Gyroscope in a curved space

[Genady]

Is it correct that if a gyroscope is moved around in a curved space then after returning to the original location it may or may not return in the original orientation, depending on the trajectory?

[Markus Hanke]

Yes, that’s correct. Whether or not there is gyroscopic drift will depend on the background metric, and, depending on absence or presence of relevant symmetries, also on the path taken by the gyroscope. The total precession will be a combination of two separate effects, called deSitter precession and Lense-Thirring precession.

At least the deSitter precession part is never zero in a curved spacetime, so there will always be some gyroscopic drift.

[Genady]

Thank you. But at least if the second half of the path traces back the first half exactly, there will be no drift?

[MigL]

Is the precession of Mercury's orbit an example of deSitter precession, or geodetic effect, as it involves a spinning, orbiting body ?

Edited May 22, 2022 by MigL

[Genady]

AFAIK, the orbital precession doesn't depend on the planet's own spin.

[MigL]

This is what a quick search shows for deSitter precession 

"The geodetic effect (also known as geodetic precession, de Sitter precession or de Sitter effect) represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body. 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. Non-spinning bodies move in geodesics, whereas spinning bodies move in slightly different orbits.^[3]From   

The difference between de Sitter precession and Lense–Thirring precession (frame dragging) is that the de Sitter effect is due simply to the presence of a central mass, whereas Lense–Thirring precession is due to the rotation of the central mass. The total precession is calculated by combining the de Sitter precession with the Lense–Thirring precession."

From    Geodetic effect - Wikipedia

So even if Mercury's orbital precession is not deSitter precession, it should still exibit some form of deSitter precession in the deeply curved Solar gravity well.

[Markus Hanke]

3 hours ago, Genady said:

Thank you. But at least if the second half of the path traces back the first half exactly, there will be no drift?

Good question. This depends on the metric - in Schwarzschild spacetime there should be no net drift for this type of motion, but in other spacetimes that don’t admit time-like Killing fields (eg Kerr), there will still be a non-zero net drift, because the two sections are of different arc lengths in spacetime (!).

Actually proving this won’t be very easy, since this path isn’t everywhere smooth.

1 hour ago, MigL said:

Is the precession of Mercury's orbit an example of deSitter precession, or geodetic effect, as it involves a spinning, orbiting body ?

No, perihelion precession is different, because only the orientation of the orbit changes, which follows directly from the geodesic equation; deSitter and Lense-Thirring on the other hand affect the orientation of the spin axis of the body (these two precessions are in different directions). There’s also a fourth kind called Pugh-Schiff precession, which is essentially spin-spin coupling.

50 minutes ago, MigL said:

So even if Mercury's orbital precession is not deSitter precession, it should still exibit some form of deSitter precession in the deeply curved Solar gravity well.

Yes, rotating bodies in orbit within a gravity well experience this. I don’t know if it has been measured for Mercury, but there’s data on this for the moon:

https://www.academia.edu/49380218/Measurement_of_the_de_Sitter_precession_of_the_Moon_A_relativistic_three_body_effect

But again, this isn’t the same as perihelion precession, they are distinct effects.