Time dilation for clock thrown up and caught back

[George Jones]

I feel your Edit... I mean pain. This to me is all the more reason to improve the terminology across physics so issues like "locality" are split into specific terms of art.

I'm on my way out with my family, so I can only make a brief comment. To add to the confusion, physicists and mathematicians often mean completely different things by "local"!

 

[Dale]

I just want to make a few comments about my earlier posts in this thread that sparked this current discussion. This is all according to my own limited knowledge:

A geodesic is a locally extremal path. The path itself may be very long, the word "local" in this context refers to the calculus of variations idea of entire functions that differ from one another by an infinitesimal amount. It in no way implies that the domain of those functions need be small. Specifically, this "local" is in no way related to the "local" of the equivalence principle. It is more closely related to the concept of "local" in optimization where a minimum may be a local minimum, but not a global minimum.

In the case of clocks A and B, they are both geodesics, so they are guaranteed to each be local maxima. The paths differ from each other by a finite amount, so there is no contradiction in one being longer than the other. Both are local maxima, but at most one could be the global maximum. If one is the global maximum then no non-geodesic clock can possibly record more proper time between the two events.