Sergei M. Kopeikin

University of Missouri | Mizzou · Department of Physics and Astronomy

This authoritative book presents the theoretical development of gravitational physics as it applies to the dynamics of celestial bodies and the analysis of precise astronomical observations. In so doing, it fills the need for a textbook that teaches modern dynamical astronomy with a strong emphasis on the relativistic aspects of the subject produce...

Post-Newtonian theory was instrumental in conducting the critical experimental tests of general relativity and in building the astronomical ephemerides of celestial bodies in the solar system with an unparalleled precision. The cornerstone of the theory is the postulate that the solar system is gravitationally isolated from the rest of the universe...

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...

This topical white paper wishes to recommend to the BPS decadal survey the state-of-the-art objectives of fundamental physics and lunar science, that are enabled by next generation single, large diameter CCRs deployed by means of NASA-CLPS missions, through the international Artemis Accords, the EL3 lunar program and the LGN during the decade 2023-...

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 perspectives. Particular attention is paid to the astrophysica...

The upcoming decade offers multiple opportunities at extending science from the Lunar Laser Ranging (LLR) experiment that would result in unprecedented data precision, which may enable detection of subtle signatures required to understand the dynamics of the Earth-Moon system and the deep lunar interior.

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 i...

The Lunar Laser Ranging (LLR) experiment has accumulated 50 years of range data of improving accuracy from ground stations to the laser retroreflector arrays (LRAs) on the lunar surface. The upcoming decade offers several opportunities to break new ground in data precision through the deployment of the next generation of single corner-cube lunar re...

The present paper derives the post-Newtonian Lagrangian of translational motion of N arbitrarily 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...

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 m...

Recent article Revisiting the 2PN Pericenter Precession in View of Possible Future Measurements published by Iorio (Universe 6(4):53, 2020. https://doi.org/10.3390/universe6040053) argues that calculations of the secular 2PN precession of the orbital pericenter of a binary system accomplished by Damour and Schäfer (Nuovo Cim B Ser 101B(2):127–176,...

Recent article "Revisiting the 2PN Pericenter Precession in View of Possible Future Measurements" published by L. 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 Cimento B, 1988) and by Kopeikin and Potapov (Astronomy Reports, 1994)...

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 velocit...

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 velocit...

Gravitational wave detectors allow us to test general relativity and to study the internal structure and orbital dynamics of neutron stars and black holes in inspiralling binary systems with a potentially unlimited rigor. Currently, analytic calculations of a gravitational wave signal emitted by inspiralling compact binaries are based on the numeri...

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 syst...

Modern geodesy is undergoing a crucial transformation from the Newtonian paradigm to the Einstein theory of general relativity. This is motivated by advances in developing quantum geodetic sensors including gravimeters and gradientometers, atomic clocks and fiber optics for making ultra-precise measurements of geoid and multipolar structure of Eart...

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 of N-body system...

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...

The central question our group is trying to answer: Can we measure the Hubble constant within the solar system? A potential signature of something interesting (cosmologically) appears to be present in Cassini Gravitational Wave Experiment data. This is ongoing work, and we’re being conservative actively searching for a mundane explanation.

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.

We report on the experimental results of testing a new physical method of determination of the gravitational potential differences and orthometric heights by measuring the relativistic effect of gravitational redshift of frequency by means of atomic clocks. The experiment was performed in the Altai Mountains between two geodetic stations, Shebalino...

Time measured by an ideal clock crucially depends on the gravitational potential and velocity of the clock according to general relativity. Technological advances in manufacturing high-precision atomic clocks have rapidly improved their accuracy and stability over the last decade that approached the level of 10−18\documentclass[12pt]{minimal} \usep...

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 g...

Lecture delivered at the Great American Solar Eclipse conference on theoretical physics held in Columbia, Missouri

The present article is written in response to the recent call of the United Nations for the enhanced international cooperation of different countries on global geodesy to build the Global Geodetic Reference Frame (GGRF). It reviews historical landmarks in the development of the State Geodetic Reference Frame on the territory of Russia over the last...

