u/Beneficial-Peak-6765

From my understanding, general relativity says that spacetime has non-zero curvature. This curvature is caused by energy-momentum density. But, when I think of a curved space, I tend to imagine something like the game Hyperbolica, which seems like the proper analog to a 2D being living on a curved surface in R^3. I think it has something to do with general relativity describing a 4D curved spacetime rather than a 3D curved space, but I don't have all the details. Also, if space is curved, why do we ordinarily think of it as Euclidean? For example, if light makes a geodesic, why do we view it as curved when it goes past a massive body? Another example is that gravitational lensing seems to be light bending around a galaxy or black hole. Space seems very much Euclidean.

Another example again would be that it that game, if you circle something around something, you will accumulate rotation around it, called holonomy. However, if I circle around something without intentionally rotating, I end in the same direction I started at. Is this holonomy effect ever observed in real life?

Here are some videos to what I am describing

https://youtu.be/yY9GAyJtuJ0?si=v-PfuDq5pZe_0nhz&t=109 (spherical geometry)

https://www.youtube.com/watch?v=VYfWfrk5P7w (hyperbolic geometry)

I've heard physicist David Tong claim that when physicists discovered quantum mechanics initially, they did not know of matrices. This seems pretty incredible to me, as matrices were known by mathematicians for a long time before that. Also, there are things like the inertia tensor, which is described using a matrix (in his book). It just seems very strange if that were true.

https://youtu.be/8RcA2BignDU?si=N1YSDN8TjFg-G92k&t=889