2007/02/28 by Erasmo Caponio, Miguel Ángel Javaloyes, Miguel Angel Javaloyes +1 · 82 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Completeness (order theory) #Energy functional #Fermat's Last Theorem #Finsler manifold #Geodesic #Geometry #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Metric (unit) #Physics #Pure mathematics #Quantum gravity #Quantum mechanics #Spacetime #Stationary spacetime #gr-qc #math.DG #msc:53C22 #msc:53C50 #msc:53C60 #msc:58E05
paper · pdf · doi:10.1007/s00208-010-0602-7
published in Mathematische Annalen 351(2), 365-392 (Springer Nature) · 23 pages, AMSLaTeX. v4 matches the published version
openalex publication_date 2010/11/16 · arxiv created 2010/12/04 · arxiv updated 2011/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais--Smale condition under the completeness of the Finsler metric. Moreover we define a Finsler metric of Randers type, which we call Fermat metric, associated to a conformally standard stationary spacetime. We shall study the influence of the Fermat metric on the causal properties of the spacetime, mainly the global hyperbolicity. Moreover we study the relations between the energy functional of the Fermat metric and the Fermat principle for the light rays in the spacetime. This allows us to obtain existence and multiplicity results for light rays, using the Finsler theory. Finally the case of timelike geodesics with fixed energy is considered.