Geodetic Precession as a Gravitomagnetic Effect

The first set of terms in the expression for $g_{0i}$ is responsible for geodetic precession, but it is clear from the above calculation that here it comes in as a gravitomagnetic effect. Let us take only the first terms,

$$g_{0i} = {\gamma+{1\over 2} \over c^3} \left[{\bf V} \cdot {\bf r} \Phi_{,i}-(\Phi_{,j} x^j) V^i \right]$$ (55)

and compute the rate of precession of a gyroscope at the origin. The gyroscope will be carried by parallel transport along the path of free fall, which happens always to be at the origin, $x^i=0$, in quasi-inertial coordinates. Thus the spin vector will satisfy

$${dS^k \over ds}=-\Gamma_{\alpha \beta}^k S^{\alpha}{dx^{\beta} \over ds}.$$ (56)

But in these coordinates, this greatly simplifies, since the path of the gyroscope is simply

$${dx^{\beta} \over ds}=\{1,0,0,0\},$$ (57)

because the gyroscope's center does not move, and $dx^0 = ds$. Therefore the equation of parallel transport reduces to

$${dS^k \over ds}=-\Gamma_{\alpha 0}^k S^{\alpha}=-\Gamma_{i 0}^k S^i,$$ (58)

because

$$\Gamma_{00}^k=0.$$ (59)

A deeper reason why the sum reduces to spatial terms only is that the spin vector satisfies the so-called "Pirani Condition", which requires that $S^0=0$ if the spin is at rest. However, we shall not go into this here. From the above equation, it is seen that the only Christoffel symbols that contribute are

$$\Gamma_{i 0}^k = {1 \over 2} \left(g_{0k,i}-g_{0i,k} \right).$$ (60)

This is enough to show what the precession is. Since from the above expressions for the gravitomagnetic components of the metric tensor we find

$$\Gamma_{i 0}^k= {\gamma+{1 \over 2} \over c^3}\left(V^i \Phi_{,k}-V^k \Phi_{,i} \right).$$ (61)

Then the equation of motion for the spin is

$${dS^k \over c dt} = {\gamma+{1 \over 2} \over c^3}\left[(V^k \Phi_{,i}-V_i \Phi_{,k}\right]S^i.$$ (62)

But this is just the form for a triple cross product:

$${d {\bf S} \over dt}={\bf\Omega} \times {\bf S},$$ (63)

$${\bf\Omega} = {\gamma+{1 \over 2} \over c^3} {\bf A} \times {\bf V},$$ (64)

where ${\bf A}$ is the acceleration of the origin of the quasi-inertial coordinates due to the gravity of the mass source:

$${\bf A} = - \nabla \Phi.$$ (65)

Even though the quantities A and V appearing in Eq. (65) are upper case, which means they are measured in the original reference frame, together they give rise to a gravitomagnetic field in the quasi-inertial frame which causes the gyroscope to precess.