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What is the shape of a gravitational wave as it hits the Earth, particularly the time portion.

Does time start at normal speed, then slow slightly, and then return to normal speed?
Or does it start at a normal speed, slow down slightly, then speed up slightly, and then return to normal speed?

Those other questions only concerned whether time dilation exists. I'm more concerned with the shape of the wave form. So not the same questions at all.

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A simple monochromatic gravitational plane wave has two possible transverse polarizations. If the wave is traveling in the $+z$ direction, then the metric for one of the polarizations can be written in the simple form

$$ds^2 = -dt^2 + (1+h_+)\,dx^2 + (1-h_+)\,dy^2 + dz^2$$

where the small metric perturbation $h_+$ is wavelike:

$$h_+ = A_+ \cos{(k z-\omega t)}.$$

The metric for the other polarization is similar, but rotated by 45 degrees in the $x$-$y$ plane.

So, at least in the coordinate system where the metric takes this form, time does not speed up or slow down and distances along the direction of the wave do not grow or shrink. But distances perpendicular to the wave’s direction do grow and shrink, in a wavelike fashion.

For this polarization, when distances in the $x$-direction grow slightly, distances in the $y$-direction shrink slightly. So a circular ring in the $x$-$y$ plane would be distorted slightly into an ellipse elongated in the $x$ direction, then back to a circle, then to an ellipse in the $y$ direction, then back to a circle, etc.

When LIGO detects a gravitational wave, it is because the length of LIGO’s two “arms” oscillates. When one gets slightly longer, the other gets slightly shorter. But the effect is very small and hard to detect, because the amplitude $A$ of gravity waves that it is detecting is roughly $10^{-21}$. This tiny metric perturbation makes LIGO’s 4-km arms grow and shrink by less than one-thousandth of the size of a proton!

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  • $\begingroup$ This is nice. I just have a couple of minor quibbles. This metric is a low-amplitude approximation, not an exact solution to the field equations. Also, I don't think it's really correct to state that "time does not speed up or slow down" as a gravitational wave passes by. I don't think the whole notion of time speeding up or slowing down at a point is a meaningful one in GR. That is, we can't even say meaningfully whether time is speeding up or slowing down at a point in Minkowski space. $\endgroup$ – user4552 Nov 22 '18 at 18:47
  • $\begingroup$ @Ben Crowell I agree, it’s definitely a low-amplitude approximation, but the amplitude of the waves LIGO is detecting is very, very low. As for my sloppy comment about time... would it be correct to say “Nearby clocks remain in synchrony despite the wave”? $\endgroup$ – G. Smith Nov 22 '18 at 18:59
  • $\begingroup$ would it be correct to say “Nearby clocks remain in synchrony despite the wave”? I'm not sure that "nearby clocks remain in synchrony" is a meaningful statement in a spacetime that isn't static. Even in a stationary spacetime, synchronization doesn't have to be transitive. $\endgroup$ – user4552 Nov 22 '18 at 19:24
  • $\begingroup$ @Ben Crowell Then I’m confused about why we can talk about what is physically happening in the $x$, $y$, and $z$ directions (and observe it) but not in the $t$ ‘direction’. Can you clarify this for me? $\endgroup$ – G. Smith Nov 22 '18 at 19:36
  • $\begingroup$ @G.Smith So going back to my original question on wave shape, is there a shape to the plane wave, where it only goes shorter, then back to normal size, or does it go shorter, longer, back to normal? $\endgroup$ – foolishmuse Nov 23 '18 at 17:17
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This is written as if the metric for a gravitational wave was something like $ds^2=(1+f(t))dt^2-dx^2-dy^2-dz^2$. It isn't. A metric of that form is just Minkowski space described in unusual coordinates. General relativity doesn't really even offer us any way of describing the notion of whether time slows down or speeds up at a particular point in space or for a particular observer. For instance, if someone asks me whether such a speeding up and slowing down of time occurs for an inertial observer in Minkowski space, I don't think the correct answer is "no, time flows at a uniform rate for that observer," it's more like "the answer is undefined," or "compared to what?"

The actual form of a gravitational wave is that it's a tidal (vacuum) distortion of spacetime that is transverse. The wave can in theory be modulated in any way whatsoever. The actual waveforms we see are determined by the properties of the source. A binary black hole inspiral makes a characteristic chirp.

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The new report here, published today, shows three very nice examples of gravitational waves coming from the merger of two black holes on page 2. So I can see the wave shapes very clearly.

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