In the context of the ESA M5 (medium mission) call we proposed a new satellite mission, Theia, based on relative astrometry and extreme precision to study the motion of very faint objects in the Universe. Theia is primarily designed to study the local dark matter properties, the existence of Earth-like exoplanets in our nearest star systems and the...

Powerpoint presentation on relativistic geodesy

Our present relativistic part of the geodetic VLBI model for Earthbound antennas is a consensus model which is considered as a standard for processing high-precision VLBI observations. It was created as a compromise between a variety of relativistic VLBI models proposed by different authors as documented in the IERS Conventions 2010. The accuracy o...

Invited Lecture for

Pulsars can be considered as very precise clocks if they are observed from the barycenter of the Solar system. Pulsar Timing Array (PTA) can be used to establish new astronomical reference frame which describes both space and time properties. Among these arrays most remarkable are the Parks Pulsar Timing Array (PPTA) and Kalyazin Pulsar Timing Arra...

We study the post-Newtonian perturbations in the orbit of a near-Earth satellite by integrating them with a high-fidelity orbit propagation software KASIOP. The perturbations of the orbital elements are evaluated for various cases from a low-Earth orbit to a geostationary one, and from an equatorial to a polar orbit. In particular, the numerical si...

We elaborate on a new proposal to determine G in deep space by measuring the period of oscillation of a retroreflector test mass moving harmonically within a cylindrical hole bored through the diameter of a solid sphere (the “gravity train” mechanism). As is well known, the oscillator's period is directly dependent upon G (mass and dimensions of th...

We report on the experimental results of testing a new physical method of determination of the gravitational potential differences and orthometric heights by measuring the relativistic effect of gravitational redshift of frequency by means of atomic clocks. The experiment was performed in the Altai Mountains between two geodetic stations, Shebalino...

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...

Responding to calls from the National Science Foundation (NSF) for new proposals to measure the gravitational constant G, we offer an interesting experiment in deep space employing the classic gravity train mechanism. Our setup requires three bodies: a larger layered solid sphere with a cylindrical hole through its center, a much smaller retrorefle...

The presentation describes the role and scientific methodology applied in modern geodesy for solving the problem of the construction of the world-wide coordinate system of heights by applying general relativistic algorithms of comparison of frequency of quantum (atomic) clocks.

We report on the experimental results of approbation of a new physical method of determination of the potential difference of the force of gravity and the normal heights with the help of measurement of the gravitational red shift of frequency by means of atomic clocks performed on the ter-ritory of Altai Mountains between two geodetic stations «She...

Chronometric geodesy applies general relativity to study the problem of the shape of celestial bodies including the earth, and their gravitational field. The present paper discusses the relativistic problem of construction of a background geometric manifold that is used for describing a reference ellipsoid, geoid, the normal gravity field of the ea...

We apply general relativity to construct the post-Newtonian background manifold that serves as a reference level surface in relativistic geodesy for conducting calculation of geoid's undulation. We chose the perfect homogeneous fluid uniformly rotating around a fixed axis as a source of the background manifold. We, then, reformulate and extend rota...

The effect of irregularities in the earth’s rotation (precession and nutation of the earth’s axis of rotation, oscillations in the modulus of the angular velocity, periodic deviations in the line of the poles, and the angular momentum of the globe) on the frequency and time of high-stability atomic clocks are examined in terms of the theory of rela...

We make use of an effective field-theoretical approach to derive post-Newtonian equations of motion of hydrodynamical inhomogeneities in cosmology. The matter Lagrangian for the perturbed cosmological model includes dark matter, dark energy, and ordinary baryonic matter. The Lagrangian is expanded in an asymptotic Taylor series around a Friedmann-L...

