On the round-trip time for a photon propagating in the field of a plane gravitational wave

Abstract

A network of large-scale laser interferometers is currently employed for searches of gravitational waves from various astrophysical sources. The frequency dependence of the dynamic response of these detectors introduces corrections to their antenna patterns which in principle can affect the outcome of the associated data-analysis algorithms. The magnitude of these corrections and the corresponding systematic errors have recently been estimated for searches of periodic and stochastic gravitational waves (Rakhmanov M, Romano J and Whelan J 2008 Class. Quantum Grav. 25 184017). However, the calculation of the detector response in that paper followed the traditional semi-rigorous approach which does not properly take into account the curved nature of spacetime. The question then arises as to whether the results will be the same if the calculation is done within the rigorous framework of general relativity. In this paper, we provide such a derivation of the response of the detectors to gravitational waves. We obtain the photon propagation time from the solution of the equation for null geodesics and calculate the corresponding phase delay by solving the eikonal equation for curved spacetime. The calculations are then extended to include phase amplification from multi-beam interference in Fabry-Perot resonators which play an important role in the formation of the signal in these detectors.