arXiv subjects

Sergei M. Kopeikin

Publications and source records attributed to Sergei M. Kopeikin.

At least 19 recordsLinked to original sources

arXiv2026

Frequency Differences between Clocks on the Earth and the Moon

Based on general relativity, clock frequency comparisons support various geodetic applications on Earth, such as determining the gravitational potential and realizing a global height system. Future lunar clocks could extend these capabilities to the Moon, connect to terrestrial clock-comparison networks, and, in principle, serve as reference clocks. Meanwhile, especially driven by future lunar navigation plans, an independent lunar time system is becoming an urgent requirement and such a system would necessarily remain linked to terrestrial time standards. To support these future applications, we establish a comprehensive model for Earth-Moon clock frequency comparisons. This paper simulates fractional frequency differences between clocks on the Earth (E) and Moon (L) through four time transformations:proper-to-coordinate time for E-clocks and for L-clocks (both linked to the local gravity potential), the conversion between the Earth and Moon coordinate times, and the time signal propagation between E- and L-clocks. Gravity potential differences between the E- and L-clocks impact the frequency difference at the 10^-10 level. The effect of the coordinate time ratio is at 10^-11 level. Contributions from static, tidal, and non-tidal potentials, body self-rotation, and different celestial bodies are evaluated. Furthermore, we quantify the Doppler, atmospheric and Shapiro delay effects in Earth-Moon time signal propagation. For a single link, the first-order Doppler term dominates at the 10^-6 level and masks the gravity-potential and coordinate-time terms. A Doppler-cancelling multi-link strategy is needed to suppress the signal-propagation effect and extract these terms.

physics.geo-ph↗

arXiv2025

Lie Group Theory of Multipole Moments and Shape of Stationary Rotating Fluid Bodies

We present a rigorous framework for determining equilibrium configurations of uniformly rotating self-gravitating fluid bodies. This work addresses the longstanding challenge of modeling rotational deformation in celestial objects such as stars and planets. By integrating classical Newtonian potential theory with modern mathematical tools, we develop a unified formalism that improves both the precision and generality of shape modeling in astrophysical contexts. Our method employs Lie group theory and exponential mapping to characterize vector flows associated with rotational deformations. We derive functional equations for perturbations in density and gravitational potential, resolved analytically using the shift operator and Neumann series. This extends Clairaut's classical linear theory into the nonlinear regime. The resulting formulation yields an exact nonlinear differential equation for the shape function, describing hydrostatic equilibrium under rotation without assuming slow rotation. This generalized Clairaut equation incorporates nonlinear effects and accommodates large rotational speeds. We validate the theory by deriving exact solutions, including the Maclaurin spheroid, Jacobi ellipsoid, and the unit-index polytrope. We also introduce spectral decomposition techniques to analyze radial harmonics of the shape function and gravitational perturbations. Using Wigner's formalism for angular momentum addition, we compute higher-order spectral corrections and derive boundary conditions for radial harmonics. This enables accurate computation of Love numbers and gravitational multipole moments, offering a comprehensive, non-perturbative approach to modeling rotational deformations in astrophysical systems.

physics.class-ph↗

arXiv2025

Local coordinates and motion of a test particle in the McVittie spacetime

We consider the orbital motion of a test particle in the gravitational field of a massive body (that might be a black hole) with mass $m$ placed on the expanding cosmological manifold described by the McVittie metric. We introduce the local coordinates attached to the massive body to eliminate nonphysical, coordinates-dependent effects associated with Hubble expansion. The resultant equation of motion of the test particle are analyzed by the method of osculating elements with application of time-averaging technique. We demonstrate that the orbit of the test particle is not subject to the cosmological expansion up to the terms of the second order in the Hubble parameter. However, the cosmological expansion causes the precession of the orbit of the test particle with time and changes the frequency of the mean orbital motion. We show that the direction of motion of the orbital precession depends on the Hubble parameter as well as the deceleration parameter of the universe. We give numeric estimates for the rate of the orbital precession with respect to time due to the cosmological expansion in case of several astrophysical systems.

gr-qc↗

arXiv2025

Revolutionizing Gravitational Potential Analysis: From Clairaut to Lie Groups

This letter introduces an advanced novel theory for calculating non-linear Newtonian hydrostatic perturbations in the density, shape, and gravitational field of fluid stars and planets subjected to external tidal and rotational forces. The theory employs a Lie group approach using exponential mappings to derive exact differential equations for large gravitational field perturbations and the shape function, which describes the finite deformation of the body's figure. This approach lays the foundation for the precise analytic determination and numerical computation of the induced body's multipole moments and Love numbers with any desired degree of accuracy.

