Geodesic: maximal aging versus extremal aging From Exploring Black Holes, by Taylor and Wheeler, page 1-7:

Purists insist that we say not maximum reading but rather extremal reading: either maximum or minimum. This book contains only examples of maximum wristwatch time for natural motion. Still, let's try to keep the purists happy!...
Principle of Extremal Aging: The path a free object takes between two events in spacetime is the path for which the time lapse between these events, recorded on the object's wristwatch, is an extremum.

What is the distinction being made by the "purists", between maximum and extremum?
Is it because a clock might simply run backwards into the future, and change the condition from a maximum to a minimum?
 A: *

*OP has a point: An extremum is by definition a maximum or a minimum, but there is no minimal wristwatch/proper time for a massive object, at least not within the set of timelike virtual paths between 2 fixed spacetime events.


*Many authors are misusing the word extremum to mean a stationary or critical path,
i.e. they are including saddle points. Be aware that this misuse of terminology is quite common in the literature.
A: The book Gravitation by Misner, Thorne, Wheeler has an explanation on page 316:
https://archive.org/details/GravitationMisnerThorneWheeler/page/n339/mode/2up

"Notice that the word “maximum” in equation (13.24) has been replaced
by “extremum” in the statement (13.25). When [events] A and B and are
widely separated, they may be connected by several different geodesics
with differing lapses of proper time (Figure 13.2). Each timelike
geodesic extremizes [proper time] with respect to nearby deformations of
itself, but the extremum need not be a maximum. When several distinct
geodesics connect two events, the typical one is not a local maximum
(“mountain peak”) but a saddle point (“mountain pass”) in such a
diagram as Figure 13.2 or 13.3."

This text apparently has the issue identified by Qmechanic, of including saddle points in the term "extremum". Confusing.
