# Gravitomagnetism and the Clock Effect

Bahram Mashhoon 1Department of Physics and Astronomy, University of Missouri, Columbia, Missouri, 65211, USA 1    Frank Gronwald 2Institut GET, Universität Magdeburg, D-39106 Magdeburg, Germany 2    Herbert I.M. Lichtenegger 3Institut für Weltraumforschung, Österreichische Akademie der Wissenschaften, A-8010 Graz, Austria3
###### Abstract

The main theoretical aspects of gravitomagnetism are reviewed. It is shown that the gravitomagnetic precession of a gyroscope is intimately connected with the special temporal structure around a rotating mass that is revealed by the gravitomagnetic clock effect. This remarkable effect, which involves the difference in the proper periods of a standard clock in prograde and retrograde circular geodesic orbits around a rotating mass, is discussed in detail. The implications of this effect for the notion of “inertial dragging” in the general theory of relativity are presented. The theory of the clock effect is developed within the PPN framework and the possibility of measuring it via spaceborne clocks is examined.

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Gravitomagnetism and the Clock Effect

## 1 Introduction

The close formal similarity between Coulomb’s law of electricity and Newton’s law of gravitation has led to a description of Newtonian gravitation in terms of a gravitoelectric field. The classical tests of general relativity can all be described via post-Newtonian gravitoelectric corrections brought about by relativity theory. Moreover, any theory that combines Newtonian gravitation and Lorentz invariance in a consistent framework must involve a gravitomagnetic field in close analogy with electrodynamics. The gravitomagnetic field is generated by the motion of matter. For instance, the mass current in the rotating Earth generates a dipolar gravitomagnetic field that has not yet been directly observed; in fact, the main objective of the GP-B is to measure this field in a polar Earth orbit via the gravitomagnetic precession of superconducting gyroscopes on board a drag-free satellite.

Gravitomagnetism had its beginning in the second half of the last century. Developments in electrodynamics led Holzmüller GHo and Tisserand FTi to postulate the existence of a solar gravitomagnetic field ande94 . In fact, attempts were made to account for the excess perihelion precession of Mercury since the planetary orbits would be affected by the gravitomagnetic field of the Sun. However, the excess perihelion precession of Mercury was successfully explained by Einstein’s general relativity theory in terms of a small relativistic correction to the Newtonian gravitoelectric potential of the Sun. It was later shown by Thirring and Lense HTh ; BM1 that general relativity also predicts a certain gravitomagnetic field for a rotating mass, but the magnitude of this field in the solar system is generally small and would lead to a retrograde precession of the planetary orbits. This Lense-Thirring precession of planetary orbits is too small to be detectable at present.

For the purposes of confronting the theory with observation, gravitomagnetic phenomena are usually described in the framework of the post-Newtonian approximation; however, it is possible to provide a fully covariant treatment of certain aspects of gravitoelectromagnetism AMa ; BJH . In fact, extensions of the Jacobi equation (i.e. the relativistic tidal equation) may be employed to identify the gravitoelectric and gravitomagnetic components of the curvature tensor in close analogy with the Lorentz force law. This analogy is incomplete, however, since the purely spatial components of the curvature tensor do not in general have an analog in the electromagnetic case; in fact, this is expected since linear gravity is a spin-2 field in contrast to the spin-1 character of the electromagnetic field.

Some of the main theoretical aspects of gravitomagnetism are discussed in Section 2. We then turn our attention to how gravitomagnetism affects the spacetime structure in general relativity. Of primary importance in this connection is the gravitomagnetic clock effect, which in its simplest form may be formulated in terms of the difference in the proper periods of two clocks moving on the same circular orbit but in opposite directions about a rotating mass. Let be the period for prograde (retrograde) motion, then for . To lowest order, this remarkable result is independent of Newton’s constant of gravitation and the radius of the orbit . The effect and its consequences are discussed in Section 3 for circular equatorial orbits in the Kerr geometry and the intimate connection between the clock effect and the gravitomagnetic gyroscope precession is demonstrated. The PPN approximation for this effect is developed in Section 4 and a brief discussion of its observability is given in Section 5. The sign of the clock effect is quite intriguing, as it implies that prograde equatorial clocks are slower than retrograde equatorial clocks. This is completely opposite to what would be expected on the basis of “inertial dragging”. In fact, gravitomagnetism is historically connected with the question of the origin of inertia as this was Thirring’s motivation in his original paper on gravitomagnetism HTh . The present status of the problem of inertia is the subject of Section 6. Finally, Section 7 contains a brief discussion.

Unless specified otherwise, we use units such that for the sake of convenience.

## 2 Gravitoelectromagnetism

This section is devoted to a brief discussion of certain essential theoretical aspects of gravitoelectromagnetism. The Larmor theorem has played an important role in the field of magnetism; therefore, we begin by an account of the gravitational analog of Larmor’s theorem.

