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*Timelike*

In Minkowski space, the proper time ##\tau=\int \sqrt{g_{ij}dx^i dx^j}## (+--- metric) is both maximized and stationary. Apparently this need not hold in some cases in GR, if there are multiple geodesics connecting two events; WP references MTW pp 316-319 on this, but I don't have my copy handy. Is there a simple example?

*Spacelike*

In Minkowski space, with n+1 dimensions for ##n\ge 2##, the proper length ##\sigma=\int \sqrt{-g_{ij}dx^i dx^j}## is stationary and is a saddle point. To see that it's a saddle point, pick a frame in which the two events are simultaneous and lie on the x axis. Deforming the geodesic in the xy plane does what we expect according to Euclidean geometry: it increases the length. Deforming the geodesic in the xt plane, however, reduces the length (as becomes obvious when you consider the case of a large deformation that turns the geodesic into a curve of length zero, consisting of two null line segments).

I don't know how much this has to be weakened for GR.

*Null*

This was discussed here last year: https://www.physicsforums.com/threads/null-geodesic-definition-by-extremisation.768196/ . I agree with some of the posts and disagree with others, and there didn't seem to be a consensus reached at the end.

In this case you have to define what "length" is. You can either take an absolute value, ##L=\int \sqrt{|g_{ij}dx^i dx^j|}##, or not, ##L=\int \sqrt{g_{ij}dx^i dx^j}##.

If you don't take the absolute value, L need not be real for small variations of the curve, and therefore you don't have a well-defined ordering, and can't say whether L is a max or min or neither.

Regardless of whether you take the absolute value, L doesn't have differentiable behavior for small variations around a null geodesic, so you can't say whether it's stationary; see https://www.physicsforums.com/threads/null-geodesic-definition-by-extremisation.768196/#post-4844366 , which seems to verify this aspect of my analysis.

If you do take the absolute value, then for the geodesic curve, the length is zero, which is the shortest possible. However, one can have nongeodesic curves of zero length, such as a lightlike helical curve about the t axis.