Response of interferometric gravitational wave detectors

The derivation of the response function of an interferometric gravitational wave detector is a paradigmatic calculation in the field of gravitational wave detection. Surprisingly, the standard derivation of the response wave detectors makes several unjustifiable assumptions, both conceptual and quantitative, regarding the coordinate trajectory and coordinate velocity of the null geodesic the light travels along. These errors, which appear to have remained unrecognized for at least 35 years, render the standard derivation inadequate and misleading as an archetype calculation. Here we identify the flaws in the existing derivation and provide, in full detail, a correct derivation of the response of a single-bounce Michelson interferometer to gravitational waves, following a procedure that will always yield correct results; compare it to the standard, but incorrect, derivation; show where the earlier mistakes were made; and identify the general conditions under which the standard derivation will yield correct results. By a fortuitous set of circumstances, not generally so, the final result is the same in the case of Minkowski background spacetime, synchronous coordinates, transverse-traceless gauge metric perturbations, and arm mirrors at coordinate rest.

Figure 1

Spacetime diagram of the optical paths in a simple Michelson interferometer. The dashed lines lie in a constant time hypersurface. The observer measures the difference in the phase of light that has traveled along the two different optical paths ($\mathsf{A}$) and ($\mathsf{B}$). In the case of a common, monochromatic light source at the beam splitter this difference is the difference in phase of the light source at times $t-{\tau }_{\mathsf{A}}$ and $t-{\tau }_{\mathsf{B}}$.