# I’m trying to understand the theory of curved space.

Why is the fundamental rule that the shortest distance between two points is a straight line thrown out the window when talking about long distances?

How could I travel in a straight line and end up at the same place?

This may be more IMHO.

Imagine a tube. You can draw a straight line along the tube’s surface, which will connect with its origin. Note that the line is only straight in 2 dimensions, but curved in 3. If space is curved in 4 dimensions like a donut or a ball, it’s possible to have a straight line that connects with itself.

As far as I know, long distances aren’t any different from short ones in this respect.

While the theory that traveling in one direction through space would eventually lead you back to your starting point if you traveled long enough was once put forward, I don’t think any mainstream scientist believes it today. But travel in that theory most emphatically was always in a straight line, technically a geodesic. A geodesic is always the shortest distance between points in any space, even if that space is curved. The geodesic of a plane is a straight line. On the surface of a globe, the shortest connector is obviously curved. Go all the way around a globe and you’ll return to your starting point. Expand that concept to a curved universe and there you are. So to speak.

Eh, nobody’s ruled out that space is shaped that way, and in fact some models make it seem rather more plausible than the alternative. But if so, then the distance you’d have to travel to wrap around is greater than the distance to the cosmological horizon, so there’s no real evidence for it. It’s not so much something that cosmologists disbelieve, as something about which we don’t know.
EDIT: I should also mention that this is a completely independent from the curvature of space. It’s possible for a non-curved space to wrap around on itself, and also possible for a curved space to not do so.

Well I can draw a line on the outside say a bicycle tire and have it connect with itself. That would be straight in one dimension. I’m a draftsman. How can a line draw around a tube be straight in 2 dimensions?

Travelling along a straight line and coming back to your starting point seems like a paradox, but it is all because “straight line” is a matter a definition in the first place.

In the classical, intuitive (and Euclidian) 3D geometry, a straigt line won’t come back to its starting point.

However, when we think about travelling on a sphere (the earth), then a straight line is, say, a line which is close to the ground and “always goes west” (for instance). Travelling along such a line would bring you back to your starting point.
But if you would just see the earth as a sphere in a classical 3D space, then you would say that such a line is actually curved (and forms a circle).

This example illustrates the fact that in various mathematical or physical problems, the definition of “space” and “straight” will change. And from the point of view of a mathematician or a physicist, our usual 3D space in which straight line don’t come back to their starting point is a special case.

No, in two dimensions. The tire is a surface (think about it, it has a surface), so the line drawn on the outside of a tire is a line in a 2D space.
A line in a 1D space would be a line drawn along a rope. Which is a little pointless since the rope itself is a line.

I find this intriguing. How would that work?

Cut open the tube and lay it flat.

The playing field in the game Asteroids wraps around, but is flat.

There’s some requirement here that I forget involving whether the corners are allowed to wrap or not. Someone else will come along any moment now…

If you can wrap a piece of paper around a surface without crumpling it, that surface is flat. So the surface of a cylinder, for instance, or the surface of a cone everywhere except the point, is flat, since you can wrap paper smoothly around those, but the surface of a sphere isn’t.

I’m following. I should have thought of an open-ended cylinder myself.

Right, we define “straight” to mean “geodesic”. But, technically, a geodesic from A to B isn’t required to actually be the shortest path from A to B; a geodesic just needs to be locally distance-minimizing. E.g., starting from some point on the Equator, I could travel 3/4 of the way around the Earth going due west, and this would be a geodesic, even though I could reach the same ending point with a shorter path which goes 1/4 of the way around the Earth going due east.

Always goes due west isn’t a great example, since this is only a geodesic along the Equator. Always stays on the same line of longitude works, though. And, indeed, a geodesic along a sphere is just one which looks like a line of longitude from some perspective. These are the paths which, if you walked along them, you wouldn’t feel like you were turning (around your head-to-toe axis).

Yes, this is exactly the example I was about to give. (But I don’t know what you are referring to with the corners; we do want the corners to “wrap around” same as everything else, but isn’t that already how it works in the game? It’d be odd to have four infinitesimal points that you’d just bump up against instead of moving through, while you still “wrap around” immediately next to them.)

Are you talking about “great circles” on a globe? It’s because you are looking at a map which is a distorted 2 dimensional representation of a 3D surface (the Earth). IOW, some parts need to be stretched on the map so the distances they represent on the map are longer than the actual distances.

Actually, a torus also has a flat metric. The best way to think about that is to consider it was a square in which the top and bottom edge are the same and similarly for its right and left edges. The square has its usual metric for points close together, although not for two points if one is close to the top and the other close to the bottom. Of course, top and bottom and left and right have no meaning except for us who appear to live in a Euclidean space. The thing that makes space flat or not is whether the sum of the angles of a triangle (that is a figure with three vertices and three geodesics joining the pairs) has angles that add to more than, less than, or exactly 180 deg. These angles are measured locally, using standard trig formulas.

For example, on the earth, there is a triangle whose vertexes are the north pole, the point where the prime meridian crosses the equator and the point on the equator at longitude 90 W (or 90 E, for that matter). All three angles of this triangle are exactly 90 and so they add up to 270.

On a saddle surface, the angle sum will be less than 180.