Out of curiosity, from someone who has never worked with non-euclidean geometry, what does it mean for a path to be curved in non-Euclidean space? My outsider understanding of curvature is that the inside of a curve is shorter than the outside of the curve, whereas a line has the same length on either side (assuming we give these curves and lines some thickness). But, if the shortest path can be curved, what do we mean by curved?
I've found it's easier to think about this stuff in two dimensions. The surface of a sphere (or the Earth!) has non-Euclidean geometry.
Imagine two people standing some distance apart from each other at the equator. They both begin walking in straight-line paths due south. At first, their paths are parallel. But as they move toward the south pole, they begin to drift closer to each other, as though their paths were curving towards each other. When they reach the south pole, they bump into each other. But they were both walking straight forward following the shortest path to the south pole the whole time. The curvature of the surface causes their initially-parallel paths to converge.[1]
On a plane (which has Euclidean geometry), initially-parallel paths never converge.
[1] Don't take this too literally; the real planet Earth is three-dimensional, and its gravity keeps us on the surface. But mathematically, it's possible to describe a curved two-dimensional space without referring to any higher dimensions. When I talk about "the surface of a sphere", that's what I mean -- the surface is the entire 2D space.
Maybe it's worth adding that in this way of thinking (intrinsic geometry of the surface), great-circle paths have exactly the property the GP brought up about straight lines: neighboring paths aren't shorter on one side and longer on the other. (If you think of them as 3-d paths then there's a shorter path below vs. longer above, but that's not part of the intrinsic geometry.)
If two people are in parallel, they will make two parallel circles. If two people aimed at a singe point, they are not in parallel.
Space-time is 4d array: array of framebuffers. You can stretch your mathematical model all day long, but you knowledge must be mapped to reality somehow. In model we have space-time, while in real world we have "physical vaccum" ("something nothing" or "phaccuum", for short). I prefer to name that thing "ether", because I like that word.
> If two people are in parallel, they will make two parallel circles. If two people aimed at a singe point, they are not in parallel.
In spherical geometry, the equivalent of a straight line is a great circle. There are no parallel great circles. That's why I used the phrase "initially parallel" -- at the starting point, both people's paths are at a 90-degree angle to the great circle connecting their locations.
I didn't want to get into "locally flat" vs. "globally curved" in something that started as an ELI5 thread.
> In spherical geometry, the equivalent of a straight line is a great circle.
Yes, of course. If we substitute parallel lines with straight lines in spherical geometry and mix 2D and 3D spaces, then our mental model will be nonsensical but cute.
We found no evidence of fourth dimension in the real world, so we cannot map this cute mathemagical model to reality.
Picture curves on the surface of the earth. They seem flat locally, but if you go a mile north, a mile east, a mile south, and a mile west, you don't end up _exactly_ where you start. (In the northern hemisphere you end up a little east of where you start; in the southern, a little west.)
Same thing in general relativity: the metric tensor measures the failure of closed loops on each axis to not close perfectly, the way they would in Euclidean space.
Basically even as a small creature on earth you can 'figure out' about the curvature by carefully measuring small-ish loops. The same is true for spacetime, but the loops' deformities are even smaller.
It's not too complicated. Get a round ball of some sort. When you draw on the surface, that's a "non-Euclidean space".
Take a straight line down from the "north pole" of your ball to its equator. Draw another straight line around a quarter of the equator. Draw a third line back to the pole. You've just drawn a triangle with 3 straight lines and the angles add to 270 degrees.
A non straight line is just not the shortest distance between two points on that surface.
Distance from center as described by the path - so if the distance to center is held, but the trajectory is changed, the line will be expressed as a curve around a tehtered point to the center.
If the 'tether' is a gravitational link (meaning that the teather, is a constant pull against the trajectory, regardless of the trajectory, the object will continue to curve around center.
I cannot be of help here but I'd say that your concept of curvature is too informal, but more formal maths can deal with it, take a look at the wikipedia page for "Geodesic", the maths are way above my head but the diagrams are cool :D