Gravitational Larmor Theorem

A century ago, Larmor established a theorem regarding the local equivalence of magnetism and rotation 1 . That is, the basic electromagnetic force on a slowly moving particle of charge and mass can be locally replaced in the linear approximation by the inertial forces that arise if the motion is referred instead to an accelerated system in the absence of the electromagnetic field. The translational acceleration of the system is related to the electric field, , and the rotational (Larmor) frequency is related to the magnetic field via . The charge-to-mass ratio is not the same for all particles; otherwise, a geometric theory of electrodynamics could be developed along the same lines as general relativity. It turns out that in general relativity one can provide an interpretation of Einstein’s heuristic principle of equivalence via the gravitational Larmor theorem 2 . This is due to the experimentally well-established circumstance that the gravitational charge-to-mass ratio is the same for all particles. Einstein’s heuristic principle of equivalence is usually stated in terms of the gravitoelectric field, i.e. the translational acceleration of the “Einstein elevator” in Minkowski spacetime. The gravitational Larmor theorem would also involve the gravitomagnetic field, i.e. a rotation of the elevator as well.

It follows from the theoretical study of the motion of test particles as well as ideal test gyroscopes in a gravitational field that in general relativity the gravitoelectric charge is , while the gravitomagnetic charge is ; in fact, since general relativity involves the tensor potential , i.e. (linear) gravitation is a spin-2 field. Thus and in this case. Indeed is the gravitomagnetic precession frequency of an ideal test gyroscope at rest in a gravitomagnetic field, i.e. far from a rotating source , where

 \boldmathΩP=GJc2r3[3(^r⋅^J)^r−^J], (1)

and is the total angular momentum of the source. Let us note that a gyro spin is in effect a gravitomagnetic dipole moment that precesses in a gravitomagnetic field. Locally, the same rotation would be observed in the absence of the gravitomagnetic field but in a frame rotating with frequency in agreement with the gravitational Larmor theorem.

It is the goal of the GP-B to measure the gravitomagnetic gyroscope precession in a polar orbit about the Earth and thereby provide direct observational proof of the existence of the gravitomagnetic field 3 .

Gravitoelectromagnetic Field

Let us consider the gravitational field of a “nonrelativistic” rotating astronomical source in the linear approximation of general relativity. The spacetime metric may be expressed as , where is the Minkowski metric. We define , where ; then, the gravitational field equations are given by

 □¯hμν=−16πGc4Tμν, (2)

where the Lorentz gauge condition has been imposed. We focus attention on the particular retarded solution of the field equations given by

 ¯hμν=4Gc4∫Tμν(ct−|x−x′|,x′)|x−x′|d3x′, (3)

where the nature and distribution of the “nonrelativistic” source must be taken into account.

We are interested in sources such that , and , where is the gravitoelectric potential, is the gravitomagnetic vector potential and we neglect all terms of order and lower including the tensor potential . It follows that is the effective gravitational charge density and is the corresponding current. Thus, far from the source

 Φ∼GMr,A∼GcJ×rr3, (4)

where and are the total mass and angular momentum of the source, respectively. It follows from the Lorentz gauge condition that

 1c∂Φ∂t+\boldmath∇⋅(12A)=0, (5)

since the other three equations all involve terms that are of and therefore neglected. The spacetime metric involving the gravitoelectromagnetic (“GEM”) potentials is then given by

 −ds2=−c2(1−2c2Φ)dt2−4c(A⋅dx)dt+(1+2c2Φ)δijdxidxj. (6)

The GEM fields are defined by

 (7)

in close analogy with electrodynamics. It follows from the field equations (2) and the gauge condition (5) that

 \boldmath∇⋅E=4πGρ, (8)
 \boldmath∇⋅(12B)=0, (9)
 \boldmath∇×E=−1c∂∂t(12B), (10)
 \boldmath∇×(12B)=1c∂∂tE+4πcGj, (11)

which are the Maxwell equations for the GEM field. Using classical electrodynamics as a guide, one can investigate the various implications of these equations 4 . A thorough approach to the determination of the gravitomagnetic field of a rotating mass (such as the Earth) is contained in the papers of Teyssandier 5 .

The fact that the magnetic parts of equations (8) - (11) always appear with a factor of as compared to standard electrodynamics is due to the circumstance that the effective gravitomagnetic charge is twice the gravitoelectric charge. That is, and are the effective gravitoelectric and gravitomagnetic charges of the source.

The linear approximation of general relativity involves a spin-2 field. This field, once its spatial components are neglected, can be interpreted in terms of a gravitoelectromagnetic vector potential. To sustain the electromagnetic analogy, however, we need to require that the gravitomagnetic charge be twice the gravitoelectric charge. This factor of 2 is a remnant of the spin-2 character of the original field, while for a pure spin-1 field (i.e. the electromagnetic field) the ratio of the magnetic charge to the electric charge is unity.

The equation of motion of a test particle of mass in this linear gravitational field can be obtained from the variational principle , where is given by

 L=−mc2[1−v2c2−2c2(1+v2c2)Φ+4c3v⋅A]1/2, (12)

using equation (6). To linear order in and , one can write (12) as

 L=−mc2(1−v2c2)1/2+mγ(1+v2c2)Φ−2mcγv⋅A. (13)

Let us note that the deviation of equation (13) from a free-particle Lagrangian is given to lowest order in by . This deviation would be of the form in electrodynamics; therefore, the slow motion of the test particle is very similar to that of a charged particle in electrodynamics except that here and as expected. It thus follows from the geodesic motion of a test particle of mass far from the source in this gravitational background that the canonical momentum of the particle is given approximately by , where is the kinetic momentum. In this electrodynamic analogy, the attractive nature of gravity is reflected in our convention of positive gravitational charges for the source and negative gravitational charges for the test particle. The gravitomagnetic charge is always twice the gravitoelectric charge as a consequence of the tensorial character of the gravitational potentials in general relativity.

