Gravitational Ultrarelativistic
SpinOrbit Interaction
And the Weak Equivalence Principle
Abstract
It is shown that the gravitational ultrarelativistic spinorbit interaction violates the weak equivalence principle in the traditional sense. This fact is a direct consequence of the MathissonPapapetrou equations in the frame of reference comoving with a spinning test body. The widely held assumption that the deviation of a spinning test body from a geodesic trajectory is caused by tidal forces is not correct.
PACS number(s): 04.20.Cv. 95.30.Sf
1 Introduction
The equivalence principle had attracted considerable attention in the past decades [1], especially after the criticism of Fock [2] and Synge [3] concerning its sense and role in Einstein’s gravitational theory. Now this principle gives rise to renewed interest, particularly in the context of new tests [4–9] (many papers have appeared for the last years [10]). In spite of the fact that in known books the equivalence principle is presented as a cornerstone of the general relativity theory, and a principle of prime importance (see, for example, [11]), the doubts generated by Fock and Synge have not disappeared completely.
The purpose of this paper is to show that there is an objective reason for the revision of the physical content and meaning of the equivalence principle in general relativity. This reason is based on results of a more careful analysis of the spinning test body deviation from the geodesic motion in the gravitational field.
As regards the influence of spin (inner rotation) of a body on its motion in the gravitational field, the prevailing opinion is presented particularly well in Ref.[11], exercise 40.8. According to this opinion the deviation of a spinning test body from the geodesic trajectory is caused by tidal forces connected with the spacetime curvature. The MathissonPapapetrou equations (MPE) [12, 13] are singled out as providing information on the behavior of a spinning test body in a gravitational field. However, the appeal to these equations in [11] is not sufficiently justified. The arguments are patently insufficient for assertion that the interaction between the spin of a test body and the spacetime curvature is reduced to the influence of tidal forces. Therefore, here we shall direct our attention to the MPE. Although these equations were extensively investigated in the 1960s and 1970s a number of problems remain. Particularly regarding the influence of spin on the world line of a test body [14]. Certainly, if one starts from general considerations, relying on the equivalence principle, then it is a priori clear that spin can only slightly deform the world line of a test body, as compared to the corresponding geodesic line. But there is another way, namely, to forget for the time being about the equivalence principle, and try to discover facts that follow directly from the MPE, without a priori restrictions.
Here we shall consider the consequences of the interaction between the spin of a test body and the curvature of a Schwarzschild’s field and compare these with properties of tidal forces in this field. In particular, it is known (see, e.g., equations (31.6) and (32.24a) in [11]) that for radial motions in this field tidal forces are determined by the nonzero components of Riemann’s tensor in the comoving frame of reference:
(in notation of [11] for local indices; is the Schwarzschild mass). Namely, the comoving frame of reference was used for the analysis of tidal forces in [11], sections 31.2 and 32.6. For the direct and correct comparison of tidal forces with the forces caused by spincurvature interaction it is expedient to consider the consequences of MPE in the comoving frame of reference also.
We shall investigate the spincurvature interaction not only for radial motions because the weak equivalence principle is formulated for general motions (see, e.g., section 2.3 of [14]). For example, it is applied to the usual falling lift, as well as to the satellites orbiting the Earth.
2 The MPE in a comoving frame of reference
The traditional form of the MPE is [12, 13]
(2) 
(3) 
where is the 4velocity of a spinning test body (STB), is the tensor of spin, and are, respectively, the mass and the covariant derivative. For the description of the test body center of mass, Eqs. (2), (3) are often supplemented by the relation [12, 15, 16]
(4) 
The correct definition of the center of mass for a STB is a subject of discussions [14, 17–22]. Below we shall investigate the MPE in the approximation linear in spin (in accordance with the linear consideration of tidal forces in [11]), when supplementary condition (4) coincides with the alternative condition proposed by Tulczyjew [19] and Dixon [20, 21]. Here we shall not consider discussions of [14, 17–22].
Besides , the 4vector of spin is also used in the literature where by definition [16]
( is the determinant of the metric tensor).
For transformations of Eqs. (2), (3) we use the known relations for the orthogonal tetrads ,
(5) 
( is the Minkowski tensor) and the conditions for comoving tetrads [16]
(6) 
(here and in the following, indices of the tetrad are placed in the parentheses; latin indices run and greek indices ). For convenience, we choose the first local coordinate axis as orientated along the spin, then
(7) 
and . is the value of the spin of a test body as measured by the comoving observer [16].
From Eq. (3), taking into account (4)–(7), we obtain , the known condition for the FermiWalker transport, where are Ricci’s coefficients of rotation [23].
