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In general relativity, relativistic gravity gradiometry involves the measurement of the relativistic tidal matrix, which is theoretically obtained from the projection of the Riemann curvature tensor onto the orthonormal tetrad frame of an observer. The observer’s 4-velocity vector defines its local temporal axis and its local spatial frame is defined by a set of three orthonormal nonrotating gyro directions. The general tidal matrix for the timelike geodesics of Kerr spacetime has been calculated by Marck [Proc. R. Soc. A 385, 431 (1983)]. We are interested in the measured components of the curvature tensor along the inclined “circular” geodesic orbit of a test mass about a slowly rotating astronomical object of mass M and angular momentum J. Therefore, we specialize Marck’s results to such a “circular” orbit that is tilted with respect to the equatorial plane of the Kerr source. To linear order in J, we recover the gravitomagnetic beating phenomenon [B. Mashhoon and D. S. Theiss, Phys. Rev. Lett. 49, 1542 (1982)], where the beat frequency is the frequency of geodetic precession. The beat effect shows up as a special long-period gravitomagnetic part of the relativistic tidal matrix; moreover, the effect’s short-term manifestations are contained in certain post-Newtonian secular terms. The physical interpretation of this effect is briefly discussed.

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... For recent work on general relativistic tidal effects in other contexts, see, for example, Refs. [16][17][18][19][20][21][22] and the references cited therein. ...

... We are interested in the curvature of Kerr spacetime as measured by the family of observers at rest. Projected onto the natural frame e µα with axes that are primarily along the Boyer-Lindquist coordinate directions, we find (E , H ). Specifically, the nonvanishing components of the tidal matrix can be obtained from [18] E 11 = −2E Furthermore, the nonzero elements of the gravitomagnetic part of the Weyl curvature can be obtained from [18] H 11 = −2H We are actually interested in the measured components of curvature along the Fermi-Walker transported tetrad frame λ µα . In this case, the measured components of the curvature tensor are given by Equation (A39), where (E , H) are related to (E , H ) via a rotation S given by Equation (A35). ...

... We are interested in the curvature of Kerr spacetime as measured by the family of observers at rest. Projected onto the natural frame e µα with axes that are primarily along the Boyer-Lindquist coordinate directions, we find (E , H ). Specifically, the nonvanishing components of the tidal matrix can be obtained from [18] E 11 = −2E Furthermore, the nonzero elements of the gravitomagnetic part of the Weyl curvature can be obtained from [18] H 11 = −2H We are actually interested in the measured components of curvature along the Fermi-Walker transported tetrad frame λ µα . In this case, the measured components of the curvature tensor are given by Equation (A39), where (E , H) are related to (E , H ) via a rotation S given by Equation (A35). ...

Relativistic tidal equations are formulated with respect to the rest frame of a central gravitational source and their solutions are studied. The existence of certain relativistic critical tidal currents are thereby elucidated. Specifically, observers that are spatially at rest in the exterior Kerr spacetime are considered in detail; in effect, these fiducial observers define the rest frame of the Kerr source. The general tidal equations for the free motion of test particles are worked out with respect to the Kerr background. The analytic solutions of these equations are investigated and the existence of a tidal acceleration mechanism is emphasized.

... For recent work on general relativistic tidal effects in other contexts, see, for example, Refs. [16][17][18][19][20][21][22][23] and the references cited therein. ...

... Specifically, the nonvanishing components of the tidal matrix can be obtained from [18] E ′11 = −2E ∆ + 1 2 a 2 sin 2 θ ∆ − a 2 sin 2 θ , E ′12 = −3 a sin θ H ∆ 1/2 ∆ − a 2 sin 2 θ , ...

... Furthermore, the nonzero elements of the gravitomagnetic part of the Weyl curvature can be obtained from [18] H ′11 = −2H ∆ + 1 2 a 2 sin 2 θ ∆ − a 2 sin 2 θ , H ′12 = 3 a sin θ E ∆ 1/2 ∆ − a 2 sin 2 θ , H ′22 = H ∆ + 2 a 2 sin 2 θ ∆ − a 2 sin 2 θ , ...