The present paper extends the Newtonian concept of the geoid in classic geodesy towards the realm of general relativity by utilizing the covariant geometric methods of the perturbation theory of curved manifolds. It yields a covariant definition of the anomalous (disturbing) gravity potential and formulate differential equation for it in the form o...

The Fresnel-Fizeau effect is a special relativistic effect that makes the speed of light dependent on the velocity of a transparent, moving medium. We present a theoretical formalism for discussing propagation of electromagnetic signals through the moving Earth atmosphere with taking into account the Fresnel-Fizeau effect. It provides the rigorous...

We study evolution of frequency of a standing electromagnetic (EM) wave in a resonant optical cavity placed to the expanding manifold described by the Robertson-Walker metric. One builds a local coordinate system in which spacetime is locally Minkowskian. However, due to the conformal nature of the Robertson-Walker metric the conventional transform...

Precise determination of geoid is one of the most important problem of physical geodesy. The present paper extends the Newtonian concept of the geoid to the realm of Einstein's general relativity and derives an exact relativistic equation for the unperturbed geoid and level surfaces under assumption of axisymmetric distribution of background matter...

We study physical consequences of the Hubble expansion of FLRW manifold on measurement of space, time and light propagation in the local inertial frame. We analyse the solar system radar ranging and Doppler tracking experiments, and time synchronization. FLRW manifold is covered by global coordinates (t,y^i), where t is the cosmic time coinciding w...

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 fir...

Relativistic celestial mechanics - investigating the motion celestial bodies under the influence of general relativity - is a major tool of modern experimental gravitational physics. With a wide range of prominent authors from the field, this two-volume series consists of reviews on a multitude of advanced topics in the area of relativistic celesti...

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 observ...

The present paper outlines theoretical principles of the post-Newtonian mechanics in the expanding universe. It is based upon the gauge-invariant theory of the Lagrangian perturbations of cosmological manifold caused by an isolated astronomical N-body system. We postulate that the background manifold is described by Friedman-Lemaitre-Robertson-Walk...

The mathematical concept of the Newtonian limit of Einstein's field equations in the expanding Friedmann universe is formulated. The geodesic equations of motion of planets and light are derived and compared.

This presentation discusses the mathematical principles of constructing coordinates on curved spacetime manifold in order to build a hierarchy of astrometric frames in the solar system and beyond which can be used in future practical applications as well as for testing the fundamentals of the gravitational physics - general theory of relativity.

Text of IAU Resolutions of 1997 Adopted at the XXIIIrd General Assembly, KyotoText of IAU Resolutions of 2000 Adopted at the XXIVth General Assembly, ManchesterText of IAU Resolutions of 2006 Adopted at the XXVIth General Assembly, PragueText of IAU Resolutions of 2009 Adopted at the XXVIIth General Assembly, Rio de Janeiro

Post-Newtonian Equations of Orbital MotionRotational Equations of Motion of Extended BodiesMotion of Spherically-Symmetric and Rigidly-Rotating BodiesPost-Newtonian Two-Body ProblemReferences

Prolegomena – Classical Mechanics in a NutshellThe N-body ProblemThe Reduced Two-Body ProblemA Perturbed Two-Body ProblemRe-examining the ObviousEpilogue to the ChapterReferences

This appendix contains sections titled:

This appendix contains sections titled:

Concepts of relativistic astrometry — such as Weyl's stellar compass or the concept of ‘flat-space plus forces’ — are discussed. To visualize effects from light deflection pictures showing the stellar sky as seen from the vicinity of a strongly gravitating source are presented. Full text is avaliable online http://www.scholarpedia.org/article/Relat...

IAU 2000 resolutions on the reference frames set up a solid theoretical foundation for implementing general relativity in astronomical data processing algorithms and for unambiguous interpretation of measured relativistic effects. We discuss possible directions for further theoretical development of the IAU resolutions aimed to take into account th...