The gauge transformations

 Φ→Φ−1c∂ψ∂t,A→A+2\boldmath∇ψ, (14)

leave the GEM fields (7) and hence the GEM equations (8)-(11) invariant. The Lorentz gauge condition (5) is also satisfied provided . However, the quantity in the Lagrangian is not invariant under the gauge transformation (14). The gauge invariance of this Lagrangian is restored, however, if the gauge function is independent of time, . In this case, we can start from a coordinate transformation in the metric (6) resulting in the gauge transformations (14) with left invariant.

The gravitational field corresponding to the metric (6) is given by the Riemann curvature tensor

 Rμνρσ=12(hμσ,νρ+hνρ,μσ−hνσ,μρ−hμρ,νσ), (15)

where and are gravitoelectric and of , while is gravitomagnetic and of . The components of the curvature tensor as measured by the standard geodesic observers are given by , where is the tetrad frame of the test observer. In the linear approximation under consideration here, is in effect equal to in the calculation of the measured curvature. The components of this tensor may be expressed in the form of a symmetric matrix , where and range over (01, 02, 03, 23, 31, 12); hence,

 R=(EBBTS), (16)

where and are symmetric matrices and is traceless. We find that the electric and magnetic components of the curvature are given by

 Eij=1c2Ej,i+O(c−4), (17)
 Bij=−1c2Bj,i+1c3ϵijk∂Ek∂t+O(c−4), (18)

and the spatial components are given by

 Sij=−1c2Ej,i+1c2(\boldmath∇⋅E)δij+O(c−4). (19)

That is traceless is consistent with equation (9) and the fact that and are symmetric is consistent with equation (10) at . It is therefore clear that gravitoelectromagnetism permeates every aspect of general relativity: the gravitational potentials (GEM potentials), the connection (GEM field) and the curvature. In the exterior of the rotating source, the spacetime is Ricci-flat and hence , is traceless and is symmetric. These restrictions on the curvature are consistent with the GEM field equations (8)-(11) in the source-free region.

The general treatment of gravitoelectromagnetism presented here has been based on a certain approximate form of the linear gravitation theory and can be used in the theoretical description of many interesting gravitational phenomena. In particular, we use this formalism below to investigate the microphysical implications of the gravitomagnetic precession of spin.

Free Fall is not Universal

The assumption that all free test particles fall in the same way in a gravitational field is reflected in general relativity via the geodesic hypothesis. That is, the worldline of a free test particle is an intrinsic property of the spacetime manifold and is independent of the intrinsic aspects of the particle. In this way, general relativity is a geometric theory of gravitation. This circumstance originates from the well-tested equality of inertial and gravitational masses.

An important consequence of Einstein’s geometric theory of gravitation is the fact that an ideal test gyroscope would precess in the gravitomagnetic field of a rotating source. Here we pose the question of whether all spins should “precess” like a gyroscope; evidently, the treatment of intrinsic spin would go beyond classical general relativity. It follows from the consideration of spin-rotation-gravity coupling that the intrinsic spin of a particle (e.g. a nucleus) would couple to the gravitomagnetic field of a rotating source (such as the Earth) via the interaction Hamiltonian

 H=\boldmathσ⋅\boldmathΩP (20)

such that the Heisenberg equations of motion for the spin would be formally the same as that of an ideal test gyro 6 . Intuitively, this interaction is due to the coupling of the gravitomagnetic dipole moment of the particle with the gravitomagnetic field just as would be expected from the electromagnetic analogy. It follows from equation (20) that the particle is subject to a gravitational Stern-Gerlach force given by

 F=−\boldmath∇(\boldmathσ⋅\boldmath% ΩP) (21)

that is purely dependent upon its spin and not its mass and therefore violates the universality of free fall.

The point is that a particle is in general endowed with mass and spin in addition to other intrinsic properties; indeed, the irreducible unitary representations of the inhomogeneous Lorentz group are characterized by mass and spin. In its interaction with a gravitational field, the mass interacts primarily with the gravitoelectric field while the spin interacts primarily with the gravitomagnetic field. Whereas the former dominant interaction is consistent with the universality of free fall, the latter is not. For instance, the bending of light by the gravitational field of a rotating source depends on the state of polarization of the radiation. The differential deflection of polarized radiation by the Sun is too small to be measurable at present. The predicted violation is also extremely small for a nucleus in a laboratory on the Earth: the weight of the particle is , depending on whether the spin is polarized vertically up or down and . Thus the predicted violation of the universality of free fall is extremely small.

