Abstract
In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.
Highlights
In this paper we present a review of the Fermat metrics associated with standard stationary spacetime and some applications for the studying of the geometrical and physical properties of such spacetimes
In this paper we have introduced the Fermat metrics associated to a standard stationary spacetime
We have recalled the Fermat principle which relates the light rays of a standard stationary spacetimes to the geodesics of the Fermat metrics and we have presented some applications to the study of the geometrical and physical properties of this class of spacetimes
Summary
In this paper we present a review of the Fermat metrics associated with standard stationary spacetime and some applications for the studying of the geometrical and physical properties of such spacetimes. If a Fermat principle holds, we can characterise light-like geodesics as critical points of a suitable functional, and apply the methods of the calculus of variations in the large to relate the multiple image effect to the topological and geometric properties of spacetime. We focus our attention on standard stationary spacetimes (or, in general, Lorentzian manifolds without any restriction on the dimension of the manifold), which are defined at Section 4 In this class of spacetimes, a Fermat principle can be stated in terms of the Fermat metrics associated to such manifolds.
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