is explained why they present significant progress over the corresponding IAU 1991 resolutions and why they are necessary in the light of present accuracies in astrometry, celestial mechanics, and metrology. In fact, most of these resolutions are consistent with astronomical models and software already in use. The metric tensors and gravitational potentials of both the Barycentric Celestial Reference System and the Geocentric Celestial Reference System are defined and discussed. The necessity and relevance of the two celestial reference systems are explained. The transformations of coordinates and gravitational potentials are discussed. Potential coefficients parameterizing the post-Newtonian gravitational potentials are expounded. Simplified versions of the time transformations suitable for modern clock accuracies are elucidated. Various approximations used in the resolutions are explicated and justified. Some models (e.g., for higher spin moments) that serve the purpose of estimating orders of magnitude have actually never been published before.

Gravitomagnetic effects in the propagation of electromagnetic waves in variable gravitational fields of arbitrary-moving and spinning bodies

Propagation of light in the gravitational field of self-gravitating spinning bodies moving with arbitrary velocities is discussed. The gravitational field is assumed to be weak everywhere. Equations of motion of a light ray are solved in the first post-Minkowskian approximation that is linear with respect to the universal gravitational constant

PROGRESS IN MEASUREMENTS OF THE GRAVITATIONAL BENDING OF RADIO WAVES USING THE VLBA

G

Parametrized post-Newtonian theory of reference frames, multipolar expansions and equations of motion in the N-body problem

. We do not restrict ourselves with the approximation of gravitational lens so that the solution of light geodesics is applicable for arbitrary locations of source of light and observer. This formalism is applied for studying corrections to the Shapiro time delay in binary pulsars caused by the rotation of pulsar and its companion. We also derive the correction to the light deflection angle caused by rotation of gravitating bodies in the solar system (Sun, planets) or a gravitational lens. The gravitational shift of frequency due to the combined translational and rotational motions of light-ray-deflecting bodies is analyzed as well. We give a general derivation of the formula describing the relativistic rotation of the plane of polarization of electromagnetic waves (Skrotskii effect). This formula is valid for arbitrary translational and rotational motion of gravitating bodies and greatly extends the results of previous researchers. Finally, we discuss the Skrotskii effect for gravitational waves emitted by localized sources such as a binary system. The theoretical results of this paper can be applied for studying various relativistic effects in microarcsecond space astrometry and developing corresponding algorithms for data processing in space astrometric missions such as FAME, SIM, and GAIA.

Microarcsecond astrometry in space - Relativistic effects and reduction of observations

We have used the Very Long Baseline Array (VLBA) at 43, 23, and 15 GHz to measure the solar gravitational deflection of radio waves among four radio sources during an 18 day period in 2005 October. Using phase-referenced radio interferometry to fit the measured phase delay to the propagation equation of the parameterized post-Newtonian formalism, we have determined the deflection parameter γ = 0.9998 ± 0.0003 (68% confidence level), in agreement with general relativity. The results come mainly from 43 GHz observations where the refraction effects of the solar corona were negligible beyond 3 deg from the Sun. The purpose of this experiment is three-fold: to improve on the previous results in the gravitational bending experiments near the solar limb; to examine and evaluate the accuracy limits of terrestrial VLBI techniques; and to determine the prospects and outcomes of future experiments. Our conclusion is that a series of improved designed experiments with the VLBA could increase the presented accuracy by at least a factor of 4.