It may still be possible to measure this relativistic quantum gravitational effect by detecting the change in the energy of a particle in the laboratory when its spin is flipped. This would require, for instance, significant refinements in modern variations of NMR and optical pumping techniques, since

 ℏΩP∼ℏGJc2R3=cJR(LPR)2∼10−28eV (22)

is a factor of below the sensitivity of recent experiments 7 . Here is the Planck length, , is the angular momentum of the Earth and is its average radius. The smallest detected energy shift is about eV corresponding to a frequency shift of 2 nHz 7 . However, it appears that significantly lower energy shifts may soon be detectable 8 .

To clarify the nature of the force (21), let us consider the motion of a classical spinning test body in a stationary gravitational field. Such a system is necessarily extended and thus couples to spacetime curvature resulting in a Mathisson-Papapetrou force

 Fα=c2RαβμνuβSμν=cRαμβνuβSμν, (23)

where is the spin tensor of the system, is the velocity vector such that and the spin vector is given by

 Sμ=12(−g)1/2ϵμνρσuνSρσ. (24)

For the calculation of , it suffices to set, in the linear approximation, , and . Then, and

 Fi≈cBijSj=−1cBj,iSj=−(\boldmathΩP)j,iSj, (25)

in agreement with equation (21). Thus the existence of the force (21) may be ascribed to the intrinsic nonlocality of a particle in the quantum theory and hence the coupling of spin to the magnetic part of the spacetime curvature in a stationary field.

It is important to remark here that our considerations are distinct from proposals to measure the classical spin-spin force as discussed in 4 . Our results ultimately follow from detailed considerations of Dirac-type wave equations in the gravitational field of a rotating mass (see the references cited in 6 ); however, one can arrive at equations (20)-(21) on the basis of certain general arguments such as the local isotropy of space, the extended hypothesis of locality and the gravitational Larmor theorem 6 .

Assuming the approximate validity of equations (20)-(21), it would be difficult to imagine a basic gravitational theory founded purely on the universality of free fall and Riemannian geometry. However, such a theoretical structure would clearly be an excellent effective theory in the macrophysical domain.

GEM Stress-Energy Tensor

Let us imagine a congruence of geodesic test particles in a gravitational field. Taking one of the test particles as the reference observer, how does the motion of the other neighboring test particles appear to the fiducial observer? The result is best expressed in a Fermi coordinate system that is set up along the reference worldline. Let be the Fermi coordinates of the test particles, while the reference observer is at the origin of spatial Fermi coordinates. Then the equation of motion of the test particles is given by the generalized Jacobi equation

 d2Xidτ2 + FR0i0jXj+2FRikj0VkXj+(2FR0kj0ViVk (26) + 23FRikjlVkVl+23FR0kjlViVkVl)Xj=0,

which is valid to first order in the relative separation and to all orders in the relative velocity . Here are components of the curvature tensor as measured by the fiducial observer, i.e. they are the projections of the Riemann tensor onto the nonrotating tetrad frame of the reference observer. Neglecting the second and third order terms in the relative rate of separation, equation (26) can be written as the GEM analog of the Lorentz force law

 md2Xdτ2=qEE+qBV×B, (27)

where , and

 Ei=FR0i0j(τ)Xj,Bi=−12ϵijkFRjk0l(τ)Xl. (28)

It is important to notice that the spacetime interval in the neighborhood of the reference worldline can be expressed in Fermi coordinates as , where

 Fg00=−1−FR0i0j(τ)XiXj+⋯, (29)
 Fg0i=−23FR0jik(τ)XjXk+⋯, (30)
 Fgij=δij−13FRikjl(τ)XkXl+⋯. (31)

Letting and , we find that

 Φ=−12FR0i0jXiXj,Ai=13FR0jikXjXk, (32)

so that the corresponding GEM fields using equation (7) agree with the results in equation (28) to linear order in the separation . One can verify directly that and , so that the source-free pair of Maxwell’s equations are satisfied along the reference worldline. Moreover, it is possible to combine the GEM fields together to form a GEM Faraday tensor ,

 Fαβ=−FRαβ0lXl, (33)

such that and as in standard electrodynamics. Then the other pair of Maxwell’s equations is given by , where is easily obtained to linear order in using equation (33). is a conserved current such that along the fiducial trajectory. This treatment should be compared and contrasted with the linear approximation developed above, in particular, the GEM current is different here.

It is now possible to develop the classical field theory of the GEM field in the Fermi frame; in particular, one can define the Maxwell stress-energy tensor and the corresponding angular momentum for the GEM field. Thus

 Tαβ(τ,X)=14π(FαγFβγ−14ηαβFγδFγδ) (34)

is the Maxwell stress-energy tensor for the GEM field that is quadratic in the spatial separation and vanishes at the location of the fiducial observer. Physical measurements do not occur at a point, as already emphasized by Bohr and Rosenfeld 9 ; moreover, the fiducial observer is arbitrary here. Therefore, a physically more meaningful quantity is obtained by averaging equation (34) over a sphere of radius in the Fermi system. Here is a constant invariant lengthscale associated with the gravitational field. We find that

 =ϵ2C0L2~Tμνρσλμ(α)λν(β)λρ(0)λσ(0), (35)

where is a constant numerical factor and

 ~Tμνρσ=12(RμξρζRξζνσ+RμξσζRξζνρ)−14gμνRαβργRαβγσ. (36)

For a Ricci-flat spacetime, reduces to the Bel-Robinson tensor ; in this case, reduces to the Weyl tensor and in equation (36)

The magnitude of depends on whether we average over the surface or the volume of the sphere; in any case, one can always absorb into the definition of . Thus the pseudo-local GEM stress-energy tensor may be defined for any observer with the tetrad frame as

 T(α)(β)=L2~Tμνρσλμ(α)λν(β)λρ(0)λσ(0). (37)

In this way, the pseudo-local GEM energy density, Poynting flux and stresses are defined up to a common multiplicative factor.