gr-qc↗

arXiv2024

Lunar Time in General Relativity

We introduce the general-relativistic definition of Lunar Coordinate Time (TCL) based on the IAU 2000 resolutions that provide a framework for relativistic reference systems. From this foundation, we derive a transformation equation that describes the relative rate of TCL with respect to Geocentric Coordinate Time (TCG) for various locations of the clock on lunar surface. This equation serves as the cornerstone for constructing a relativistic TCL--TCG time conversion algorithm. Using this algorithm, we can compute both secular and periodic variations in the rate of an atomic clock placed on the Moon, relative to an identical clock on Earth. The algorithm accounts for various effects, including time dilation caused by the Moon's orbital motion around Earth, gravitational potentials of both Earth and Moon, and direct and indirect time dilation effects due to tidal perturbations caused by the Sun and other major planets of the solar system. Our approach provides exquisite details of the TCL--TCG transformation, achieving a precision of several nanoseconds within the spatial volume dominated by the Earth's gravitational field known as the Hill sphere. This sphere extends from the Earth to a distance of approximately 1.5 million km, substantially encompassing the Moon's orbit. To validate our methodology for lunar coordinate time, we compare it with the mathematical formalism of local inertial frames applied to the Earth-Moon system and confirm their equivalence.

gr-qc↗

arXiv2022

The Orbital Pericenter Precession in the 2PN Approximation

Recent article "Revisiting the 2PN Pericenter Precession in View of Possible Future Measurements" published by Iorio (Universe, 2020) argues that calculations of the secular 2PN precession of the orbital pericenter of a binary system accomplished by Damour and Schaefer (Nuovo Cim. B, 1988) and by Kopeikin and Potapov (Astron. Rep., 1994) with different mathematical techniques are inconsistent, differ from each other and do not agree with the result obtained by Iorio (Universe, 2020). The purpose of this communication is to demonstrate that the article by Iorio is erroneous, suffers from misconceptions and cannot be trusted.

gr-qc↗

arXiv2021

The Science of Fundamental Catalogs

This review paper discusses the science of astrometric catalogs, their current applications and future prospects for making progress in fundamental astronomy, astrophysics and gravitational physics. We discuss the concept of fundamental catalogs, their practical realizations, and future prospects. Particular attention is paid to the astrophysical implementations of the catalogs such as the measurement of the Oort constants, the secular aberration and parallax, and asteroseismology. We also consider the use of the fundamental catalogs in gravitational physics for testing general theory of relativity and detection of ultra-long gravitational waves of cosmological origin.

astro-ph.IM↗

arXiv2020

Post-Newtonian Lagrangian of an N-body System with Arbitrary Mass and Spin Multipoles

The present paper derives the post-Newtonian Lagrangian of translational motion of N arbitrary-structured bodies with all mass and spin multipoles in a scalar-tensor theory of gravity. The multipoles depend on time and evolve in accordance with their own dynamic equations of motion. The Lagrangian is retrieved from the post-Newtonian equations of motion by solving the inverse problem of the Lagrangian mechanics and generalizes a well-known Lagrangian of pole-dipole-quadrupole massive particles to the particles of higher multipolarity. Analytic treatment of the higher-order multipole contributions is important for more rigorous computation of gravitational waveform of inspiralling compact binaries at the latest stage of their orbital evolution before merger when tidal and rotational deformations of stars are no longer small and rapidly change in time. The Lagrangian of an N-body system with arbitrary mass and spin multipoles is instrumental for formulation of the post-Newtonian conservation laws of energy, momenta and the integrals of the center of mass.

gr-qc↗

arXiv2019

PEXO: a global modeling framework for nanosecond timing, microsecond astrometry, and $μ$m/s radial velocities

The ability to make independent detections of the signatures of exoplanets with complementary telescopes and instruments brings a new potential for robust identification of exoplanets and precision characterization. We introduce PEXO, a package for Precise EXOplanetology to facilitate the efficient modeling of timing, astrometry, and radial velocity data, which will benefit not only exoplanet science but also various astrophysical studies in general. PEXO is general enough to account for binary motion and stellar reflex motions induced by planetary companions and is precise enough to treat various relativistic effects both in the solar system and in the target system. We also model the post-Newtonian barycentric motion for future tests of general relativity in extrasolar systems. We benchmark PEXO with the pulsar timing package TEMPO2 and find that PEXO produces numerically similar results with timing precision of about 1 ns, space-based astrometry to a precision of 1 $μ$as, and radial velocity of 1 $μ$m/s and improves on TEMPO2 for decade-long timing data of nearby targets, due to its consideration of third-order terms of Roemer delay. PEXO is able to avoid the bias introduced by decoupling the target system and the solar system and to account for the atmospheric effects which set a practical limit for ground-based radial velocities close to 1 cm/s. Considering the various caveats in barycentric correction and ancillary data required to realize cm/s modeling, we recommend the preservation of original observational data. The PEXO modeling package is available at GitHub (https://github.com/phillippro/pexo).

