In Hartle's book 'Introduction to General Relativity', he says that straight-line paths between two time-like separated points are the longest paths. He uses this in the case of Twin paradox, where the moving observer does not have a straight path between the same points, hence its 'distance' or the proper time is shorter. This is understood clearly using the time-dilation formula, but how are straight-line in non-Euclidean space longer? Any geometrical or mathematical explanation would do.
[Physics] Straight lines and longest distance
general-relativitygeodesicsmetric-tensorspacetime
Best Answer
The "distance" between two points(events) in space-time is given by,
$ds^2 = dt^2-d\bar{x}^2$
Where $d\bar{x}^2$ is the spatial part of the space-time. Now moving in a timelike straight line you can always choose a co-ordinate such that this spatial part is zero. And then in that co-ordinate $ds^2 = dt^2$. Now for any other path between this two events the extra term $d\bar{x}^2$ comes with a negative sign and therefor decreases the distance. Thus the longest distance will be straight line one.