It is important to note that the spatial components of the curvature have been ignored in our construction of the GEM tensor . For a Ricci-flat spacetime, however, the spatial components of the curvature are simply related to its electric components; therefore, the pseudo-local tensor defined via equation (37) using the Bel-Robinson tensor contains the full (Weyl) curvature tensor and is thus the gravitational stress-energy tensor.

It follows from a simple application of these results to the field of a rotating mass that there exists a steady Poynting flux of gravitational energy in the exterior field of a rotating mass.

Oscillations of a Charged Rotating Black Hole

Imagine a black hole of mass , charge and angular momentum that is perturbed by external radiation. The black hole is stationary and axisymmetric; therefore, the perturbation is expressible in terms of eigenmodes
, where depends upon the frequency of the radiation, the total angular momentum parameters of the eigenmode and the spin of the external field. It turns out that for a Fourier sum of such eigenmodes, the response of the black hole far away and at late times is dominated by a superposition of certain damped oscillations of the form , where with . For these quasinormal modes, the amplitude depends, among other things, on the strength of the perturbation while depends only on the black hole parameters . Moreover, these black hole oscillations are in general denumerably infinite and are numbered as , such that is least damped and increases with . The intrinsic ringing of a black hole is due to the fact that once perturbed, the black hole undergoes characteristic damped oscillations in order to return to a stationary state.

The fundamental modes of oscillations of black holes were originally found by numerical experiments and initial attempts to explain the numerical results via the properties of black hole effective potentials were unsuccessful 10 . The solution of the problem was first given around 1980 11 . This work provided the stimulus for many subsequent investigations by a number of authors 12 . A detailed discussion of black hole oscillations is contained in 13 .

For the modes of oscillation of a charged rotating black hole, the only reliable results are for . Expressions for have been obtained in the case of for a general Kerr-Newman black hole; however, the results have been generalized to the case of only for a slowly rotating charged black hole 14 . To express in terms of in the latter case, let be the “Keplerian” frequency for the motion of a neutral particle in a circular geodesic orbit of radius about a Reissner-Nordström black hole of mass and charge . Here we use Boyer-Lindquist type of coordinates for the Kerr-Newman geometry. Timelike circular geodesic orbits exist down to a null orbit of radius such that . Let , then it can be shown that for a slowly rotating black hole

 ωBH≈±jωN+mjΩN, (38)

where

 ΩN=Jr3N(1−Q2MrN)rN+MrN−M, (39)

is an effective black hole rotation frequency. The -fold degeneracy in the spectrum of oscillations of the spherical black hole is removed by its rotation. We note that is proportional to the gravitomagnetic precession frequency at . Moreover,

 ΓBH≈(n+12)(2−3MrN)1/2(ωN∓mjjQ∗ΩN), (40)

where is the mode number and . It is interesting to note that if the black hole is charged, the rotation removes the degeneracy of the damping factor as well. Moreover, in the formulas (38) and (40) if is a ringing mode, then so is . These results are independent of the spin of the perturbing field, since they are valid for states of high total angular momentum .

## 3 Structure of Time and Relativistic Precession

Let us now return to the gravitomagnetic temporal structure around a rotating source. The gravitomagnetic clock effect involves a coupling between the orbital motion of clocks and the rotation of the source. On the other hand, the gravitomagnetic gyroscope precession occurs even for a gyroscope at rest in the exterior field of a rotating source. Nevertheless, there is a general physical connection between relativistic precession and temporal structure. This is not surprising since the operational definition of time ultimately involves counting a definite period and simple precession is uniform periodic motion. It is the purpose of this section to explain this relationship. We do this in several steps in the context of Kerr geometry with

 −ds2=−dt2+ΣΔdr2+Σdθ2+(r2+a2)sin2θdϕ2 +RgrΣ(cdt−asin2θdϕ)2, (41)

where and . Here is the gravitational radius of the source and the Kerr parameter is a lengthscale characteristic of the rotation of the source. For and , the spacetime given by (41) is flat as expected. For and , the metric (41) represents the Schwarzschild geometry. Finally, for and we have the metric of an inertial frame expressed in spherical coordinates .

