  It is important to emphasize that the members of \(\mathcal{P}\) represent possible or actual paths of arbitrary massive particles that may have some internal structure such as higher order gravitational and electromagnetic multipole moments and that may therefore interact in complicated ways with various physical fields. In order to constructively establish the projective structure of spacetime, it is necessary to single out a subset of \(\mathcal{P}\), namely \(\mathcal{P}_{f}\), the set of possible or actual paths of spherically symmetric, electrically neutral particles (that is, the world lines of freely falling particles). However, the set \(\mathcal{P}_{f} \subset \mathcal{P}\), can be properly  characterized only after a coordinate system (differential structure) is available. Consequently, one must employ arbitrary  particles in the statement of those axioms that lead to the local  differential structure of spacetime. 