  The measurement of frequency shifts for light beams exchanged between two
test masses nearly in free fall is at the heart of gravitational wave
detection. It is envisaged that the derivative of the frequency shift is in
fact limited by differential forces acting on those test masses. We calculate
the derivative of the frequency shift with a fully covariant, gauge-independent
and coordinate-free method. This method is general and does not require a
congruence of nearby beams' null geodesics as done in previous work. We show
that the derivative of the parallel transport is the only means by which
gravitational effects shows up in the frequency shift. This contribution is
given as an integral of the Riemann tensor --the only physical observable of
curvature-- along the beam's geodesic. The remaining contributions are: the
difference of velocities, the difference of non-gravitational forces, and
finally fictitious forces, either locally at the test masses or non-locally
integrated along the beam's geodesic. As an application relevant to
gravitational wave detection, we work out the frequency shift in the local
Lorentz frame of nearby geodesics.
