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The measurement of frequency shifts for light beams exchanged between twotest masses nearly in free fall is at the heart of gravitational wavedetection. It is envisaged that the derivative of the frequency shift is infact limited by differential forces acting on those test masses. We calculatethe derivative of the frequency shift with a fully covariant, gauge-independentand coordinate-free method. This method is general and does not require acongruence of nearby beams' null geodesics as done in previous work. We showthat the derivative of the parallel transport is the only means by whichgravitational effects shows up in the frequency shift. This contribution isgiven as an integral of the Riemann tensor --the only physical observable ofcurvature-- along the beam's geodesic. The remaining contributions are: thedifference of velocities, the difference of non-gravitational forces, andfinally fictitious forces, either locally at the test masses or non-locallyintegrated along the beam's geodesic. As an application relevant togravitational wave detection, we work out the frequency shift in the localLorentz frame of nearby geodesics.

Derivative of the light frequency shift as a measure of spacetime curvature for gravitational wave detection