Really interesting stuff. I had a random question, that I'm not sure makes sense. In the exaggerated illustration of the wave on a ring of dust in a vacuum, if I am one of those specks of dust, I would notice the other speck moving farther away and then closer it seems. Does it have an effect on my clock as well if I'm staring at an atomic clock?
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That is a tough question actually.
If I am not completely mistaken gravitational waves do not cause gravitational time dilation in principle (time components in g are unaffected). However they do change the spatial geometry.
But I think the wave would have to have a frequency comparable to atomic transition energies to affect the quantum chemistry of atoms in the atomic clock. I'm not sure.
If you measure time by dividing distance over speed of light then it should affect your measurement if it is quick enough because distances do change
Oh interesting. Yes I was not thinking too specifically about the clock. So it sounds like there's no extra time component to this wave. I guess I was asking since I was imagining that the wave took place on the time-space manifold... Whatever that means.... Weird!
@galotta: watch out that you seem to confuse coordinate time and proper time. The proper time is given by the integration of the line element (defined from the metric). So, a change of the spatial geometry (for one specific observer) does also lead to a time dilatation for that observer. In general, keep in mind that talking about the space component of the geometry or about the time component is observer depedent.
I never mentioned time dilation as an observable effect of gravitational waves though (which is also not the case for a stationary observer ). You are right in that it is observer dependent, I am implicitly referring to first order perturbation theory to minkowski spacetime for a stationary observer on earth which is just the simplest model assumption for the purpose of explaining and detecting gravitational waves.
If we are talking about atomic clocks you would have to solve a Schrödinger equation in explicitly time dependent curved spacetime which I would not want to make an uninformed guess about ^^
And if you measure e.g. the time it takes for light to travel an infinitesimal distance you still measure coordinate time i.e. proper time from the perspective of a stationary observer in only spacially perturbed minkowski space.
@galotta sorry I should have been more specific. I was referring to your comment to @eonwarped "If I am not completely mistaken gravitational waves do not cause gravitational time dilation in principle (time components in g are unaffected)."