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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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