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In an inertial frame of reference in flat space-time, the path of maximal aging will appear to be a straight line in that frame. In an accelerating frame , such as that of an accelerating spaceship, the same path through space-time will appear to be curved in space. One can explain why the curved path has maximal aging in an accelerated frame by invoking what is usually called "gravitational time dilation", which interacts with the usual velocity-dependent "time dilation" in special relativity to generate the path which satisfies the condition of maximal aging. Different frames have different "explanations" with various degrees of complexity, but everyone agrees on the end result.

The "straightforwards" approach is to compute the total elapsed time along any timelike path, then use variational principles to find the differential equation that represents the path of maximal aging.

Taylor wrote a fair number of papers on the principle of maximal aging and the related principle of "least action", some discussion and bibliographical references are available on his website https://www.eftaylor.com/leastaction.html.

It is worth noting that in some cases, one needs to use the principle of extremal or stationary aging, rather than a true maximum, but that's probably best left for another thread.