Relativistic tidal equations are formulated with respect to the rest frame of a central gravitational source and their solutions are studied. The existence of certain relativistic critical tidal currents are thereby elucidated. Specifically, observers that are spatially at rest in the exterior Kerr spacetime are considered in detail; in effect, these fiducial observers define the rest frame of the Kerr source. The general tidal equations for the free motion of test particles are worked out with respect to the Kerr background. The analytic solutions of these equations are investigated and the existence of a tidal acceleration mechanism is emphasized.

... The resulting radial and tangential components of the spatial frame, namely, λ µ1 and λ µ3 , respectively, turn out to be periodic in τ with period 2π/ω 0 . The difference between the orbital frequency (26) and the Keplerian frequency ω 0 leads to a combination of prograde geodetic and retrograde gravitomagnetic precessions of these frame components with respect to static inertial observers at spatial infinity in the asymptotically flat Kerr spacetime [38]. ...

... The tidal matrix is obtained as a certain symmetric and traceless projection of the Riemann curvature tensor evaluated along the orbit. The nonzero components of the tidal matrix consist of constant terms proportional to ω 2 0 plus terms that are periodic in τ with frequency 2 ω 0 and can be expressed as [38] ...

... The exterior gravitational field is represented by the Kerr metric linearized in the angular momentum parameter a or, equivalently, the Schwarzschild metric plus the Thirring-Lense term. The symmetric and traceless tidal ma-trix can be obtained from [38] ...

Gravity gradiometry within the framework of the general theory of relativity involves the measurement of the elements of the relativistic tidal matrix, which is theoretically obtained via the projection of the spacetime curvature tensor upon the nonrotating orthonormal tetrad frame of a geodesic observer. The behavior of the measured components of the curvature tensor under Lorentz boosts is briefly described in connection with the existence of certain special tidal directions. Relativistic gravity gradiometry in the exterior gravitational field of a rotating mass is discussed and a gravitomagnetic beat effect along an inclined spherical geodesic orbit is elucidated.

... where we have assumed that τ = 0 at t = 0. The natural orthonormal tetrad frame e µα of these observers is given by [24] e0 = u , ...

... Next, the curvature of Kerr spacetime as measured by static observers using their natural tetrad system has been discussed in detail in Appendix B of Ref. [24]. Along the axis of rotation (θ = 0, π), we find that the tidal matrix is diagonal such that [24] ...

... Next, the curvature of Kerr spacetime as measured by static observers using their natural tetrad system has been discussed in detail in Appendix B of Ref. [24]. Along the axis of rotation (θ = 0, π), we find that the tidal matrix is diagonal such that [24] ...

Within the framework of general relativity, we investigate the tidal acceleration of astrophysical jets relative to the central collapsed configuration ("Kerr source"). We neglect electromagnetic forces throughout. The rest frame of the Kerr source is locally defined via the set of hypothetical static observers in the spacetime exterior to the source. Relative to such a fiducial observer fixed on the rotation axis of the Kerr source, jet particles are tidally accelerated to almost the speed of light if their outflow speed is above a certain threshold, given roughly by one half of the Newtonian escape velocity at the location of the reference observer; otherwise, the particles reach a certain height, reverse direction and fall back toward the gravitational source.

... We recommend [1,2] for historical review and references. Various approaches to the issues of the field have recently been proposed, and we quote a non-exhaustive list of papers concerning, by way of example, gravitomagnetic effects [3][4][5][6], the relation of GEM to special relativity [2,7], tidal tensors [8,9], weak-field approximation [10][11][12], the Lorentz violation [13,14], teleparallel gravity [15,16], the Mashhoon-Theiss effect [17], quantum gravity [18,19], gravitational waves [20,21], the relation of GEM to electro-dynamics in curved spacetime [22,23], gravitational field of astrophysical objects [24,25], the Sagnac effect [26,27], torsion gravity [28], the Schrödinger-Newton equation [29], non-commutative geometry [30], spin-gravity coupling [31], gravity and thermodynamics [32], the Casimir effect [33], gauge transformations [34] and, quantum field gravity [35,36]. It is commonly known that GEM is a source of new ideas and a guide for research into new physics. ...

... and equations (15)(16)(17)(18) hold. On the other hand, (22) changes to ...