Let us first imagine an accelerated observer in an inertial frame. Suppose that this observer carries along its worldline an ideal pointlike test gyroscope so that there is no net torque on the gyroscope and its spin axis is therefore nonrotating. To simplify matters, let us first assume that the path is a circle of radius in the -plane with its center at the origin of coordinates. According to the standard static observers in the inertial frame, the accelerated observer moves with uniform frequency . The transformation between the inertial frame and the rest frame of the rotating observer involves a simple rotation of frequency ; therefore, from the viewpoint of the standard (i.e. static) observers in the rotating frame a natural operational way to keep the direction of the gyroscope spin axis nonrotating is to imagine fixing this axis at some initial time with respect to the axes of the rotating frame, but then continuously rotating it backward with frequency . In this way, the spin direction would remain fixed in the inertial frame if the rotation of the observer were virtual. In reality, however, the observer’s proper time is related to by , where ; hence, the backward rotation of the spin occurs with respect to the rotating observer’s proper time, i.e. with frequency . From the standpoint of the standard inertial observers, the time dilation causes the spin direction to overcompensate and hence the spin direction is not fixed but precesses with the Thomas precession frequency . This amounts to a precession of frequency in a sense that is opposite to that of the rotation of the comoving observer; moreover, the generalization to arbitrary acceleration can be simply carried out by means of the Frenet procedure. That is, a Frenet frame can be set up along the path of the observer in space; then, , where is the radius of the curvature at each instant of time .

The intimate connection between time dilation and Thomas precession in Minkowski spacetime can be extended to a gravitational field. Therefore, let us imagine next that the motion described above is the geodesic motion of a free test particle carrying an ideal test gyroscope around a spherical mass . Let be the Keplerian frequency as perceived by static inertial observers at infinity; then, , where is the Schwarzschild radius of the circular orbit. The proper frequency in this case is , where . The gravitational time dilation involves the static “gravitational redshift” effect of in the Schwarzschild geometry as well as the azimuthal motion resulting in the factor of 3 in . This situation is reminiscent of the spin-orbit coupling in the motion of the electron around the nucleus in the hydrogen atom; however, there are subtle differences between the electromagnetic and gravitational cases. In this case, the spin precession frequency is given by the Fokker frequency , and the sense of precession is in the same sense as the orbital motion. This gravitational analog of the Thomas precession has a simple and transparent explanation in terms of Einstein’s principle of equivalence. According to this heuristic principle, an observer in a gravitational field is locally equivalent to an observer in Minkowski spacetime with an acceleration that is equal in magnitude but opposite in direction to the Newtonian gravitational “acceleration” of . The gravitational (Fokker) precession is thus locally equivalent to Thomas precession with the direction of acceleration reversed. It follows that the Fokker precession is in the same sense as the orbital motion. For an arbitrary accelerated observer in a gravitational field with velocity and acceleration , the nonrotating equation of motion for the torque-free pointlike spin vector is

 dSμdτ+ΓμαβuαSβ=uμaνSν, (42)

so that both Fokker and Thomas precessions would be present for accelerated motion in Schwarzschild geometry.

The gravitomagnetic precession of a gyroscope is in a similar way related to the temporal structure brought about by the rotation of the source. However, the situation here is more complicated than the gravitoelectric Fokker precession since the temporal structure is affected by the coupling of orbital motion with the angular momentum of the source.

Specifically, let us imagine stable circular geodesic orbits in the equatorial plane of the Kerr source (41). It can be shown that

 (43)

where is the Keplerian period of the orbit and . Here the upper (lower) sign refers to a prograde (retrograde) orbit. It follows that the orbital period is given by

 t±=TK(1±α),τ±=TK(1±2α−3Rg/2r)1/2. (44)

Let us first note that and . Since , we find that . In fact, monotonically decreases as a function of and approaches as . Thus a prograde clock moves more slowly than a retrograde clock according to comoving observers as well as the standard asymptotically inertial observers at infinity; moreover, for . This remarkable gravitomagnetic clock effect has been discussed in some detail in recent publications JMC -WBB . This classical effect is in some sense the gravitomagnetic analog of the topological Aharonov-Bohm effect; in fact, let us note that far from a finite rotating source is nearly a constant independent of the “distance” and the gravitational coupling constant . These aspects of the clock effect have been discussed in detail elsewhere BFD .

Let us now imagine two clocks moving in opposite directions on a stable circular geodesic orbit of radius in the equatorial plane of the Kerr source. Suppose that at , they are both at ; let us denote the event at which the clocks next meet again by . It follows from equation (43) that for the prograde clock and for the retrograde clock. Thus and . Moreover, , which is negligibly small for clocks in orbit about astronomical sources in the solar system; in fact, . The next time the clocks meet is at , which bears the same relationship to as to ; therefore, and . In general, the nth time the clocks meet is at with modulo and .

Consider now the behavior of the diametrical line joining these events to the origin of the spatial coordinates. For , i.e. in the Schwarzschild case, this line is fixed as the clocks repeatedly meet at two diametrically opposite points. However, for the line precesses in the opposite sense as the rotation of the source with the precession frequency given approximately by (cf. Fig. 1)

 π−ϕ1τ+(ϕ1)=GJc2r3+O(c−4), (45)

which at this order is in agreement with the precession frequency of an ideal torque-free gyroscope that is fixed in the equatorial plane of the Kerr source mits70 . Figure 1: Gravitomagnetic precession of the diametrical lines indicating the points (ϕ0,ϕ1,ϕ2,…) at which the clocks would meet.

