1999/05/13 by Fabio Giannoni, Paolo Piccione, Giannoni, Fabio +3
Mathematics · Physics and Astronomy · #53C22 #53C50 #58E05 #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #math-ph #math.MP #msc:53C22 #msc:53C50 #msc:58E05
paper · pdf · doi:10.48550/arxiv.math-ph/9905007
arxiv created 1999/05/13 · openalex publication_date 1999/05/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.