From (2), after corresponding calculations, we find
(8) 
It is important that Ricci’s coefficients of rotation have a direct physical meaning, namely, as the components of the 3acceleration of a STB relative to geodesic free fall as measured by the comoving observer (one can see this fact from the equation of geodesic world lines in a vector tetrad space ). Therefore, by Eq. (8) we have
(9) 
3 Spincurvature interaction in
Schwarzschild’s field
For simplicity, we shall restrict ourself to the case of equatorial motions of a STB in Schwarzschild’s field when spin is orthogonal to the motion plane. Using the components of the metric tensor in the standard coordinates , , , and relations (5), (6) it is easy to find the nonzero components of the comoving tetrads:
(10) 
The nonzero components of the Riemann tensor in the standard coordinates for are given by
(11) 
(signature —,—,—,+). For calculation of the local components of the Riemann tensor we use the general relation
(12) 
Inserting Eqs. (10) and (11) into Eq. (12) we obtain the expressions for the components :
(13) 
By Eq. (9) these local components of the Riemann tensor determine the force of the spincurvature interaction from the point of view of a comoving observer.
Now we can compare components (13) with local (comoving) components of the Riemann tensor (1) which determine tidal forces for radial motions. It is easy to see that for radial motions, when , all components (13) are equal 0. That is, in this case the spincurvature interaction does not deviate the motion of a spinning test body from the geodesic radial motion. (This fact is also known from the partial solution of Eqs. (2)–(4) in the Schwarzschild field). At the same time, all components (1) and tidal forces for radial motions are not equal 0. [The correspondence between the local indices in Eq. (1) and the notation in (13) for radial motions is given by , , , . Then in accordance with (1) such components of the Riemann tensor in our notation are not equal 0:
(14) 
(). The expressions for these components follow directly from relation (10) (at ), (11), (12) and one can check that they coincide with the corresponding righthand sides of Eq. (1)].
We emphasize that each component of (13) has only one time local index, whereas such components are absent among those of Eq. (14). This fact is very important because the components of the Riemann tensor with one time index correspond to the ”gravitomagnetic” components of the gravitational field [24, 25]. The components of the Riemann tensor with two time indices correspond to the ”gravitoelectric” components of the gravitational field. According to the unnumbered equation preceding Eq. (32.24b) of [11] namely the ”gravitoelectric” components cause tidal forces. (The deep analogy between the ”gravitomagnetic moment” of a spinning test body in general relativity and the magnetic moment in electromagnetism was studied in [22]).
So, we cannot consider tidal forces as the reason for the STB deviation from the geodesic motion.
We can indicate two other arguments in support of the conclusion that the MPE do not contain tidal forces. For example, let us suppose that the tidal forces are present in the MPE. Then these forces cannot disappear if we make the spin equal to zero (more exactly, the angular velocityof the inner rotation) in the MPE, because the tidal forces are connected with the dimension of a test body and its nonrotating state does not remove these forces. However, if one puts , Eqs. (2), (3) pass to the geodesic eqs. and do not to the eqs. of the geodesic deviation. The geodesic eqs. do not contain tidal forces (in contrast to the geodesic deviation eqs., which do) and therefore the assumption that tidal forces are present in the MPE is not correct.
It is necessary to remember that the tidal forces will be taken into account if we consider two close world lines (see Ref.[26], Chap. 6, Sec. 10, where the clear procedure for derivation of geodesic deviation eqs. is given). However, the MPE, as well as the geodesic equations, describe only one world line. Therefore, we can point out the place in the procedure of the MPE derivation where the tidal forces were neglected: when the world tube of a test body was replaced with only one world line (see, e.g., Ref.[13], page 250).
The second additional argument, which refutes the assumption that presence of tidal forces is the reason of the STB deviation from the geodesic motion, is connected with the known fact that the MPE are the classical limit of the general relativistic Dirac equation. In a number of publications it is shown that the righthand side of Eq. (2) describes the interaction of a quantum electron with a gravitational field [27]. Obviously, one cannot speak about tidal forces in the Dirac equation.
We stress that the authors of Ref.[11] do not provide a proof of the statement that the tidal forces are present in the MPE; it is an assumption (hypothesis) only. The appearance of the Riemann tensor in the righthand side of Eq. (2) is not a sufficient argument for this statement, because this tensor has a number of physically different components and only part of them is connected with tidal forces. Our rigorous consideration, presented above, gives a direct proof of the conclusion that the righthand side of Eq. (9) does not contain any components of the Riemann tensor that are connected with tidal forces.