astro-ph.EP↗

arXiv2019

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

This paper employs the post-Newtonian approximations of scalar-tensor theory of gravity along with the Cartesian STF tensors and the Blanchet-Damour multipole formalism to derive translational and rotational equations of motion of N extended bodies with arbitrary distribution of mass and velocity. We assume that spacetime can be covered by a global coordinate chart which is Minkowskian at infinity. We also introduce N local coordinate charts adapted to each body and covering a finite domain of space around the body. Gravitational field of each body is parametrized by an infinite set of the body's mass and spin multipoles and tidal multipoles of external N-1 bodies. The origin of the local coordinates is set moving along accelerated worldline of the center of mass of the body by an appropriate choice of the internal and external dipole moments of its gravitational field. Translational and rotational equations of motion are derived by integrating microscopic equations of matter and applying the method of asymptotic matching. The matching is also used for separating the post-Newtonian self-field effects from the external gravitational environment and for 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. 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 taken into account. The equations of translational and rotational motion derived in the present paper include the infinite set of mass and spin multipoles of the bodies and can be used for much more accurate prediction of orbital dynamics of extended bodies in inspiraling binary systems before they merge.

gr-qc↗

arXiv2019

Covariant Equations of Motion Beyond the Spin-Dipole Particle Approximation

The present paper studies the post-Newtonian dynamics of N bodies 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 of N-body system along with the mathematical apparatus of the Cartesian STF 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 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 manifold 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.

gr-qc↗

arXiv2018

Carter-like Constant of Motion in Newtonian Gravity is the Vinti Integral

We compare the Vinti integral of the classic celestial mechanics with a conserved Carter-like integral of motion for an axially-symmetric body in the Newtonian theory that has been recently found by Clifford Will. We demonstrate that the integrals are identical. It sheds new light on the Newtonian limit of the Kerr geometry.

gr-qc↗

arXiv2018

Normal gravity field in relativistic geodesy

Modern geodesy is subject to a dramatic change from the Newtonian paradigm to Einstein's theory of general relativity. This is motivated by the ongoing advance in development of quantum sensors for applications in geodesy including quantum gravimeters and gradientometers, atomic clocks and fiber optics for making ultra-precise measurements of the geoid and multipolar structure of the Earth's gravitational field. At the same time, VLBI, SLR, and GNSS have achieved an unprecedented level of accuracy in measuring coordinates of the reference points of the ITRF and the world height system. The main geodetic reference standard is a normal gravity field represented in the Newtonian gravity by the field of a Maclaurin ellipsoid. The present paper extends the concept of the normal gravity field to the realm of general relativity. We focus our attention on the calculation of the first post-Newtonian approximation of the normal field that is sufficient for applications. We show that in general relativity the level surface of the uniformly rotating fluid is no longer described by the Maclaurin ellipsoid but is an axisymmetric spheroid of the forth order. We parametrize the mass density distribution and derive the post-Newtonian normal gravity field of the rotating spheroid which is given in a closed form by a finite number of the ellipsoidal harmonics. We employ transformation from the ellipsoidal to spherical coordinates to deduce the post-Newtonian multipolar expansion of the metric tensor given in terms of scalar and vector gravitational potentials of the rotating spheroid. We compare these expansions with that of the normal gravity field generated by the Kerr metric and demonstrate that the Kerr metric has a fairly limited application in relativistic geodesy. Finally, we derive the post-Newtonian generalization of the Somigliana formula for the gravity field on the reference ellipsoid.

gr-qc↗

arXiv2017

Post-Newtonian celestial mechanics in scalar-tensor cosmology

Applying the recently developed dynamical perturbation formalism on cosmological background to scalar-tensor theory, we provide a solid theoretical basis and a rigorous justification for phenomenological models of orbital dynamics that are currently used to interpret experimental measurements of the time-dependent gravitational constant. We derive the field equations for the scalar-tensor perturbations and study their gauge freedom associated with the cosmological expansion. We find a new gauge eliminating a prohibitive number of gauge modes in the field equations and significantly simplifying post-Newtonian equations of motion for localized astronomical systems in the universe with time-dependent gravitational constant. We identify several new post-Newtonian terms and calculate their effect on secular cosmological evolution of the osculating orbital elements.

