A Local Variational Theory for the Schmidt metric

TL;DR

This paper develops a local variational framework for the Schmidt metric, demonstrating that minimal b-length causal curves are geodesics and generalizing a theorem about cluster curves in space-time.

Contribution

It introduces a variational approach to analyze b-length in space-time, extending classical geodesic results and generalizing Schmidt's theorem.

Findings

Causal curves of minimal b-length are geodesics in small globally hyperbolic sets.

Generalization of Schmidt's theorem on cluster curves of inextendible curves.

Examples showing cluster curves need not be closed or incomplete.

Abstract

We study local variations of causal curves in a space-time with respect to b-length (or generalised affine parameter length). In a convex normal neighbourhood, causal curves of maximal metric length are geodesics. Using variational arguments, we show that causal curves of minimal b-length in sufficiently small globally hyperbolic sets are geodesics. As an application we obtain a generalisation of a theorem by B. G. Schmidt, showing that the cluster curve of a partially future imprisoned, future inextendible and future b-incomplete curve must be a null geodesic. We give examples which illustrate that the cluster curve does not have to be closed or incomplete. The theory of variations developed in this work provides a starting point for a Morse theory of b-length.