# Laser interferometric gravitational wave detectors

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 so I was trying to solve this excercise: Now I was able to find the eq. of geodetics (or directly by Christoffel formulas calculation or by the Lagrangian for a point particle). And I verified that such space constant coordinate point is a geodetic. Now, for the second point I considered $$ds^2=0$$

to separate the

$$dt=0$$

and find the separation time. But I don't know how to solve for a generic path of a light ray. So I considered that maybe the text wants a light ray travelling along x axis and the second along y axis.

I checked in other sources and all people make the same, by considering a light ray along x-axis and then setting

$$dy=dz=0$$

.

But when I substitute these in my geodesic equations it turns out that they are not true even at first order in A! So these people that consider a light ray travelling along x-axis, such as in an interferometer, are not considering a light geodesic. All of this if and only if my calculations are true.

So I know that if

$$ds^2=0$$

I have a light geodesic. And so it should solve my eq. of geodesics. But if I restrain my motion on x axis what I can say is that the

$$ds^2=0$$

condition now is on a submanifold of my manifold. So, the light wave that I consider doesn't not move on a geodesic of the original manifold but on one of the x axis. This is the only thing that came in my mind.

Is there any way to say that I can set

$$dy=dz=0$$

without worring? And if I can't set it how can I solve the second point? asked Sep 1, 2016
recategorized Sep 1, 2016

You don't have to substitute y' and z' in the geodesic equation but in the metrics to get the coordinate speed ( of a null geodesic moving in the x-direction ) and integrate it on the paths after a few approximations. There is a detailed answer to the 2nd question in Gravitational Waves Notes for Lectures at the Azores School on Observational Cosmology by B F Schutz. "Gravitational Waves and Their Mathematics" Arxiv is a good complement. THE MATHEMATICS OF GRAVITATIONAL WAVES answers to the why questions about the accepted methods.

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