|Speaker:||Miguel Angel Javaloyes (University of Murcia, Spain)|
|Title:||Interplay between Randers metrics and the causal geometry of stationary spacetimes. Almost isometries.|
|Date (JST):||Fri, Feb 14, 2014, 11:00 - 12:00|
"We obtain some results, in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure of standard stationary spacetimes on M=SxR and Randers metrics on S. In particular:
(1) For sSTATIONARY SPACETIMES: we give a simple characterization on when SxR is causally continuous or globally hyperbolic (including in the latter case, when S is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived.
(2) For FINSLER GEOMETRY: Causality allows to determine that the natural sufficient condition for the convexity (i.e., geodesic connectedness by minimizing geodesics) of any Finsler manifold is the compactness of the symmetrized closed balls. Then, we show that for any Randers metric R with compact symmetrized closed balls, there exists another Randers metric with the same pregeodesics and geodesically complete.
As a further application, we will develop the basics of a theory of almost isometries for spaces endowed with a quasi-metric. Having in mind the interplay explained before, the case of non-reversible Finsler (more specifically, Randers) metrics becomes of particular interest, and it will be studied in more detail. Again, the main motivation arises from General Relativity, and more specifically in stationary spacetimes, in which case K-conformal diffeomorphisms correspond to almost isometries of the Fermat metric defined in the spatial part. A series of results on the topology and the Lie group structure of K-conformal maps will be discussed."