This paper describes gravito-electromagnetism (GEM) as a constrained field theory. Equations of motion, continuity equation, energy conservation, field tensor, energy-momentum tensor, constraints and Lagrangian formulation are presented as a simple and unified formulation that can be useful for future research.

... Atomic clocks and gravity field determination for geodesy by many researcher 17,18 . In USA, many theory work [19][20][21] was done with the plan of NASA 22 . In German, the plan of gravitational clock compass 23 based on the early direction study of the curvature 24 was done. ...

... With respect to the static observers, the nonvanishing components of the tidal matrix are given by [72] E11 = − 2E ∆ + 1 2 a 2 sin 2 θ ∆ − a 2 sin 2 θ , E12 = − 3a sin θ B ∆ 1/2 ∆ − a 2 sin 2 θ , E22 = E ∆ + 2 a 2 sin 2 θ ∆ − a 2 sin 2 θ , ...

Mass currents in astrophysics generate gravitomagnetic fields of enormous complexity. Gravitomagnetic helicity, in direct analogy with magnetic helicity, is a measure of entwining of the gravitomagnetic field lines. We discuss gravitomagnetic helicity within the gravitoelectromagnetic (GEM) framework of linearized general relativity. Furthermore, we employ the spacetime curvature approach to GEM in order to determine the gravitomagnetic helicity for static observers in Kerr spacetime.

... One drawback of harmonic frames is that the harmonic gauge condition does not admit rigidly rotating frames [51, chapter 8]. Other recent approaches are based on a perturbed Schwarzschild metric [52], or on the Kerr metric [53] in the different context of a slowly rotating astronomical object. Following the pioneering works, a set of Resolutions was adopted at the IAU General Assembly in Manchester in the year 2000 [54]: We summarize here very briefly these resolutions. ...

In section 2, we introduce fundamental concepts of GR concerning the measurement of time, relativistic reference systems and we review the recent literature of chronometric geodesy. In section 3 we introduce the theory of frequency standard comparisons, beginning with the Einstein equivalence principle, followed by the description of the frequency techniques, and finally, we describe clock syntonization and the realization of timescales. Section 4 describes the geodetic methods for determining the gravity potential, namely the geometric levelling approach and the GNSS/geoid approach, as well as considerations about the uncertainties of these methods. In section 5 we describe the European project ITOC where unified relativistic redshift corrections were determined for several clocks in European national metrology institutes. Finally, in section 6 we present numerical simulations exploring what could be the contribution of clock comparisons for the determination of the geoid.

... which is orthogonal to e 0 and is parallel propagated along the geodesic orbit. Marck then completed these first two vector fields to an orthonormal frame adapted to U by adding the two vector fields whose corresponding 1-forms (indicated by ♭) at the equatorial plane are [40,41] ...

The precession of a test gyroscope along unbound equatorial plane geodesic orbits around a Kerr black hole is analyzed with respect to a static reference frame whose axes point towards the "fixed stars." The accumulated precession angle after a complete scattering process is evaluated and compared with the corresponding change in the orbital angle. Limiting results for the non-rotating Schwarzschild black hole case are also discussed.

... which is orthogonal to e 0 and is parallel propagated along the geodesic orbit. Marck then completed this to an orthonormal frame by adding the two vector fields whose corresponding 1-forms (indicated by ♭) at the equatorial plane are [27] ...

The precession of a test gyroscope along stable bound equatorial plane orbits around a Kerr black hole is analyzed and the precession angular velocity of the gyro's parallel transported spin vector and the increment in precession angle after one orbital period is evaluated. The parallel transported Marck frame which enters this discussion is shown to have an elegant geometrical explanation in terms of the electric and magnetic parts of the Killing-Yano 2-form and a Wigner rotation effect.

This paper describes general relativity at the gravito-electromagnetic precision level as a constrained field theory. In this novel formulation, the gravity field comprises two auxiliary fields, a static matter field and a moving matter field. Equations of motion, continuity equation, energy conservation, field tensor, energy-momentum tensor, constraints, and Lagrangian formulation are presented as a simple and unified formulation that can be useful for future research.

