  In some recent papers, the relations existing between the metric properties
of Randers spaces and the conformal geometry of stationary Lorentzian manifolds
were discovered and investigated. In this note, we focus on 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 between two points, in connection
with infinite dimensional techniques in Morse Theory.
