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If one could hypothetically stretch / squash / in some way distort a piece of space, an outside observer could tell by looking at the distorted object/space that it had changed because the difference is evident relative to their own undistorted space. I.e., I can tell you have been distorted because I can from the outside compare you to my lack of distortion.

So, how would an object or observer inside the distorted bit of space be able to tell that they had been stretched or squashed? The space around them, and thus the coordinates relative to themselves, are distorted with them and are unchanged from their point of view (this is assuming they cannot view the outside, undistorted space). Right? E.g. if I could only see myself and my ruler and none of the rest of the universe, and both my ruler and I were scaled in exactly the same way, could I tell?

Secondarily: this stemmed from me trying to wrap my head around whether mass and energy distort only spacetime, therefore creating curved paths for objects in that space to follow, or if objects are also distorted. I always assumed the latter, thinking that spacetime and objects were not independent, but then started to doubt because explanations of GR make it seem like spacetime is bent and then objects fall into these bent paths, themselves unchanged (meaning that they are indeed independent). Then, I thought about spaghettification around black holes and that contradicts this!

Please help me organise this mess of thoughts! I love thinking about GR semantics and subtleties. I've just started learning about it and it is just beautiful :)

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So, how would an object or observer inside the distorted bit of space be able to tell that they had been stretched or squashed?

It is important to understand that spacetime curves, which is not the same sort of deformation as stretching or compressing. Curved spacetime can lead to compressed or stretched matter as different parts of an extended object try to follow diverging or converging paths through spacetime. But that material compression is just ordinary pressure and can be measured with ordinary pressure sensors.

The curvature of spacetime describes tidal gravity. Meaning that curved spacetime is non-uniform gravity. So to detect spacetime curvature requires a large enough region of spacetime to see the non-uniformity of gravity. Such a region is called non-local in this context, even if it doesn’t access any information from outside the region.

For example, if you have a ball of non-interacting test particles, often described as coffee grounds, then in the presence of spacetime curvature you can see that ball stretch on one axis and compress on the other two axes, in response to tidal gravity across the ball. This and other similar measurements can show the presence of curved spacetime.

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What you have to keep in mind is that General Relativity is built upon the Equivalence Principle. The strong version of this principle is that locally, whichever spacetime you're working with will be Minkowski spacetime, i.e. it will be locally flat. That's the key aspect of the (strong) Equivalence Principle, it is local. Therefore a local observer will not be able to tell if they're present in a curved spacetime, this observer will have to make non-local measurements in order to tell if their spacetime is curved.

Take the gravitational field caused by the Earth. Locally, a free falling observer A will be in inertial motion (considering that their mass is small compared to the Earth's mass) since they're following a geodesic. However, consider another free falling observer B, and put A and B relatively near each other but far from the surface of the Earth (for the sake of simplicity, let's suppose they're equally far away from the Earth's surface). You'll notice that they will approach each other as they're falling towards the Earth, almost as if there's a "force" that's pulling them together. This is a tidal force that is present due to the curvature the Earth induces on spacetime. In contrast, if A and B are falling near the Earth's surface, they will not seem to approach each other because, locally, there is no apparent curvature and spacetime is flat.

As an extreme example, consider you're falling towards a sufficiently small black hole, so that your soles are facing towards the event horizon. Since the curvature of this black hole is so extreme, as you're free falling towards it, the nearer you are to it the stronger the pull is on your feet than it is on your head, i.e. there will be a tidal force that will eventually stretch you violently and you'll be spaghettified before you reach the horizon. In this example, you're not a local observer since the curvature is so extreme, i.e. locally you're not in Minkowski spacetime.

Edit: I'd like to add to your last point that matter is not external to spacetime, it is embedded in it. The reason why this curvature seems to leave this matter unchanged, is because gravity is very weak compared to the other three forces. You would need to have an extreme gravitational field in order for matter to be affected by it. That's why spaghettification happens. The electromagnetic forces that hold you together aren't strong enough to hold against the tidal forces caused by the black hole's curvature.

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  • $\begingroup$ This cleared it up perfectly for me - thank you! $\endgroup$
    – aac
    Nov 20 '21 at 16:46
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How did even the ancients discover that the earth is spherical ? ? By using a mathematical model, that showed compared to measurements that the earth was not flat.

In the same way, we are able to compare data with two models, the Newtonian gravity one and the General Relativity one , and see experimentally that the GR curvature exists.GPS is used by innumerable people , and they rely on the accuracy of its predictions. Those predictions use both special and General relativity in order to calculate coordinates. So it is evident that one can know if the space is distorted or not, even in a local environment.

Gravity even if modeled with General Relativity is not the only force in our universe. There are three other forces the electromagnetic, the strong and the weak. A general statement is that in bound systems,at their center of mass the curvature of space is too small to change the energy levels.The whole data gathered from the cosmos to create our current cosmological model depend that in the center of mass of the atom the energy levels do not change.

It is not simple and you will learn about it in your studies.

spaghettification around black holes

happens when the tidal gravitational forces induced on a body falling into a black hole are stronger than the binding forces of electromagnetism and the strong force, breaking the bindings.

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