The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric

In a series of recent papers by the author and collaborators, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. These relations were called Stationary-to-Randers Correspondence (SRC). In this paper we focus on one aspect of SRC, the equality between the index of a geodesic in a Randers space and that of its lightlike lift in the associated conformal stationary spacetime. Moreover we make some remarks about regularity of the energy functional of a Finsler metric on the infinite dimensional manifold of $H^1$ curves connecting two points, in connection with infinite dimensional techniques in Morse Theory.