Notation

To save putting a lot of primes on quantities, large and small letters will be used for quantities measured in two different reference frames. In the model to be used here, there is a single massive source $M$ at the origin of the frame $S$, in which the coordinates are denoted by:

$$X^{\mu} = \{X^0,X^1,X^2,X^3\}.$$ (1)

In this reference frame, the metric tensor will be

$$G_{00}=-(1+{2 \Phi \over c^2}); \quad G_{0i}=0; \quad G_{ij}=\delta_{ij}(1-{2 \gamma \Phi \over c^2});$$ (2)

where latin indices run from 1 to 3 and greek indices run from 0 to 3 as usual. The factor $\gamma$ in the spatial part of the metric tensor is inserted so that later I can identify whether a particular contribution to gyroscopic precession comes from space curvature (the term in $\gamma$) or the ordinary gravitational potential, in $G_{00}$. The fact that there are no space-time terms in the metric tensor means that in this frame of reference there are no gravitomagnetic effects.

I must specify also the order to which the calculations will be carried.Let me adopt the convention that a velocity term of the order $V/c$ or $v/c$ is of "Order(1)", or $O(1)$. In the 00-component of the metric tensor, arising from solutions of the field equations, only even orders occur. In the problem at hand, it is not necessary to go to fourth order, only terms of $O(2)$ need be considered. Similarly in the spatial part of the metric tensor, only terms of $O(2)$ will be kept. The space-time cross terms have to be computed to $O(3)$, however. This approximation scheme should be kept in mind during the calculations to follow.