  The geometric concept of geodesic completeness depends on the choice of the
metric field or "metric frame". We develop a frame-invariant concept of
"generalised geodesic completeness" or "time completeness". It is based on the
notion of physical time defined by counting oscillations for some physically
allowed process. Oscillating solutions of wave functions for particles with
varying mass permit the derivation of generalised geodesics and the associated
notion of completeness. Time completeness involves aspects of particle physics
and is no longer a purely geometric concept.
