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When a gravitational wave is present in LIGO, how does the splitter mirror work to send out a different frequency than the laser frequency on the arm perpendicular to the laser direction? What are the EM boundary conditions observed at the mirror surface that allow this?

The standard explanation (reference) of how LIGO detects a GW is in Local Lorentz Coordinates (which I think implies an unaffected observer outside the region of the GW). For a properly oriented GW, the observer sees one arm of LIGO compressed and the other expanded. These strains are thought of as approximately static, requiring just a few bounces of laser light along the L=4 km arms to measure the strain at that instant of time for the $\lambda _{GW}=3000 \ km$ GW. Furthermore, the explanation states the wave length of the laser light is strained just like the arm of LIGO the light is travelling along is strained, and that the speed of the light along each arm is the same value c. LIGO then detects the GW because a laser wave front takes a short time to travel back to the splitter along the shorter arm and a longer time along the longer arm, thus changing the interference pattern. Since the observer sees a light wave obeying $\lambda = c/ \nu$ along each arm, $$ \frac {L_1}{L_2}=\frac {\lambda _1}{\lambda _2}=\frac{c/ \nu _1}{c/ \nu _2} $$ Therefore, the observer sees different frequencies $\nu _1$ and $\nu _2$ along the two arms. How does the observer explain the details of the splitter mirror that puts out different frequencies (electric field peaks/sec) in the two directions? If no such mirror process can be conceived of, what is wrong with the standard explanation of what a GW does to LIGO?

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The mirror doesn't change the frequency of the light, the GWs do. If you want to work out the EM boundary conditions, you would have to do EM in a curved background spacetime. The GWs change the differential operators in Poisson's equation.

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  • $\begingroup$ Thank you for your answer. Rederiving BCs at the mirror using covariant derivatives in Maxwell's equations might well be the answer. I have not solved this problem before. However, I guess that the BC would have to be time dependent to change frequencies. The BC must also show ratio of frequencies is the ratio of the strain amplitudes done to the wavelengths. I now think the answer to my question is that the GW does not strain the light wavelengths, and thus the light freqs along the two arms are the same. Straining the wavelengths is not necessary for LIGO's time delay argument to work. $\endgroup$
    – Gary Godfrey
    Commented May 26, 2020 at 9:15
  • $\begingroup$ I have not done this calculation either, so I don't know how it will work out in the end. One thing to remember is that in the TT gauge there is no coordinate acceleration due to a passing GW. As a GW passes two test masses, the coordinate location of each does not change, but the proper separation between them does. In the TT gauge the coordinates move with the wave. $\endgroup$
    – Paul T.
    Commented May 26, 2020 at 11:58
  • $\begingroup$ Yes, we could think in TT where the coordinate difference between the mirrors does not change, but instead the metric is strained (and hence c is different along each arm). Then the light wavelengths on the two arms must change to keep the two freqs the same to avoid the mirror BC quandary. So, in LL mirrors move, wavelengths unchanged, metric (c) unchanged, and prop sep changes. In TT mirrors don't move, wavelengths change, the metric (c) changes, and prop sep changes. Wavelength and mirror position not straining together seems strange, but I can't think of a GR reason why they should. $\endgroup$
    – Gary Godfrey
    Commented May 26, 2020 at 17:29

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