The theory of general relativity was born more than one hundred years ago, and since the beginning has striking prediction success. The gravitational redshift effect discovered by Einstein must be taken into account when comparing the frequencies of distant clocks. However, instead of using our knowledge of the Earth’s gravitational field to predict frequency shifts between distant clocks, one can revert the problem and ask if the measurement of frequency shifts between distant clocks can improve our knowledge of the gravitational field. This is known as chronometric geodesy. Since the beginning of the atomic time era in 1955, the accuracy and stability of atomic clocks were constantly ameliorated, with around one order of magnitude gained every ten years. Now that the atomic clock accuracy reaches the low \(10^{-18}\) in fractional frequency, and can be compared to this level over continental distances with optical fibres, the accuracy of chronometric geodesy reaches the cm level and begins to be competitive with classical geodetic techniques such as geometric levelling and GNSS/geoid levelling. Moreover, the building of global timescales requires now to take into account these effects to the best possible accuracy. In this chapter we explain how atomic clock comparisons and the building of timescales can benefit from the latest developments in physical geodesy for the modelization and realization of the geoid, as well as how classical geodesy could benefit from this new type of observable, which are clock comparisons that are directly linked to gravity potential differences.

Gravity gradiometry within the framework of the general theory of relativity involves the measurement of the elements of the relativistic tidal matrix, which is theoretically obtained via the projection of the spacetime curvature tensor upon the nonrotating orthonormal tetrad frame of a geodesic observer. The behavior of the measured components of the curvature tensor under Lorentz boosts is briefly described in connection with the existence of certain special tidal directions. Relativistic gravity gradiometry in the exterior gravitational field of a rotating mass is discussed and a gravitomagnetic beat effect along an inclined spherical geodesic orbit is elucidated.

With continuous advances in related technologies, precision tests of modern gravitational theories with orbiting gradiometers becomes feasible, which may naturally be incorporated into future satellite gravity missions. In this work, we derive, at the post-Newtonian level, the new secular gravity gradient signals from the non-dynamical Chern–Simons modified gravity for satellite gradiometry measurements, which may be exploited to improve the constraints on the mass scale \(M_{CS}\) or the corresponding length scale \({\dot{\theta }}\) of the theory with future missions. For orbiting superconducting gradiometers, a bound \(M_{CS}\ge 10^{-7}\ \mathrm{eV}\) and \({\dot{\theta }} \le 1\ \mathrm{m}\) could in principle be obtained, and for gradiometers with optical readout based on the similar technologies established in the LISA PathFinder mission, an even stronger bound \(M_{CS}\ge 10^{-6}\)–\(10^{-5}\ \mathrm{eV}\) and \({\dot{\theta }} \le 10^{-1}\)–\(10^{-2} \ \mathrm{m}\) might be expected.

A theoretical study of testing nonlocal gravity in its Newtonian regime with gravity gradient measurements in space is given. For certain solutions of the modification to Newton’s law in nonlocal gravity, a null test and a lower bound on related parameters may be given with future high precision multi-axis gravity gradiometers along elliptic orbits.

A summary of the main relativistic effects in the motion of the moon is
presented. The results are based on the application of a novel approach
to the restricted three-body problem in general relativity to the lunar
motion. It is shown that the rotation of the sun causes a secular
acceleration in the relative earth-moon motion. This might appear to be
due to a temporal 'variation' of the gravitational constant.

A new general relativistic many-body effect is described. It results in an unexpectedly large relative acceleration between neighboring test particles that follow an inclined orbit about a rotating mass. The effect vanishes if the orbit coincides with the equatorial plane of the rotating mass. The existence of this effect is due to a small divisor involving the deviation of the orbital frequency measured by a comoving clock from the frequency measured by an inertial clock. The influence of the rotation of the Sun on the Earth-Moon system is investigated, and it is shown that the new effect causes a harmonic variation in the Earth-Moon separation with an amplitude of order 1 m and dominant periods of 18.6 yr, 1/2 yr, 1 month, and 1/2 month. The confirmation of these results by the lunar laser ranging experiment would provide a significant new test of general relativity and a measurement of the angular momentum of the Sun.

