Gauge invariance and the detection of gravitational radiation

Gravitational radiation is one of the most important predictions of Einstein’s general theory of relativity. It has been detected indirectly through its effect on the orbital period of the Hulse-Taylor binary pulsar but it has not yet been directly detected. Several gravitational radiation detectors have been built in an attempt to perform such a direct detection. These detectors are essentially large laser interferometers, and the usual explanation of how they work goes as follows: an interferometer, by using the interference of light beams that travel along different paths is very sensitive to changes in the lengths of those paths. According to general relativity, gravity is a distortion in the geometry of space. When a gravitational wave passes through the detector, it changes the lengths of the arms of the interferometer and this change is detected through its effect on the the relative phase of the two light rays. At first glance, this explanation sounds simple and clear. But on reflection some issues arise: one issue comes from thinking about the usual explanation for cosmoligical redshift, which is that the expansion of the universe causes a corresponding expansion in the wavelength of light. Applying this concept to the interferometer, if the wavelength of the light expands as much as the interferometer arm does, then there should be no change in phase and therefore no detection. Other issues arise from the fact that general relativity, as a theory of gravity, doesn’t just give predictions for the geometry of space, but also for the propagation of light and the motion of material objects. In addition to changes in the lengths of the interferometer arms then, one might expect additional effects due to the effect of gravity on the propagation of the light as it moves along the interferometer arms. Furthermore, the mirrors at each end of the arms are also subject to gravity, so one might expect an additional effect due to motion of these mirrors under the effects of the gravitational wave. Why are these additional effects not discussed in the usual explanation of how gravitational wave detectors work? Are these additional effects small enough to be negligible? But if so, then why are they? Are these additional effects absent? But again, if so, why are they absent? It turns out that these questions can be answered by a careful consideration of the role of coordinate invariance in general relativity. Though coordinate invariance in general relativity is not a subject easily at the command of most physicists, it turns out that it is analogous to gauge invariance in electrodynamics. In fact the line of reasoning used to resolve the issue of the properties of gravitational wave detectors is the same as that used to understand the role of gauge invariance in the Aharonov-Bohm effect. In this paper, we will first look at the issues that come up for the Aharonov-Bohm effect and the resolution of those issues. We will then show how the same line of reasoning applied to gravitational wave detectors serves to resolve the issues raised here.