Elementary question about non-Euclidean geometry in general relativity: “cannot move about without changing shape”

Question

One basic result of general geometry (from math) in curved spaces or on curved surfaces is that if you are in a surface of variable curvature, things like the Euclidean congruence postulates and theorems for triangles and other simple figures fail, and the reason this is is because those are actually strong statements about the homogeneity of the curvature of the space. Being non-homogeneous, they break down, and since as Euclid shows these are related to the notion of "moving something from place to place in space without altering its size or shape", it follows that in such a space, something CANNOT be moved about from place to place in the space without altering its size and shape.

It would therefore stand to reason this also holds in the variably-curved spaces of general relativity. So my question is, physically, what is the effect of being unable to move about from point to point without altering your size and shape?

Answer 1

It isn't necessarily objects that change. This is a 4 dimensional space with 3 "space" dimensions and 1 time dimension.

The definition of curvature can be stated as when you go around what should be a rectangle, you don't come back to the same place. Or equivalently, if you take the two different paths to the opposite corner, you wind up in two different places.

In a gravity well, time runs slower when you are deeper. So you can see curvature by considering a rectangle in the time-radial plane. Start at an event - a point in space-time. Wait 1 second and move 1 meter down, and note the event. Start over. Move one meter down and then wait 1 (slower) second. You arrive at the same place, but at a later event.