If tidal forces are not the reason for the deviation of a STB from the geodesic trajectory, then what is the reason? Considering (9) and (13) it is easy to answer this question. Indeed, for the value of 3acceleration of a STB relative to geodesic free fall, where
using (9), (13) we find for all cases of equatorial motions (not only for the circular orbits)
(15) 
where is the tangential component of the test body 4velocity. Even though Eq. (13) contains the radial velocity, when we calculate , the terms with cancel out due to the relation . (By (9), (13) one can check that vector is orientated towards the source of the Schwarzschild field). In accordance with Eq. (15) is nonzero only if . When the velocity of a STB is much less than the velocity of light, i.e. when , Eq. (15) corresponds to the expression (44) from paper [22], where the spinspin and spinorbit gravitational interactions were investigated in the lowest approximation in the velocity of a STB. One may consider Eq. (15) as the generalization of Eq. (44) from [22] for any velocities of a STB. This generalization is not trivial and contains significant new information on gravitational spinorbit interaction. Namely, when after Eq. (15) and the condition for a STB [22]
(16) 
we have , where is the Newtonian value of the free fall acceleration. In this case, if the dimension of a STB (and its value ) are sufficiently small, the is negligible and we can say that gravitational spinorbit interaction obeys the weak equivalence principle. However, another situation in principle exists in the ultrarelativistic region of velocities, when . Then, according to (15), for any small we can indicate such sufficiently large value for which the value of acceleration of a STB measured by the comoving observer will not be negligible. Therefore, the ultrarelativistic gravitational spinorbit interaction violates the weak equivalence principle in the traditional sense. Here we accent that the usual formulation of the weak equivalence principle is not restricted to the special case of the radial fall. This principle is applied to any motion in any gravitational field. The partial result that for from Eq. (15) we have cannot remove the necessity of reinterpretation of the weak equivalence principle. Expression Eq. (15) demonstrates the limit of validity of this principle in the traditional formulation.
Thus, in accordance with (15) and (16), the above two limiting processes are essentially different in their physical consequences.

The dimension of a test body and its spin tend to whereas the body velocity is limited from above and fixed (but as close as one likes to the velocity of light);

The dimension and spin of a test body is as small as one likes, but fixed, but larger and larger velocity is given to a body. In the first case, the motion of a STB tends to the geodesic motion, while in the second case a STB is moving away from it more and more. If in the first case the weak equivalence principle is fulfilled, then in the second case it is obviously violated. In regard to the corresponding control experiment one cannot assert that it does not depend on the velocity of the free falling device of Ref. [14], Sec. 2.3.
One can rewrite Eq. (15) in the form
(17) 
where is the orbital momentum of a test body. In Ref.[28] R. Micoulaut has investigated the connection between the orbital momentum and the velocity of a spinning test body for equatorial motions in a Schwarzschild field. From the results of [28] it is clear that is arbitrarily large for any solution of the MPE with the sufficiently large initial value of the tangential component of the test body 4–velocity. So, the MPE admit the motions with arbitrarily large . On the whole, for these motions is not const. We emphasize that Eqs. (15) and (17) are valid for all cases of equatorial motions of a STB in a Schwarzschild field, including
Naturally, under our earthly conditions we do not have the possibility to launch a macroscopic ultrarelativistic STB with the comoving observer for the registration of acceleration (15). For the ultrarelativistic elementary particles we cannot realize a comoving ultrarelativistic falling laboratory. [It follows from Eq. (15) that the electron flying near the surface of the Earth feels the acceleration equal to , i.e. to the Newtonian acceleration for the Earth, if the velocity of an electron corresponds to the energy of its free motion . One may obtain this value substituting in Eq. (15)]. Nevertheless, in principle, the possibility exists for a nondirect examination of Eq. (15) by an earthly observer, because for such an observer an ultrarelativistic electron or proton must change its electromagnetic radiation due to the additional nongeodesic acceleration connected with Eq. (15).
4 Conclusions
The importance of the expression (15) is not restricted to the assertion that the gravitational spinorbit interaction in the ultrarelativistic limit indicates the necessity for a more careful wording of the weak equivalence principle. Namely, according to Eq. (15) we cannot treat this principle at all as one that excludes the gravitational field as a force field and which, at the same time, interprets all gravitational actions as inertial actions. Quite on the contrary, we see a closer analogy between gravitation and electromagnetism. In electromagnetism for the demonstration of the presence of a magnetic field it is necessary to have, for example, a magnetic needle. Similarly, in the case of the gravitational field, the presence of a spinning test body, serving as a specific ”gravitomagnetic needle”, allows us to show that in a free falling frame of reference the gravitational field does not disappear. Only a test body without spin does not feel this field.
Thus, the traditional interpretation of the weak equivalence principle must be revised in principle. In this connection, we remember the considered judgement of Fock that the kinematic interpretation of gravity plays a heuristic role only. According to Fock, the true logical foundation of Einstein’s gravitational theory is not the equivalence principle but other two ideas, namely, the idea of spacetime unification in the united 4dimensional chronogeometrical manifold with an indefinite metric and giving up the ”rigidity” of the metric, which allowed to unite it with the phenomenon of gravity.
It is necessary to analyze the gravitational phenomena for which the weak equivalence principle is violated as a consequence of the gravitational ultrarelativistic spinorbit interaction. In all probability, the necessity will arise for a modification of the traditional picture of the gravitational collapse because it is based on the analysis of the geodesic world lines only.
Acknowledgment
The author would like to thank Prof. O.M.Bilaniuk for his support and encouragement and for a helpful reading of this manuscript.
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