Article

Relativistic gravity gradiometry

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Abstract

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). ...
Article
Full-text available
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 , ...
Preprint
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] ...
Preprint
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] ...
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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 ...
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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 , ...
Preprint
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. ...
Preprint
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... 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] ...
Article
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] ...
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Chapter
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