Geodesic (in)completeness in general metric frames

TL;DR

This paper introduces a frame-invariant concept of geodesic completeness based on physical time derived from oscillations in wave functions, linking geometry with particle physics.

Contribution

It develops a novel, frame-invariant notion of geodesic completeness using physical time from wave oscillations, extending geometric concepts to include particle physics aspects.

Findings

Frame-invariant generalised geodesic completeness defined

Physical time based on wave oscillations established

Links geometry with particle physics concepts

Abstract

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.