Astrometric Effects of Secular Aberration

Abstract Post-Newtonian relativistic theory of astronomical reference frames based on Einsteins general theory of relativity was adopted by General Assembly of the International Astronomical Union in 2000. This theory is extended in the present paper by taking into account all relativistic effects caused by the presumable existence of a scalar field and parametrized by two parameters, β and γ , of the parametrized post-Newtonian (PPN) formalism. We use a general class of the scalar-tensor (Brans-Dicke type) theories of gravitation to work out PPN concepts of global and local reference frames for an astronomical N-body system. The global reference frame is a standard PPN coordinate system. A local reference frame is constructed in the vicinity of a weakly self-gravitating body (a sub-system of the bodies) that is a member of the astronomical N-body system. Such local inertial frame is required for unambiguous derivation of the equations of motion of the body in the field of other members of the N-body system and for construction of adequate algorithms for data analysis of various gravitational experiments conducted in ground-based laboratories and/or on board of spacecrafts in the solar system. We assume that the bodies comprising the N-body system have weak gravitational field and move slowly. At the same time we do not impose any specific limitations on the distribution of density, velocity and the equation of state of the bodys matter. Scalar–tensor equations of the gravitational field are solved by making use of the post-Newtonian approximations so that the metric tensor and the scalar field are obtained as functions of the global and local coordinates. A correspondence between the local and global coordinate frames is found by making use of asymptotic expansion matching technique. This technique allows us to find a class of the post-Newtonian coordinate transformations between the frames as well as equations of translational motion of the origin of the local frame along with the law of relativistic precession of its spatial axes. These transformations depend on the PPN parameters β and γ , generalize general relativistic transformations of the IAU 2000 resolutions, and should be used in the data processing of the solar system gravitational experiments aimed to detect the presence of the scalar field. These PPN transformations are also applicable in the precise time-keeping metrology, celestial mechanics, astrometry, geodesy and navigation. We consider a multipolar post-Newtonian expansion of the gravitational and scalar fields and construct a set of internal and external gravitational multipoles depending on the parameters β and γ . These PPN multipoles generalize the Thorne-Blanchet-Damour multipoles defined in harmonic coordinates of general theory of relativity. The PPN multipoles of the scalar-tensor theory of gravity are split in three classes— active , conformal , and scalar multipoles. Only two of them are algebraically independent and we chose to work with the conformal and active multipoles. We derive the laws of conservations of the multipole moments and show that they must be formulated in terms of the conformal multipoles. We focus then on the law of conservation of bodys linear momentum which is defined as a time derivative of the conformal dipole moment of the body in the local coordinates. We prove that the local force violating the law of conservation of the bodys linear momentum depends exclusively on the active multipole moments of the body along with a few other terms which depend on the internal structure of the body and are responsible for the violation of the strong principle of equivalence (the Nordtvedt effect). The PPN translational equations of motion of extended bodies in the global coordinate frame and with all gravitational multipoles taken into account are derived from the law of conservation of the bodys linear momentum supplemented by the law of motion of the origin of the local frame derived from the matching procedure. We use these equations to analyze translational motion of shperically symmetric and rigidly rotating bodies having finite size. Spherical symmetry is defined in the local frame of each body through a set of conditions imposed on the shape of the body and the distribution of its internal density, pressure and velocity field. We prove that our formalism brings about the parametrized post-Newtonian EIH equations of motion of the bodies if the finite-size effects are neglected. Analysis of the finite-size effects reveal that they are proportional to the parameter β coupled with the second and higher-order rotational moments of inertia of the bodies. The finite-size effects in the translational equations of motion can be appreciably large at the latest stage of coalescence of binary neutron stars and can be important in calculations of gravitational waveform templates for the gravitational-wave interferometers. The PPN rotational equations of motion for each extended body possessing an arbitrary multipolar structure of its gravitational field, have been derived in bodys local coordinates. Spin of the body is defined phenomenologically in accordance with the post-Newtonian law of conservation of angular momentum of an isolated system. Torque consists of a general relativistic part and the PPN contribution due to the presence of the scalar field. The PPN scalar-field-dependent part is proportional to the difference between active and conformal dipole moments of the body which disappears in general relativity. Finite-size effects in rotational equations of motion can be a matter of interest for calculating gravitational wave radiation from coalescing binaries.

The speed of gravity in general relativity and theoretical interpretation of the Jovian deflection experiment

The framework of General Relativity is applied to the problem of reduction of high-precision astrometric observations of the order of 1 μarcsec. Such precision is expected to be attained in the not so distant future by means of the space optical interferometers orbiting the Earth. Theoretical methods are described enabling one to construct astrometric reference systems involving the barycentric system, geocentric system, and satellite (observer) system. The relativistic transformations between the employed reference systems have been derived. The equations of geometric optics for the nonstationary gravitational field of the Solar system have been deduced

