Regardless of an expanding universe

[Yessem101]

In our universe, is the shortest distance between two points a straight line or a hyperbolic curve? (I'm debating this with my teacher)

Of course you're going to ask me for the definition of a straight line but I don't think I can define that relative to a curved line without getting unnecessarily rigorous, so just use the definition of a straight line that you would use synonymously with "a curve that has a constant slope."

Don't pick apart my definition and just answer me this quick one!

 

[Luuk Steitner]

I guess you're describing a straight line through curved space like in this image?

Although this grid/space is curved, the rectangles are still rectangles (correct me if I'm wrong). The line is not passing straight through the rectangles, which means the distance is actually longer going this way.
This is just how I think it is so maybe someone will correct me.

 

[Chalnoth]

I guess you're describing a straight line through curved space like in this image?

Although this grid/space is curved, the rectangles are still rectangles (correct me if I'm wrong). The line is not passing straight through the rectangles, which means the distance is actually longer going this way.
This is just how I think it is so maybe someone will correct me.

That's not quite it. Think, rather of a sphere. There just isn't any such thing as a perfectly straight line on the surface of a sphere, because the surface itself is curved. So what we use instead are called "geodesics", which are the shortest distance between two points (geodesics are sometimes also called straight lines, though perhaps this is misleading...geodesics are only actually straight when the surface itself is flat). On a sphere, a geodesic turns out to be a great circle (it's quite literally what you'd get if you were standing on the sphere and just picked any direction and walked straight that way, assuming you could actually walk straight).

So the answer is a geodesic, which is a generalization of a straight line. It's not going to be a hyperbolic curve except in some specially-chosen geometry.

 

[Yessem101]

So the answer is a geodesic, which is a generalization of a straight line. It's not going to be a hyperbolic curve except in some specially-chosen geometry.

I know what a geodesic is, but it seems like you're defining a straight line as the shortest distance between two points regardless of the space its in.

Let me ask my question in this way. The shortest distance between two points in some flat space forms a straight line, and every straight line in this flat space can be recursively defined through another straight line in the flat space. Converting the flat space to a curved space, without converting the straight lines, these straight lines still connect two points in the curved space, but are they the shortest distance between the two points in the curved space?

 

[Luuk Steitner]

I know what a geodesic is, but it seems like you're defining a straight line as the shortest distance between two points regardless of the space its in.

Let me ask my question in this way. The shortest distance between two points in some flat space forms a straight line, and every straight line in this flat space can be recursively defined through another straight line in the flat space. Converting the flat space to a curved space, without converting the straight lines, these straight lines still connect two points in the curved space, but are they the shortest distance between the two points in the curved space?

Isn't that what I pointed out with the image? In this situation your line would only look straight from the outside, but actually it's the line that's hyperbolic/curved relative to the space. Since you're traveling in this space - and not another dimension outside this space - traveling along this line would take longer.

 

[Chalnoth]

I know what a geodesic is, but it seems like you're defining a straight line as the shortest distance between two points regardless of the space its in.

Well, this is the way we often think of geodesics as being: the generalization of the straight line. In any case, whether you call a geodesic a straight line or not is just words. Obviously in the special case of flat space-time, a geodesic is what we normally think of as a straight line. In curved space (or space-time, depending), straight lines (in the usual sense) are impossible.

Let me ask my question in this way. The shortest distance between two points in some flat space forms a straight line, and every straight line in this flat space can be recursively defined through another straight line in the flat space. Converting the flat space to a curved space, without converting the straight lines, these straight lines still connect two points in the curved space, but are they the shortest distance between the two points in the curved space?

That entirely depends upon how you do the conversion. And bear in mind that doing such conversion always requires a stretching of said space In some cases, it can only be done with tearing (generally, you can always go from a flat space to a small region of curved space without tearing, but it's often impossible to do it globally; for instance, it's not possible to take a single sheet of paper and make an entire sphere surface without tearing).

The basic issue here is that there are a great many ways to deform a flat surface to make a curved one. Even if you could do it in such a way that the geodesics in some small area conform to straight lines in the flat surface before deformation, you'd have to do it specially, and even then it's typically going to be impossible for the mapping to hold for any significant distance.