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

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 nongravitational forces, and finally fictitious forces, either locally at the test masses or nonlocally 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.