I am taking a course in introductory general relativity and came up on this question, which a google search didn't answer. The rotation of the Earth can be measured using a Foucault pendulum. The moon also rotates to always keep the same side facing the Earth, and it seems therefore that we should be able to measure this rotation using a pendulum on the moon. However, I have just learnt that a particle falling freely in a gravitational field is really following a geodesic curve through space-time. This got me wondering whether the apparent rotation of the moon is an artifact of the curvature of space-time, or if it is a real physical effect. I think my question can be phrased as in the title - would a Foucault pendulum rotate on the moon?
Another way of phrasing the question which may be clearer: If a (non-rotating) satellite were held still above the surface of the Earth, and then given a sudden sideways kick, putting it into orbit: would its orientation remain fixed relative to the stars, or would it rotate to always face the same side towards the Earth?
Simplifying assumptions: The Earth is a perfect gravitational point source, and the orbits are perfectly circular.