The angular momentum of the Earth produces gravitomagnetic components of the Riemann curvature tensor, which are of the order of 10/sup /minus/10/ of the Newtonian tidal terms arising from the mass of the Earth. These components could be detected in principle by sensitive superconducting gravity gradiometers currently under development. We lay out the theoretical principles of such an experiment by using the parametrized post-Newtonian formalism to derive the locally measured Riemann tensor in an orbiting proper reference frame, in a class of metric theories of gravity that includes general relativity. A gradiometer assembly consisting of three gradiometers with axes at mutually right angles measures three diagonal components of a 3/times/3 ''tidal tensor,'' related to the Riemann tensor. We find that, by choosing a particular assembly orientation relative to the orbit and taking a sum and difference of two of the three gradiometer outputs, one can isolate the gravitomagnetic relativistic effect from the large Newtonian background.

The precession of a test gyroscope along stable bound equatorial plane orbits around a Kerr black hole is analyzed and the precession angular velocity of the gyro's parallel transported spin vector and the increment in precession angle after one orbital period is evaluated. The parallel transported Marck frame which enters this discussion is shown to have an elegant geometrical explanation in terms of the electric and magnetic parts of the Killing-Yano 2-form and a Wigner rotation effect.

The existence of relativistic secular tidal effects along orbit motions will largely improve the measurement accuracies of relativistic gravitational gradients with orbiting gradiometers. With the continuous advances in technologies related to gradiometry and the improvements in their resolutions, it is feasible for future satellite gradiometry missions to carry out precision relativistic experiments and impose constraints on modern theories of gravity. In this work, we study the theoretical principles of measuring directly the secular post-Newtonian (PN) tidal effects in semi-conservative metric theories with satellite gradiometry missions. The isolations of the related PN parameters in the readouts of an orbiting three-axis gradiometer is discussed.

With continuous advances in technology, future satellite gradiometry missions will be capable of
performing precision relativistic experiments and imposing constraints on modern gravity theories. To this end, the full first-order post-Newtonian tidal tensor under inertially guided and Earth-pointing local frames along post-Newtonian orbits is worked out. The physical picture behind the “Mashhoon-Theiss anomaly” is explained at the post-Newtonian level. The relativistic precession of the local frame with respect to the sidereal frame will produce modulations of Newtonian tidal forces along certain bases, which gives rise to two different kinds of secular tidal tensors. The measurements of the secular tidal force from the frame- dragging effect is also discussed.

The relativistic corrections to the Newtonian tidal accelerations generated by a rotating system are studied. The possibility of testing the relativistic theory of gravitation by measuring such effects in a laboratory in orbit around the Earth is considered. A recent proposal to measure a rotation-dependent tidal acceleration as an alternative to the Stanford gyroscope experiment is critically examined and it is shown that such an experiment does not circumvent the basic difficulties associated with the gyroscope experiment.

Tidal forces caused by a slowly rotating spherical mass are calculated using the relativistic theory of tides. The possibility of measuring the gravitational 'magnetic' contribution of the earth to the tidal acceleration between test masses in a satellite is considered. It is shown that in such an experiment unexpectedly large secular 'magnetic' contributions can occur. Already after about 20 days of observation this new effect would reach a value which is larger than the total error of measurement by a factor of 100.

Large secular contributions can occur in the angular momentum dependence of the tidal matrix for a system in free fall about a slowly rotating spherical mass (as shown by Theiss). The frame dragging PPN parameters Delta1 and Delta2 are shown here to be associated with these contributions, so that the proposed gravity gradiometer experiment of Chan and Paik (1987) can be interpreted as measurements of Delta1 and Delta2.

An experiment has been performed to demonstrate a new source-independent null test of the inverse square law of gravitation. A single-axis superconducting gravity gradiometer was rotated into three orthogonal orientations to measure the Laplacian of the gravitational potential produced by a 1600-kg lead pendulum at an average distance of 2.3 m. The result is that if one assumes a potential of the form ϕ(r)=-(GM/r)[1+αexp(-μr)], the value of α is + 0.024±0.036 at μ-1=1 m.

A new approach to the relativistic theory of the motion of the moon is presented. In this framework, the contribution of the gravitational ``magnetic'' field of the sun to the tidal force acting on the earth-moon system is determined. Furthermore, the main relativistic perturbations in the lunar motion are estimated.

