Quasi-Inertial Coordinates

A frame of reference that is freely falling along some geodesic G (see the diagram in the previous lecture) will be introduced, with axes which are non-rotating. I refer to such non-rotating coordinates as "quasi-inertial", because the origin is still in free fall, but the rotation matrix $\Omega^{ij}$ introduced in the previous lecture will be identically zero. A gyroscope will be placed at the origin of this freely falling frame. Coordinates in the new frame will be denoted by

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

so that the position of the gyroscope will be at $x^k=0$ in this frame. The gyroscope will still precess, because no restrictions have been placed on the gyroscope itself. The precession is only being described from a different frame of reference, in which the reference frame axes are non-rotating. In the Gravity Probe B experiment, precession of the gyroscopes will be measured with reference to sighting lines to distant stars, so the reference frame we are constructing corresponds very closly to that experiment. The analysis shows that the only significant terms in the metric tensor near $x^k=0$ will be from $g_{0i}$. The precession of the gyrosope will therefore be entirely gravitomagnetic in origon. This discussion shows that gravitomagnetism depends very much on the reference frame used to describe phenomena.

A physical example of the effect of using different reference frames comes from transforming to a rotating reference frame fixed in the earth. In this reference frame, "fictitious" Coriolis forces come in as a consequence of the rotation. If phenomena near earth are viewed instead from a reference frame which is has its origin at the center of mass of the earth but which does not rotate, there are no such Coriolis forces.