# Is a straight line the shortest distance between two points?

Quite simply, I heard a lot of talk about how a straight line isn't necessarily the shortest distance between two points.

Is this true, and if it isn't, how would that work?

## migrated from physics.stackexchange.comMar 23 '16 at 3:29

This question came from our site for active researchers, academics and students of physics.

• That's not a physics question, it's pure geometry. The entire field of Riemannian geometry is about spaces where the shortest line is not "straight". – ACuriousMind Mar 22 '16 at 13:31
• I would agree that this is not a physics question but argue that a closely related one. At least as physical as say calculating the trajectory of a projectile. – linuxick Mar 22 '16 at 13:42
• @bytec0de : Define 'straight line'. – Qmechanic Mar 22 '16 at 13:45
• In terms of physics, the path of ray of light in vacuum is the shortest distance between two points. – Dirk Bruere Mar 22 '16 at 13:51
• I'm voting to close this question as off-topic because it is not about physics, and I consider it of too low quality to migrate it. – ACuriousMind Mar 23 '16 at 0:10

Here is a very non-technical answer: If our space was Euclidean then a straight line would be the shortest distance between two points. And until Einstein, through his general theory of relativity, showed that the space can actually be bent everybody believed and treated the space as Euclidean.

But now we know that the "physical" space is not Euclidean and therefore a straight line is not necessarily the shortest distance between two points. Consider for example being on the surface of a solid (impenetrable) sphere. The shortest distance between two points on the sphere is not a straight line.

• "The shortest distance between two points on the sphere is not a straight line." It's not straight when embedded in a 3D Euclidean space, but on the surface of a sphere those lines are as straight as it gets. – Emil Mar 22 '16 at 14:09
• Actually, even Special Relativity requires a non-Euclidean space-time. Euclidean space and the space of Special Relativity have different metrics. Special Relativity space-time is given the name Minkowski Space. – K7PEH Mar 22 '16 at 15:42
• I'm pretty sure that space is Euclidean in special relativity. Its spacetime that's non-Euclidean. – goblin Mar 23 '16 at 3:40
• I think an example with some math would be a nice addition. – inf3rno Nov 23 '17 at 8:06

If I recall what I've seen from Neil deGrasse Tyson correctly, he said that for what we currently have observed in the universe, a straight line is the shortest distance between two points. However, something that we have very healthily theorized about is worm holes. If worm holes exist, then you can travel through the worm hole to potentially travel less distance to get to the same point.

Think about it like a piece of paper. Of course, if you draw two points on opposite ends of the piece of paper, the least distance pathway is to draw a straight line connecting the two. What wormholes would do is what he describes as folding the piece of paper. So, if you fold the piece of paper so the points are closer, the least distance pathway would be using the worm hole instead of travelling along the piece of paper.

• The first point is already incorrect, since geodesics in GR are mostly not straight lines. The eye will perceive them as straight lines, but that's because our brain implicitly makes the assumption that light travels in straight lines. – Martin Mar 22 '16 at 14:47
• Also if you have a wormhole you can just draw a new straight line which is shorter so the statement still holds. – Jaywalker Mar 22 '16 at 15:29
• @Martin: your "geodesics in GR are mostly not straight lines" contradicts " Insofar as the term straight line has any meaning in curved spacetime it means a geodesic" here - this confuses me. – RedGrittyBrick Mar 22 '16 at 15:48
• @RedGrittyBrick: No, it doesn't. "Insofar as the term straight line has any meaning". That's just it - it rarely has any meaning. Hence geodesics are no straight lines. The problem here lies with the definition of "straight line". If you don't have any idea about non-Euclidean geometry, a straight line means a Euclidean straight line. The thing you draw on a flat piece of paper with a ruler. In non-Euclidean geometry, this is often not possible, so people sometimes might define "straight line" to refer to geodesics, i.e. shortest connections between two points. – Martin Mar 22 '16 at 16:00
• I mean, it's not possible to draw a straight line as in "that thing you draw with a ruler on a piece of paper" on a sphere. You'll have to draw great circles - and circles are definitely not the Euclidean version of "straight line". – Martin Mar 22 '16 at 16:04