A LOCAL VARIATIONAL THEORY FOR THE SCHMIDT METRIC

We study local variations of causal curves in a space-time with respec t to b-length (or generalized affine parameter length). In a convex no rmal neighbourhood, causal curves of maximal metric length are geodesi cs. Using variational arguments, we show that causal curves of minimal b-length insufficiently small globally hyper belie sets are geodesics . As an application we obtain a generalization of a theorem by Schmidt , showing that the cluster curve of a partially future imprisoned, fut ure inextendible and future b-incomplete curve must be a null geodesic . We give examples which illustrate that the cluster curve does not ha ve to be closed or incomplete. The theory of variations developed in t his work provides a starting point for a Morse theory of b-length. (C) 1997 American Institute of Physics.