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I never had a problem accepting that spacetime is curved as a result of matter, until I learned the LIGO experiments showed that evidently the curvature of spacetime can be measured. This, to me, is very strange.

Suppose the entire universe is empty except for two people, floating in space 100 meters apart. Since their masses are so small, spacetime is almost entirely flat compared to the conditions on the surface of Earth. Then suddenly, some incredibly huge cosmic event happens, and unthinkably large gravitational waves pass through the area. Somehow this doesn't kill them. Spacetime goes all spaghetti, and these two dudes are just shaking all over like a jellyfish.

But if space itself is changing, how would they know? They should observe, at all times, that they are still 100 meters apart, since they are not moving in space, but space is moving with them in it... right? Or maybe would each guy think that he is remaining still while the other one just goes totally nuts?

If the changing curvature of spacetime can be measured, what if these two guys were each holding one end of a 99 meter pole? Seems to me being measurable would imply that or one (or both) would have to lose grip and then have to dodge the end as it comes back.

This is probably a well-explored question and I just don't know the answer. I looked around on physics.se and saw a lot of questions like this very informative one and this one that assumes the effects are noticeable but none of them seem to be quite what I'm looking for.

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    $\begingroup$ phys.org/news/2016-02-ligo.html Obviously it would be impossible to detect a gravitational wave with only one object, there would be no difference observed. $\endgroup$
    – Stuckelberator
    Commented Feb 8, 2017 at 21:27
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    $\begingroup$ How about Einstein's famous predictions about the precession of Mercury and bending of light by the Sun - both explained by curved spacetime. $\endgroup$
    – jamesqf
    Commented Feb 9, 2017 at 0:19
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    $\begingroup$ Are you also confused about how tidal forces can tear you apart? The principle is rather similar. It takes considerable force to force your molecules apart, which requires a huge spacetime curvature. The waves we're observing are barely detectable with some of the most sensitive equipment on Earth - we're measuring extremely tiny dilation differences between points that are quite a bit far away. They're nowhere near curved enough to "go spaghetti". All those animations you've probably seen are massively exaggerated - LIGO measured a change 10000 times smaller than a proton in 4km arms. $\endgroup$
    – Luaan
    Commented Feb 9, 2017 at 10:32

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Analogy: how do we know that the surface of the Earth is curved? Well, we could e.g. draw a triangle on the surface of the Earth, and check the sum of the corner angles. If the Earth was flat, you'd always find that the sum of these angles was 180°, so it would be impossible to e.g. create a triangle with two 90° corners. However, since the Earth is curved, this is indeed possible; you could e.g. draw a triangle where one edge follows the equator, and the two others follow meridians from the equator to the north pole.

The same concept would apply to spacetime: simple geometric relationships such as e.g. the sum of corner angles in a triangle would be different in flat spacetime from curved spacetime, and these relationships should be possible to measure to figure out the curvature of spacetime itself.

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    $\begingroup$ To nitpick a bit, how can you distinguish between a curved surface on earth and a flat surface on earth that happens to lie in curved space? $\endgroup$
    – orlp
    Commented Feb 9, 2017 at 2:14
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    $\begingroup$ @orlp If one is nitpicking in this manner, and if I understand correctly what you mean, there is no such thing as a flat surface on Earth that happens to lie in curved space. The surface of the Earth is curved, and as a consequence, any part of that surface - in your words, any surface that is on Earth - is curved. It may still be possible to have a flat surface that touches the Earth's surface, but the flat surface will stick out into space. $\endgroup$
    – David Z
    Commented Feb 9, 2017 at 6:35
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    $\begingroup$ @orlp It's not just space that is curved, it's the spacetime. People tend to forget that. The geometry of spacetime is far more complex than the geometry of a sphere. A flat surface on Earth that happens to lie in curved space would give you gravitational attraction in an unexpected direction (in fact, as far as we know, it is outright impossible - it would exhibit gravitational repulsion, still entirely unobserved phenomenon), and to get enough curvature in spacetime would make the force great indeed, as well as substantial time dilation etc. $\endgroup$
    – Luaan
    Commented Feb 9, 2017 at 10:24
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    $\begingroup$ @DavidZ A flat surface touching Earth smaller than 8010km in its longest dimension won't stick out into space. - assuming it doesn't bend. $\endgroup$
    – Mindwin Remember Monica
    Commented Feb 9, 2017 at 11:19
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    $\begingroup$ There was a story I read where they used a similar idea to measuring the angles of triangles. In it advanced aliens used cosmic strings as weapons and to detect the presence of cosmic strings nearby the human protagonists measured changes to the value of PI. $\endgroup$
    – slebetman
    Commented Feb 10, 2017 at 1:46
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For the two observers you describe, it's of course true that if the observers moved about in space they would see each other moving about in space. It's also true that if they remained 'floating' in a space that was itself 'moving', they would also see each other moving about in space. That is to say, upon the passage of a large gravitational wave between them, each observer would 'feel' nothing, providing tidal forces can be neglected, but they would see the other observer moving around.

So what if these two observers were each holding the end of a pole? As the gravitational wave passes through the pole, they would each feel it pushing and pulling on them. As you suggest, indeed one or both would lose grip of the pole if this cosmic event was violent enough. To give a more 'down-to-earth' analogy (pun to become apparent), if you placed two observers a very large distance from the earth with a very long pole connecting them, then as they fell towards the earth in the radial direction, their trajectories would tend towards one another, and each would feel the pole actively pushing them outwards. The situation here is quite analogous.

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Then suddenly, some incredibly huge cosmic event happens, and unthinkably large gravitational waves pass through the area... But if space itself is changing, how would they know?

Richard Feynman's sticky bead argument:

It is simply two beads sliding freely (but with a small amount of friction) on a rigid rod. As the wave passes over the rod, atomic forces hold the length of the rod fixed, but the proper distance between the two beads oscillates. Thus, the beads rub against the rod, dissipating heat.

See this question for more details. Here is an animated visualization:

enter image description here

Here is another way that more closely matches the way gravitational waves are detected in practice, based on interferometry:

enter image description here

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