We introduce the Jacobi coordinates adopted to the advanced theoretical analysis of the relativistic Celestial Mechanics of the Earth-Moon system. Theoretical derivation utilizes the relativistic resolutions on reference frames adopted by the International Astronomical Union (IAU) in 2000. The resolutions assume that the Solar System is isolated an...

Millisecond and binary pulsars are the most stable natural standards of astronomical time giving us a unique opportunity to search for gravitational waves (GW) and to test General Relativity. GWs from violent events in early Universe and from the ensemble of galactic and extragalactic objects perturb propagation of radio pulses from a pulsar to obs...

We introduce the Jacobi coordinates adopted to the advanced theoretical analysis of the relativistic celestial mechanics of the Earth-Moon system. Theoretical derivation utilizes the relativistic resolutions on the local reference frames adopted by the International Astronomical Union in 2000. The advantage of the local frames is in a more simple...

Since VLBI techniques give microarcsecond position accuracy of celestial objects, tests of GR using radio sources as probes of a gravitational field have been made. We present the results from two recent tests using the VLBA: In 2005, the measurement of the classical solar deflection; and in 2002, the measurement of the retarded gravitational defle...

Since VLBI techniques give microarcsecond position accuracy of celestial objects, tests of GR using radio sources as probes of a gravitational field have been made. We present the results from two recent tests using the VLBA: In 2005, the measurement of the classical solar deflection; and in 2002, the measurement of the retarded gravitational defle...

We report on recent VLBA/VERA/IVS observational tests of General Relativity. First, we will summarize the results from the 2005 VLBA experiment that determined gamma with an accuracy of 0.0003 by measuring the deflection of four compact radio sources by the solar gravitational field. We discuss the limits of precision that can be obtained with VLBA...

We derive explicit Lorentz-invariant solution of the Einstein and null geodesic equations for data processing of the time delay and ranging experiments in gravitational field of moving gravitating bodies of the solar system - the Sun and major planets. We discuss general-relativistic interpretation of these experiments and the limitations imposed b...

Millisecond and binary pulsars are the most stable astronomical standards of frequency. They can be applied to solving a number of problems in astronomy and time-keeping metrology including the search for a stochastic gravitational wave background in the early universe, testing general relativity, and establishing a new time-scale. The full explora...

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...

This note suggests nomenclature for dealing with the units of various astronomical quantities that are used with the relativistic time scales TT, TDB, TCB and TCG. It is suggested to avoid wordings like "TDB units" and "TT units" and avoid contrasting them to "SI units". The quantities intended for use with TCG, TCB, TT or TDB should be called "TCG...

We construct a set of reference frames for description of the orbital and rotational motion of the Moon. We use a scalar-tensor theory of gravity depending on two parameters of the parametrized post-Newtonian (PPN) formalism and utilize the concepts of the relativistic resolutions on reference frames adopted by the International Astronomical Union...

We report on VLBA/VERA/geodetic observational tests of General Relativity. First, we will summarize the results from recent VLBA experiments which measured gamma from the bending of radio waves of quasars by the solar gravitational field, and the experiment that measured the the aberration of gravity using the Jovian gravitational field. We will th...

We construct a set of reference frames for description of the orbital and rotational motion of the Moon. We use a scalar-tensor theory of gravity depending on two parameters of the parametrized post-Newtonian (PPN) formalism and utilize the concepts of the relativistic resolutions on reference frames adopted by the International Astronomical Union...

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 October 2005. Using phase-referenced radio interferometry to fit the measured phase delay to the propagation equation of the parameterized post-Newtonian (PPN) formali...

The primary scientific goal of this proposal is to test for technical capabilities to measure the time of quasar disappearance/reappearance during occultation by Moon with a precision of about 100 microsec of time. These occultations of different quasars can be observed by different VLBI sites. An accurate determination of the moment of ingress/egr...

