Geodetic effect

For example, the vector could be the angular momentum of a gyroscope orbiting the Earth, as carried out by the Gravity Probe B experiment.

The geodetic effect was first predicted by Willem de Sitter in 1916, who provided relativistic corrections to the Earth–Moon system's motion.

De Sitter's work was extended in 1918 by Jan Schouten and in 1920 by Adriaan Fokker.

[1] It can also be applied to a particular secular precession of astronomical orbits, equivalent to the rotation of the Laplace–Runge–Lenz vector.

[2] The term geodetic effect has two slightly different meanings as the moving body may be spinning or non-spinning.

The geodetic effect was verified to a precision of better than 0.5% percent by Gravity Probe B, an experiment which measures the tilting of the spin axis of gyroscopes in orbit about the Earth.

This observer, however, sees a vector positioned at some other value of r as rotating at a different rate, due to relativistic time dilation.

Now, the metric is in the canonical form From this canonical form, we can easily determine the rotational rate of a gyroscope in proper time where the last equality is true only for free falling observers for which there is no acceleration, and thus

This leads to Solving this equation for ω yields This is essentially Kepler's law of periods, which happens to be relativistically exact when expressed in terms of the time coordinate t of this particular rotating coordinate system.

Then The −2m/r term is interpreted as the gravitational time dilation, while the additional −m/r is due to the rotation of this frame of reference.

For the acceleration due to uniform circular motion in flat Minkowski spacetime, Fermi Walker transport gives the Thomas precession.