On Fermat's principle for causal curves in time oriented Finsler   spacetimes
  In this work, a version of Fermat's principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the secondvariation of the {\it time arrival functional} along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of the critical points of the time arrival functional is investigated and a Morse index theorem in the context of Finsler spacetime is presented. 
Ricardo Gallego Torrom\'e, Paolo Piccione, Henrique Vit\'orio
math.DG gr-qc