Morse Theory for the Travel Time Brachistochrones in Stationary Spacetimes

Abstract

The travel time brachistochrone curves in a general relativistic framework are timelike curves, satisfying a suitable conservation law with respect to a an observer field, that are stationary points of the travel time functional. In this paper we develop a global variational theory for brachistochrones joining an event $p$ and the worldline of an observer $γ$ in a stationary spacetime $M$. More specifically, using the method of Lagrange multipliers, we compute the first and the second variation of the travel time functional, obtaining two variational principles relating the geometry of the brachistochrones with the geometry of geodesics in a suitable Riemannian structure. We present an extension of the classical Morse Theory for Riemannian geodesics to the case of travel time brachistochrones, and we prove a Morse Index Theorem for brachistochrones. Finally, using techniques from Global Analysis, we prove the Morse relations for the travel time functional and we establish some existence and multiplicity results for brachistochrones.