A detailed treatment of the mathematical theory of black holes is
presented. The analytical methods on which the theory is based are
reviewed, and a space-time of sufficient generality to encompass the
different situations arising in the study of black holes is developed.
The Schwarzschild space-time and the perturbations of the Schwarzschild
black hole are addressed. The Reissner-Nordstrom solution, the Kerr
metric, geodesics in Kerr space-time, electromagnetic waves in Kerr
geometry, gravitational perturbations of the Kerr black hole, and
spin-1/2 particles in Kerr geometry are discussed. Other solution and
methods are examined.

A gyroscope in orbit about a central rotating mass undergoes relativistic nutational oscillations in addition to the well-known precessional motions. The amplitude of the oscillation is proportional to the angular momentum of the rotating mass and its period is the Fokker period of geodetic precession. The amplitude is maximum for a polar orbit and vanishes if the orbit is equatorial. This nodding effect is due to a small divisor phenomenon involving the Fokker frequency, and its existence implies that the applicability of the post-Newtonian approximation of general relativity is limited in time. The dynamical significance of the new effect for the relative motion of neighboring test masses in the field of a rotating mass as well as for the restricted three-body problem in general relativity is investigated and the possibility of its detection is briefly discussed.

The angular momentum of the Earth produces gravitomagnetic components of the Riemann curvature tensor, which are of the order
of 10−10 of the Newtonian terms arising from the mass of the Earth. Due to the dragging of the local inertial frame by the spinning
Earth, there are also secular terms, which grow in time. These fields can be detected in principle by a set of orbiting superconducting
gravity gradiometers. The Riemann tensor components for various spacecraft orientations have been computed and the principle
of detecting the gravitomagnetic tidal force has been published. In this paper, we review the conclusions of the earlier analyses
and discuss the feasibility of the gravity gradiometer experiment.

The main relativistic effects in the motion of the Moon are caused by the gravitational influence of the Sun on the Earth-Moon
orbit. In view of the recent measurement of the de Sitter precession of the lunar node and perigee by Shapiroet al., we further analyze relativistic three-body effects that may become measurable in the near future. Of particular interest
is a 6 cm variation in the Earth-Moon distance due to the post-Newtonian gravitoelectric field of the Sun. Moreover, the tidal
influence of the solar gravitomagnetic field on the Earth-Moon system is similar to a temporal variation of the Newtonian
constant of gravitation at the level of ∼10−16 per year.

A new relativistic tidal effect of a deformed axisymmetric mass is described and the possibility of measuring it in an earth satellite is considered. The calculations show that the quadrupole moment of the earth causes a large secular relativistic contribution to the relative acceleration between two test masses placed in a satellite around the earth. Already after ≈10 hours of observation this new relativistic effect would exceed the corresponding newtonian quadrupole effect by a factor of 10.

A sensitive gravity gradiometer can provide much needed gravity data of the earth and improve the accuracy of inertial navigation. Superconductivity and other properties of materials at low temperatures can be used to obtain a sensitive, low-drift gravity gradiometer; by differencing the outputs of accelerometer pairs using superconducting circuits, it is possible to construct a tensor gravity gradiometer which measures all the in-line and cross components of the tensor simultaneously. Additional superconducting circuits can be provided to determine the linear and angular acceleration vectors. A tensor gravity gradiometer with these features is being developed for satellite geodesy. The device constitutes a complete package of inertial navigation instruments with angular and linear acceleration readouts as well as gravity signals.

Relativistic effects in the motion of parallel-transported axes and their contribution to the tidal forces are considered. To this end we analyze the motion of a local inertial frame and the projected tidal-force tensor in the field of a slowly rotating mass in the first post-Newtonian approximation. It is demonstrated that the Mashhoon-Theiss ``anomaly'' appears already at this level and can be understood as arising from an interplay between the geodetic (Fokker) precession and Lense-Thirring (Schiff) precession. Hence, there is no ``new'' relativistic (resonant) effect related to rotating masses. Applications and orders of magnitude of these effects are discussed for space gradiometry and the motion of the Earth-Moon system.

The main general relativistic effects in the motion of the Moon are briefly reviewed. The possibility of detection of the solar gravitomagnetic contributions to the mean motions of the lunar node and perigee is discussed.

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