# Decomposing geodetic/de Sitter effect into Thomas precession and spatial curvature

According to Rindler the geodetic effect can be considered as consisting of Thomas precession combined with the effect of moving through curved space.

Wolfgang Rindler (2006) Relativity: special, general, and cosmological (2nd Ed.) p234

However according to Misner, Thorne, and Wheeler, Gravitation, p. 1118, Thomas precession does not come into play for a freely moving satellite.

I think that although a freely moving satellite doesn't feel gravity, it's relation to an observer is still subject to lorentz transformations and hence Thomas precesssion.

So who is right ? Rindler or Misner, Thorne, and Wheeler ?

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The difference between Rindler's wording and the MTW wording is just a difference in the choice of coordinates.

Thomas precession in STR

First, what is the Thomas precession? It is a special relativistic effect so the original derivation of the Thomas precession only applies in flat spacetimes. In other words, Rindler's application of the Thomas precession in the context of general relativity requires one to specify how the flat special relativistic spacetime is identified with, or embedded into, the curved spacetime in general relativity.

The Thomas precession is a change of the angular momentum that a gyroscope undergoes when it is attached to another object whose velocity is changing and going along a curve in the space of velocities. Why does it occur?

Well, in special relativity, the velocity is a vector $u^\mu$ normalized so that $u^\mu u_\mu=1$. The space of such vectors is a two-part hyperboloid in a Minkowski space (its intrinsic signature is purely spacelike). Now, the angular momentum of a gyroscope moving together with an object is given by an antisymmetric tensor $J_{\mu\nu}$ that satisfies $J_{\mu\nu}u^\nu=0$. Now, if you try to parallel transport this tensor $J_{\mu\nu}$ along a path inside the hyperboloid, it will not return to itself because the hyperboloid is a curved submanifold. Instead, it will rotate by an angle around an axis (one that depends on the path in the space of velocities) - and this is what we mean by the Thomas precession.

General relativity

In the case of a gyroscope attached to a satellite in a gravitational field described by general relativity, we may describe the vicinity of the world line of the satellite as a piece of flat Minkowski spacetime. After a whole orbit, we return to the same place in space. However, the orbit won't be quite periodic in our "flat, thin, and long Minkowski cylinder" surrounding the world line. Instead, we will induce both a rotation of the velocity space as well as a rotation of the ordinary position space, caused by the actual curvature of the space. Rindler probably calculates the full effect by summing these two contributions - from the monodromy in the uniformly curved velocity space and from the monodromy in the actual spacetime curved by the presence of matter.

So I am pretty confident that Rindler did his job correctly and avoided any sign errors. Gravity Probe B has confirmed the result, after all.

MTW are arguably able to calculate the total result correctly, too. But they organize their calculation differently, attributing the whole effect to the curvature of space. Effectively, their coordinates for the "cylinder surrounding the satellite's world line" differ by a "twist" (a time-dependent rotation) from Rindler's cylinder - the two groups of relativists use different coordinates.

The MTW proposition seems to directly contradict Rindler's calculational strategy and I think that the MTW, in the very satellite context that is relevant and with the Rindler's choice of coordinates, is incorrect. A statement that would be true and similar to the MTW proposition is that if the satellite were freely moving in space and had "no intrinsic rotation" relatively to a chosen system of coordinates, then the velocity could be viewed as a "constant" and the Thomas effect would be exactly zero.

However, the coordinates chosen by Rindler contradict the assumption because the satellite itself - not just the gyroscope - is rotating in this system of coordinates (the gyroscope is also rotating, and differently). So the velocity itself is non-constant even in the flattened cylinder surrounding the world line, and the Thomas precession contribution is nonzero.

As you can see, the existence or absence of the Thomas precession depends on whether or not we consider the velocity of the satellite to be "constant" and this question depends on whether or not our "natural" (there is no natural one!) coordinate system is rotating relatively to other people's natural systems. If there is no Thomas precession in one system, it's because we embedded the special relativistic spacetime in such a way that the velocities stay "constant". However, if we choose a "twisted" coordinate system that is rotating with respect to the first one, the velocities will be non-constant and the precession will receive contributions from the Thomas precession.

By changing the coordinate systems, arbitrary parts of the overall precession after one period - that everyone has to agree with - may be moved from the Thomas precession contribution to the curvature-of-space contribution.

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Excellent answer. This is an example of something very common in relativity: the observed fact (precession) can "look like" different things in different coordinate systems. MTW are correct that, in a certain choice of coordinates, there is no Thomas precession, but to the extent that they seem to mean that this is a universal fact about the world rather than a fact about a particular choice of coordinates, their statement is misleading. – Ted Bunn Apr 4 '11 at 18:29
Thanks Luboš Motl for the detailed answer, and Ted Bunn for the helpful comment. – user2929 Apr 5 '11 at 14:52