Lunar Laser Ranging (LLR) measurements are crucial for advanced exploration of the evolutionary history of the lunar orbit, the laws of fundamental gravitational physics, selenophysics and geophysics as well as for future human missions to the Moon. Current LLR technique measures distance to the Moon with a precision approaching one millimeter that...

Given the impressive investment by the nation in observational Astronomy and Astrophysics facilities coming on line now and in the near future, we advocate for an increased investment in applied and fundamental research on Astrophysical and Celestial Dynamics (ACD). Specifically we call for a) continued and expanded support for applied research in...

ASTROD I isx a planned interplanetary space mission with multiple goals. The primary aims sre: to test general relativity with an improvement in sensitivity of over three orders of magnitude, improving our understanding of gravity and aiding the development of a new quantum gravity theory; to measure key solar system parameters with increased accur...

We construct a set of post-Newtonian reference frames for a comprehensive study of the orbital dynamics and rotational motion of the Moon and Earth by means of lunar laser ranging (LLR) with the precision of one millimeter. We also derive the post-Newtonian coordinate transformations between the frames and analyze the residual gauge freedom, which...

The proper motion of a binary pulsar gradually changes the apparent geometrical orientation of its orbital plane. Consequently, the derived values of the projected semimajor axis of the orbit and the longitude of orbital periastron have systematic errors that limit the precision of fundamental tests of relativistic gravity in the strong gravity fie...

We compare two theoretical approaches to the data analysis of the Cassini relativity experiment based on the Doppler tracking and the time delay technique that were published correspondingly by Kopeikin et al. [S.M. Kopeikin, A.G. Polnarev, G. Schäfer, I.Yu. Vlasov, Phys. Lett. A 367 (2007) 276] and by Bertotti et al. [B. Bertotti, N. Ashby, L. Ies...

Bertotti et al (2008 Class.Quant.Grav. 25 045013) studied the effect of the motion of the Sun on the gravitational time delay of light in interplanetary relativity experiments and, somehow, found a discrepancy with our paper (Kopeikin et al 2007 Phys. Lett. A 367 276). For this reason, Bertotti et al spontaneously decided that our paper gives wrong...

Gravitomagnetic field is covariantly split in the {\it intrinsic} and {\it extrinsic} parts, which are generated by rotational and translational currents of matter respectively. The {\it intrinsic} component has been recently discovered in the LAGEOS spacecraft experiment. We discuss the method of detection of the {\it extrinsic} tidal component wi...

ABSTRACT Lunar Laser Ranging (LLR) measurements are crucial for advanced exploration of the laws of fundamental gravitational physics and geophysics as well as for future human and robotic missions to the Moon. The corner-cube reflectors (CCR) currently on the Moon require no power and still work perfectly since their installation during the projec...

Modern theories of the fundamental particles and the interactions among them have achieved a simple and coherent description of an unprecedented range of natural phenomena, but our new understanding raises intriguing new questions. Though there is a growing confidence in our ability to unify electroweak and strong interactions in a single gauge the...

We introduce a linearized bimetric theory of gravity with two metrics. The metric gαβ describes null hypersurfaces of the gravitational field while light moves on null hypersurfaces of the optical metric gαβ. Bimetrism naturally arises in vector–tensor theories with matter being nonminimally coupled to gravity via long-range vector field. We derive...

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 accurac...

Lunar Laser Ranging (LLR) measurements are crucial for advanced exploration of the laws of fundamental gravitational physics and geophysics. Current LLR technology allows us to measure distances to the Moon with a precision approaching 1 millimeter. As NASA pursues the vision of taking humans back to the Moon, new, more precise laser ranging applic...

The propagation of electromagnetic signals of pulsars through the non-stationary gravitational field of the stellar globular clusters formed by an ensemble of arbitrarily distributed stars are discussed. The expression for the relativistic time delay of pulsars radiation in such fields are derived taking into account the negligible aberration corre...