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 🙂
Best Answer
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.