On Imprisoned Curves and b-length in General Relativity

Details On Imprisoned Curves and b-length in General Relativity On Imprisoned Curves and b-length in General Relativity

2000-06-13

arxiv.org/abs/gr-qc/0006050v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

26 pages, 7 figures, LaTeX 2e with AMSLaTeX 1.2 and AMSFonts, submitted to J. Math. Phys

Scientific paper

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.

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