Our treatment of the clock effect has been limited thus far to circular orbits in the equatorial plane of the source. Off this plane, even circular orbits are not generally closed and the discussion of the clock effect as well as its intimate connection with the gravitomagnetic gyroscope precession becomes more complicated. In fact, the clock effect can be extended to such orbits using the notion of azimuthal closure BFD .

Finally, it should be mentioned that the general motion of an ideal pointlike torque-free gyroscope in the Kerr field would, in accordance with equation (42), involve Thomas and Fokker precessions as well as a complicated gravitomagnetic motion that consists of both precession and nutation. Indeed, the notion of relativistic nutation has been introduced in the post-Schwarzschild approximation scheme in order to describe the nutational part of the gravitomagnetic spin motion BDS . The complex gravitomagnetic spin motion reduces to the simple (Schiff) precession in the lowest post-Newtonian order.

## 4 Clock Effect in the PPN Approximation

In view of the possibility of detecting the gravitomagnetic clock effect via spaceborne clocks, it is interesting to develop the theory of the clock effect within the parametrized post-Newtonian (PPN) framework. The PPN formalism contains a set of parameters that characterize different metric theories of gravitation in the post-Newtonian approximation. The general form of the PPN metric is described in misn73 ; will93 ; it includes theories and effects that are not of primary interest for our treatment of the clock effect. Therefore we will start from a simplified PPN metric of the form

 g00 = −1+2U−2βU2, (46) g0i = −14(4γ+4+α1)H∗i, (47) gij = (1+2γU)δij, (48)

which describes a rotating body in an underlying Cartesian coordinate system with . In the following, we will use spherical coordinates ; the isotropic radial coordinate should not be confused with the Schwarzschild radial coordinate . In general relativity, the PPN parameters , and are given by and .

The PPN metric (46)-(48) is restricted to theories that exhibit conservation laws for total momentum and ignores the Whitehead and preferred-frame effects mash89 . We assume that the gravitational source is a uniformly rotating and nearly spherical body that is symmetric about the axis of rotation (i.e. the -axis). We are interested in the exterior gravitational field of the source when its center of mass is at the origin of spatial coordinates. The gravitoelectric potential is given in this case by 5

 U(ϱ,θ)=GMc2ϱ[1+∞∑n=2Jn(ϱeϱ)nPn(cosθ)], (49)

where is the equatorial radius of the source, is in effect the asymptotically measured mass of the source, is the Legendre polynomial of degree and

 Jn:=1Mϱne∫μ(ϱ′,θ′)ϱ′nPn(cosθ′)d3ϱ′. (50)

Here denotes the effective mass-energy density of the source. In a similar way the multipole expansion of the gravitomagnetic vector potential can be expressed as 5

 H∗i(ϱ,θ)=G(J×ϱ)ic3ϱ3[1+∞∑n=1Kn(ϱeϱ)nP′n+1(cosθ)], (51)

where is in effect the asymptotically measured angular momentum of the source and . Here

 Kn:=22n+3Mϱ2eJ(Ln+Jn+2) (52)

and

 Ln:=1Mϱn+2e∫μ(ϱ′,θ′)ϱ′n+2Pn(cosθ′)d3ϱ′. (53)

The derivation of the clock effect involves the computation of the proper time over a complete azimuthal cycle along geodesic orbits about the source. For simplicity, we limit our discussion to circular geodesic orbits in the equatorial plane, i.e.  and .

The radial geodesic equation, corresponding to a circular orbit in the equatorial plane, is given by

 Γϱαβdxαdτdxβdτ=0, (54)

which can be written as

 (cdtdϕ)2+2(cdtdϕ)Γϱ0ϕΓϱ00+ΓϱϕϕΓϱ00=0. (55)

It is straightforward to show that and . Using equations (46)-(48), we find that

 g0ϕ=−14(4γ+4+α1)H(ϱ,θ)sin2θ, (56)

where

 H(ϱ,θ)=GJc3ϱ[1+∞∑n=1Kn(ϱeϱ)nP′n+1(cosθ)], (57)

and . The solution of equation (55) can then be written as

 dtdϕ = ±∣∣c2ϱU,ϱ∣∣−12[1+(β+γ)U+12γϱU,ϱ] (58) +18c(4γ+4+α1)H,ϱU,ϱ+O(c−3).

It follows from the PPN metric that

 (dτdϕ)2=(1−2U)(dtdϕ)2−1c2ϱ2+O(c−3). (59)

Using equation (58), we find after some algebra that

 dτdϕ = ±∣∣c2ϱU,ϱ∣∣−12[1+(β+γ−1)U+12ϱ(γU,ϱ−|U,ϱ|)] (60) +18c(4γ+4+α1)H,ϱU,ϱ+O(c−3).

Integration of this equation immediately yields ; hence,

 τ+−τ−=π2c(4γ+4+α1)H,ϱU,ϱ+O(c−3) (61)

gives the gravitomagnetic clock effect within the restricted PPN framework adopted here. The explicit dependence of the gravitomagnetic clock effect on the PPN parameters is through the proportionality factor of ; in fact, the clock effect has this feature in common with other main gravitomagnetic effects mash89 .

