On Imprisoned Curves and b-length in General Relativity

TL;DR

This paper addresses issues in the theory of imprisoned curves and b-length in General Relativity, providing corrected proofs, analyzing the structure of b-length neighborhoods, and clarifying previous results in the field.

Contribution

It offers a new proof regarding imprisoned curves, explores the geometry of b-length neighborhoods, and corrects prior literature on b-length relations in spacetime geometry.

Findings

Provided a new proof for the existence of null geodesic cluster curves.

Analyzed the structure of b-length neighborhoods and their geometric implications.

Corrected previous results linking b-lengths of curves in the frame bundle.

Abstract

This paper is concerned with two themes: imprisoned curves and the b-length functional. In an earlier paper by the author, it was claimed that an endless incomplete curve partially imprisoned in a compact set admits an endless null geodesic cluster curve. Unfortunately, the proof was flawed. We give an outline of the problem and remedy the situation by providing a proof by different methods. Next, we obtain some results concerning the structure of b-length neighbourhoods, which gives a clue to how the geometry of a spacetime is encoded in the pseudo-orthonormal frame bundle equipped with the b-metric. We also show that a previous result by the author, proving total degeneracy of a b-boundary fibre in some cases, does not apply to imprisoned curves. Finally, we correct some results in the literature linking the b-lengths of general curves in the frame bundle with the b-length of the corresponding horizontal curves.