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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.