It is interesting to note that in general relativity the gravitomagnetic clock effect in the post-Newtonian approximation is given by

 τ+−τ−≈4πJMc2[1+(32J2−92K2)ϱ2eϱ2], (62)

when the source is assumed to be symmetric about the equatorial plane and all moments higher than the quadrupole are neglected. Using data given in 5 , we find that for the Earth and , so that gives the relative contribution of the oblateness of the Earth to the clock effect for a near-Earth equatorial orbit.

## 5 Detection of the Gravitomagnetic Temporal Structure

According to Eq. (43) and the discussion following it, the orbital motion of free clocks around a rotating mass gives rise to the gravitomagnetic clock effect which shows up in the difference between the proper orbital periods of co- and counter-orbiting clocks. This is given by for equatorial trajectories. Inserting the specific angular momentum of the Earth ( m) yields an amazingly ”large” value of s.

Despite this seemingly large effect, the actual measurement of this time difference encounters severe practical difficulties. Since the two clocks are assumed to move along opposite but identical orbits, their Kepler periods exactly cancel upon forming the difference , thereby revealing the gravitomagnetic clock effect. In reality, however, clocks cannot be injected into identical trajectories and the resulting difference in the Kepler periods will readily exceed the time difference induced by the rotation of the Earth. Since for near-Earth orbits a radial separation of 0.1 mm of the clocks involves a time difference in the Kepler periods of the same order of magnitude as the gravitomagnetic clock effect, the position of the clocks has to be known at the submillimeter level in order to filter the effect which is caused by the rotation of the Earth out of the data. Similary, as the satellite moves just under 1 mm within s along its track (or milliarcseconds in angular distance), the azimuthal position has to be known at the same accuracy as the radial one. On the other hand, the gravitomagnetic time difference accumulates with the number of revolutions and after hundreds or thousands of periods a knowledge of the position of the clocks at the centimeter level will be sufficient to overcome the difference in the Kepler periods.

Another difficulty arises from the determination of all the forces that act on the satellites carrying the clocks. Since the period of revolution for orbits of km altitude is of the order of s, accelerations as weak as m/s will already cover the gravitomagnetic clock effect. Among these forces, gravitational perturbations due to the nonsphericity of the Earth, solid and ocean Earth tides as well as the interaction with the Sun, Moon and planets will cause the most significant deviations from an ideal orbit. Depending on the altitude of the satellites, the atmospheric drag effect can also considerably change the shape of the orbit. Moreover, this latter effect is quite difficult to model because it strongly depends on the atmospheric density which is not only correlated to the orbital height but also subject to temporal variations. Other non-gravitational perturbations like solar and terrestrial radiation pressure, thermal thrust, charged particle drag etc. must also be taken into account despite their less distinct influence, since they likely induce accelerations in excess of m/s.

In practice, the effect of all these perturbations will be modeled by determining a precise orbit based on the actual spacecraft observations. This will be accomplished by generating an orbital trajectory following Newton’s equations of motion and by including all perturbing forces acting on the satellite, using the most accurate models available. In the next step, this predicted orbit has to be best fitted to the one observed, where some force parameters may be solved for during the orbit adjustment process in order to obtain an improved or tailored force model for the specific mission. From the resulting precise orbit the effects of the individual perturbations are removed step by step thus yielding a quasi-Keplerian orbit, but still carrying the signatures of the relativistic effects. Finally, a comparison with the corresponding clock predictions for an appropriate synthetic orbit is performed which is expected to confirm the clock effect being investigated.

Therefore, in order to meet the very stringent conditions for the observation of the clock effect, many tiny perturbing sources have to be considered and investigated that are usually absent in most of the present orbit determination systems.

## 6 Quantum Origin of Inertia

The sign of the gravitomagnetic clock effect has a remarkable consequence that will be elucidated in this section. It follows from and that the uniform motion of the prograde clock is slower than that of the retrograde clock. Thus in comparison with motion around a nonrotating mass, a rotating mass would “drag” free test particles along such that it would take longer (shorter) to go once around it on an equatorial circular orbit in the prograde (retrograde) direction. We may call this circumstance virtual “inertial antidragging,” since it is the exact opposite of what would be expected on the basis of the so-called “inertial dragging” AE1 . In fact, as Fig. 2 clearly demonstrates, for a given (i.e. fixed orbital radius), the faster the Kerr source spins, the slower the prograde motion and the faster the retrograde motion. Figure 2: Plot of the clock rate versus the rotation of the source. Imagine an ensemble of Kerr fields with the same mass but different angular momenta. For a fixed stable circular geodesic orbit with “radius” r, τ+(τ−) monotonically increases (decreases) over the ensemble with increasing angular momentum of the source. Stable orbits of this type occur for r≥r±, where r±−3Rg±4a(2Rg/r±)1/2=3a2/r±. Let us note that r−≥r+ and the equality occurs for a=0 and r±=3Rg; moreover, r±/a→(3)1/2 as 2a/Rg→∞. The graph illustrates the behavior of τ± for an ensemble of Kerr black holes with 2a≤Rg; for 2a=Rg, r+=a and r−=9a. In the graph, τ0=τ±(a=0) and the radius r is chosen to be 5Rg.
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