So, when we are talking about geodetic precession, my intuition tells me that the angular momentum vector of the spinning particle should be parallel transported along it's orbit. Because parallel transport of a vector around a closed loop doesn't reproduce the same vector, then the parallel transported vector will not point in the same direction as the original, vector, giving precession. Is this the correct intuitive understanding of geodetic precession? Also, I just finished reading Wald's chapter 4, and in the problems he introduces the Lense-Thirring effect. In the homework you show that a space-like vector S at the origin inside a spherical shell of mass rotating at angular frequency omega parallel transported along the geodesic (purely time-like, i.e., you are "at rest" at the origin) will precess due to the rotation of the outside spherical shell. I understand if the vector is like, e.g. an angular momentum vector and it would precess, but what if I just put a solid rod "free falling" (so it's really really short) at the center? Would the rod start to get dragged along with the rotation? (i.e. if I placed it somehow "at rest" w.r.t. the outside stars who's gravitational fields are negligible inside my bubble, would it start to rotate with the bubble?) Would this be a confirmation of Mach's principle, in that the exterior gravitational field determines what it means to rotate or not, rather than there being absolute rotation?