Are pairs of events characterized by spacetime intervals even if the spacetime containing these events is not flat?

In Wikipedia, spacetime intervals are presented explicitly under the heading "Spacetime intervals in flat space"; apparently including a presentation of spacetime intervals for (all) pairs of events in flat spacetime.

But is it correct and understood that spacetime interval values, $s^2$, can also be attributed (unambiguously, up to a common non-zero factor) to (all) pairs of events in a (any) spacetime which is not flat?

is it correct and understood that spacetime interval values, $s^2$, can also be attributed (unambiguously, up to a common non-zero factor) to (all) pairs of events in a (any) spacetime which is not flat?

No. There can be multiple paths between the same events. And while each path can be broken into small sections that have a well defined interval and you could take the square root of each section of them and add them all up (getting a proper length of the path) and then take the square you will get different answers for the different paths.

And if you try to limit your paths to geodesics, some events have no geodesics between them and others have multiple geodesics between them with different proper lengths for different geodesics. (The proper length is the sum of the square roots of the $s^2$ for each little piece of path.)

for all all-timelike paths I'd evaluate the plain extremum

Consider the set $$\{(v,w,x,y,z)\in\mathbb R^5: v^2+w^2=1\}$$ and give the larger space the metric $ds^2=dz^2+dy^2+dx^2-dw^2-dv^2.$ This induces an obvious metric on the 4d spacetime as an isometrically embedded surface. Then there are arbitrarily long timelike curves and arbitrarily short timelike curves between any two events, I don't even know what you could mean by a plain extremum.

• isn't geodesic the shortest path? how can there be multiple geodesics of different length? Sep 9 '15 at 22:32
• @AliMoh The geodesic is not the shortest path. There is a unique geodesic between two events if they are sufficiently close (topologically). But two paths can both be geodesics as long as they aren't too close. Consider the most well known case of gravitational lensing. Two paths start out in different directions curve about a mass and get bent towards each other and cross. Or more at home notice that every line of longitude is a geodesic from the north pole to the south pole. Sep 9 '15 at 23:26
• @AliMoh Consider, e.g. the sphere. You can go the "long way" or the "short way" around a great circle. The "short way" is a minimum, the "long way" is a saddle point in curve space. Sep 10 '15 at 2:29
• Timaeus: "[...] sections that have a well defined interval and take the square root [...]" -- Do you know how to name the result of this operation, in general (then please consider contributing to physics.stackexchange.com/q/186745). "And if you try to limit your paths to geodesics" -- I wouldn't get involved with that before having evaluated intervals between all event pairs. Null intervals seem obvious anyways; for all all-timelike paths I'd evaluate the plain extremum; the all-spacelike paths would first have to be preselected. Sep 10 '15 at 5:17
• @user12262 If you have too many paths each with different length, there is no obvious interval. Limiting to geodesics was an attempt to have fewer paths, so that you might get an obvious interval. It fails because it can still have too many and can sometimes have none. Sep 10 '15 at 5:24