gr-qc↗

arXiv2014

Dynamic Field Theory and Equations of Motion in Cosmology

We discuss a field-theoretical approach based on variational principle to derive the field and hydrodynamic equations of motion of baryonic matter governed by cosmological perturbations of dark matter and dark energy. The action depends on the gravitational and matter Lagrangian. The gravitational Lagrangian depends on the metric tensor and its first and second derivatives. The matter Lagrangian includes dark matter, dark energy and the ordinary baryonic matter. The total Lagrangian is expanded in an asymptotic Taylor series around the background manifold defined as a solution of Einstein's equations in the form of the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric tensor. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the series expansion is gauge-invariant and all of them together form a basis for the successive post-Friedmannian approximations. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does not require the post-Friedmannian perturbations to be small though computationally it works the most effectively when the perturbed metric is close enough to the background FLRW metric. The temporal evolution of the background metric is governed by dark matter and dark energy and we associate the large scale inhomogeneities in these two components as those generated by the primordial cosmological perturbations. The small scale inhomogeneities are generated by the condensations of baryonic matter considered as the bare perturbations. We explicitly work out the covariant field equations of the successive post-Friedmannian approximations of Einstein's equations and derive equations of motion of large and small scale inhomogeneities of dark matter and dark energy. We apply these equations to derive the post-Friedmannian equations of motion of the baryonic matter.

gr-qc↗

arXiv2014

Einstein's equivalence principle in cosmology

We study physical consequences of the Einstein equivalence principle (EEP) for a Hubble observer in FLRW universe. We introduce the local inertial coordinates with the help of a special conformal transformation. The local inertial metric is Minkowski-flat and materialized by a congruence of time-like geodesics of static observers. The static observers are equipped with the ideal clocks measuring the proper time that is synchronized with the clocks of the Hubble observer. The local inertial metric is used for physical measurements of spacetime intervals with the ideal clocks and rulers. The special conformal transformation preserves null geodesics but does not keep invariant time-like geodesics. Moreover, it makes the rate of the local time coordinate dependent on velocity of the particle which makes impossible to rich the uniform parameterization of the world lines of static observers and light geodesics with a single parameter - they differ by the conformal factor of FLRW metric. It tells us that the metric on the light cone is not Minkowski-flat but depends on the scale factor of FLRW universe and it can be interpreted as a weak violation of EEP for photons. The importance of this violation for gravitational physics is that some of local experiments conducted with freely-propagating electromagnetic waves may be sensitive to the Hubble expansion. We show that the Hubble constant H can be measured within the solar system by means of high-precision spacecraft Doppler tracking as a blue shift of frequency of radio waves circulating in the Earth-spacecraft radio link. We also analyze the behavior of the standing wave in a microwave resonator and show that the standing wave is insensitive to the Hubble expansion.

gr-qc↗

arXiv2009

Beyond the Standard IAU Framework

We discuss three conceivable scenarios of extension and/or modification of the IAU relativistic resolutions on time scales and spatial coordinates beyond the Standard IAU Framework. These scenarios include: (1) the formalism of the monopole and dipole moment transformations of the metric tensor replacing the scale transformations of time and space coordinates; (2) implementing the parameterized post-Newtonian formalism with two PPN parameters - beta and gamma; (3) embedding the post-Newtonian barycentric reference system to the Friedman-Robertson-Walker cosmological model.

gr-qc↗

arXiv2007

Gravimagnetism, Causality, and Aberration of Gravity in the Gravitational Light-Ray Deflection Experiments

Experimental verification of the existence of gravimagnetic fields generated by currents of matter is important for a complete understanding and formulation of gravitational physics. Although the rotational `intrinsic' gravimagnetic field has been extensively studied and is now being measured by the Gravity Probe B, the `extrinsic' gravimagnetic field generated by the translational current of matter is less well studied. The present paper uses the post-Newtonian parametrized Einstein and light geodesics equations to show that the `extrinsic' gravimagnetic field generated by the translational current of matter can be measured by observing the relativistic time delay and/or light deflection caused by the moving mass. We prove that the `extrinsic' gravimagnetic field is generated by the relativistic effect of the aberration of the gravity force caused by the Lorentz transformation of the metric tensor and the Levi-Civita connection. We show that the Lorentz transformation of the gravity field variables is equivalent to the technique of the retarded Lienard-Wiechert gravitational potentials predicting that a light particle is deflected by gravitational field of a moving body from its retarded position so that both general-relativistic phenomena -- the aberration and the retardation of gravity -- are tightly connected and observing the aberration of gravity proves that gravity has a causal nature. We explain in this framework the 2002 deflection experiment of a quasar by Jupiter where the aberration of gravity from its orbital motion was measured with accuracy 20%. We describe a theory of VLBI experiment to measure the gravitational deflection of radio waves from a quasar by the Sun, as viewed by a moving observer from the geocentric frame, to improve the measurement accuracy of the aberration of gravity to a few percent.

gr-qc↗