A Local Variational Theory for the Schmidt metric

Details A Local Variational Theory for the Schmidt metric A Local Variational Theory for the Schmidt metric

1996-12-02

arxiv.org/abs/gr-qc/9612005v1

J.Math.Phys. 38 (1997) 3347-3357

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

14 pages, LaTeX 2.09, REVTeX, AMSFonts, EPSF, 2 figures

Scientific paper

10.1063/1.532047

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.

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