On a Fermat principle in general relativity. A Morse theory for light rays
Among all lightlike curves joining a point \(p\) to a timelike curve \(\gamma\) in a Lorentzian manifold, the geodesics are characterized by stationary arrival time. Here, ``arrival time'' refers to the parametrization of \(\gamma\), and ``stationary'' refers to a minimum, a maximum or a saddle-point. This variational principle can be viewed as a general relativistic version of Fermat's principle. In the paper under review, the authors use this variational principle to develop a Morse theory for lightlike geodesics, similar to the classical Morse theory for Riemannian geodesics in its infinite-dimensional version à la Palais and Smale. To that end the Lorentzian manifold has to satisfy several assumptions that imply, in particular, global hyperbolicity. The authors briefly indicate that their results might have applications to the gravitational lens effect.
scientific article; zbMATH DE number 923666
scientific article