 For a path that is everywhere timelike, i.e., that does not involve speeds at or above that of light, it is natural to take as the parameter the proper time a particle or observer would experience along the path, that is, the time measured along the path by a natural clock, such as one based on the vibrational frequency of an atom. (There are also natural choices that one can make for spacelike paths, e.g., those that consist of points at a single “time”, and for null paths, those followed by light signals; however, because the spacelike and null cases add yet another level of technical complexity, we shall not discuss them here.) The physical interpretation of this sort of incompleteness for timelike paths is more or less straightforward: a timelike path incomplete with respect to proper time in the future direction would represent the possible trajectory of a massive body that would never age beyond a certain point in its existence. (An analogous statement can be made, mutatis mutandis, if the path were incomplete in the past direction.) 