Astrometric and timing effects of gravitational waves from localized sources

One of the main endeavors of fundamental astrometry is to establish a practical realization of a nonrotating, inertial reference frame anchored to celestial objects whose positions are defined in the barycentric coordinates of the solar system matching the current level of astrometric observational accuracy. The development of astrometric facilities operating from space at a microarcsecond level of precision makes the nonuniformity of the Galactic motion of the barycenter an observable and nonnegligible effect that violates the desired inertiality of the barycentric frame of the solar system. Most of the observable effect is caused by the nearly constant (secular) acceleration of the barycenter with respect to the center of the Galaxy. The acceleration results in a pattern of secular aberration that is observable astrometrically as a systematic vector field of the apparent proper motions of distant quasars. We employ the classic approximations of planar epicycle and vertical harmonic oscillation for the Suns Galactic motion to estimate the magnitude of secular acceleration components in the Galactic coordinates and to show that the peculiar accelerations are smaller than the main galactocentric component. We employ the vector spherical harmonic formalism to describe the predicted field of proper motions and evaluate the amplitude of this field at each point on the celestial sphere. We show that the pattern of secular aberration is fully represented by three low-order electric-type vector harmonics; hence, it is easily distinguishable from the residual rotations of the reference frame and other possible effects, such as the hypothetical long-period gravitational waves, which are described by other types of vector or tensor harmonics. Comprehensive numerical simulations of the grid astrometry with Space Interferometry Mission (SIM) PlanetQuest are conducted assuming that 110 optically bright quasars are included as grid objects and observed on the same schedule, but to lower precision due to their faintness, as regular grid stars. The full covariance matrix of the simulated grid solution is used to evaluate the covariances of the three electric harmonic coefficients, representing the secular aberration pattern of proper motions. This is the only reliable method to estimate such sample-based statistics in view of the considerable star-to-star correlations in SIM global astrometry. We conclude that the grid astrometry with SIM PlanetQuest will be sensitive to the main galactocentric component of secular acceleration, arising from the circular motion of the local standard of rest (LSR) around the Galactic center, while the peculiar acceleration of the Sun with respect to the LSR is expected to be too small to be detected with this astrometric space interferometer.

Celestial ephemerides in an expanding universe

According to Einstein, the notions of geodesic, parallel transport (affine connection) and curvature of the spacetime manifold have a pure geometric origin and do not correlate with any electromagnetic concepts. At the same time, curvature is generated by matter which is not affiliated with the spacetime geometric concepts. For this reason, the fundamental constant c entering the geometric and matter sectors of the general theory of relativity has different conceptual meanings. Specifically, the letter c on the left-hand side of the Einstein equations (geometric sector) entering the Christoffel symbols and its time derivatives is the ultimate speed of gravity characterizing the upper limit on the speed of its propagation as well as the maximal rate of change of time derivatives of the metric tensor, that is gravitational field. The letter c on the right-hand side of the Einstein equations (matter sector) is the maximal speed of propagation of any other field rather than gravity. Einsteins general principle of relativity extends his principle of special relativity and equates the numerical value of the ultimate speed of gravity to that of the speed of light in the special theory of relativity but this general principle must be tested experimentally. To this end, we work out the speed of gravity parametrization of the Einstein equations (cg-parametrization) to keep track of the time-dependent effects associated with the geometric sector of general relativity and to separate them from the time-dependent effects of the matter sector. Parametrized post-Newtonian (PPN) approximation of the Einstein equations is derived in order to explain the gravitational physics of the Jovian deflection VLBI experiment conducted on 8 September 2002. The post-Newtonian series expansion in the cg-parametrized general relativity is with respect to a small parameter that is proportional to the ratio of the characteristic velocity of the bodies to the speed of propagation of the gravitational interaction cg. The Einstein equations are solved in terms of the Li?nard?Wiechert tensor potentials which are used for integrating the light-ray propagation equations. An exact analytic expression for the relativistic time delay in the propagation of a radio wave from a quasar to an observer is calculated under the assumption that the light-ray deflecting bodies move with constant velocities. A post-Newtonian expansion of the time delay proves that in general relativity the time delay is affected by the speed of gravity already to the first order in 1/cg beyond the leading (static) Shapiro term. We conclude that recent measurements of the propagation of the quasars radio signal past Jupiter are directly sensitive to the time-dependent effect from the geometric sector of general relativity which is proportional to the speed of propagation of gravity cg but not the speed of light. It provides a first confirmative measurement of the fundamental speed c of the Einstein general principle of relativity for gravitational field. A comparative analysis of our formulation with the alternative interpretations of the experiment given by other authors is provided.

Comment on "gravitomagnetic influence on gyroscopes and on the lunar orbit".

Localized astronomical sources like a double stellar system, rotating neutron star, or a massive black hole at the center of the Milky Way emit periodic gravitational waves. For a long time only a far-zone contribution of gravitational fields of the localized sources (plane-wave-front approximation) were a matter of theoretical analysis. We demonstrate how this analysis can be extended to take into account near-zone and intermediate-zone contributions as well. The formalism is used to calculate gravitational-wave corrections to the Shapiro time delay in binary pulsars and low-frequency (LF) pulsar timing noise produced by an ensemble of double stars in our galaxy.