17 research outputs found
Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric
On the energy functional on Finsler manifolds and applications to stationary spacetimes
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.Comment: 23 pages, AMSLaTeX. v4 matches the published versio
Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric
We show that the index of a lightlike geodesic in a conformally standard
stationary spacetime is equal to the index of its spatial projection as a
geodesic of a Finsler metric associated to the spacetime. Moreover we obtain
the Morse relations of lightlike geodesics connecting a point to an integral
line of the standard timelike Killing vector field by using Morse theory on the
associated Finsler manifold. To this end, we prove a splitting lemma for the
energy functional of a Finsler metric. Finally, we show that the reduction to
Morse theory of a Finsler manifold can be done also for timelike geodesics.Comment: AMSLaTeX, 22 pages. v5: Appendix B removed and replaced by
arXiv:1211.3071 [math.DG
An account on links between Finsler and Lorentz Geometries for Riemannian Geometers
Some links between Lorentz and Finsler geometries have been developed in the
last years, with applications even to the Riemannian case. Our purpose is to
give a brief description of them, which may serve as an introduction to recent
references. As a motivating example, we start with Zermelo navigation problem,
where its known Finslerian description permits a Lorentzian picture which
allows for a full geometric understanding of the original problem. Then, we
develop some issues including: (a) the accurate description of the Lorentzian
causality using Finsler elements, (b) the non-singular description of some
Finsler elements (such as Kropina metrics or complete extensions of Randers
ones with constant flag curvature), (c) the natural relation between the
Lorentzian causal boundary and the Gromov and Busemann ones in the Finsler
setting, and (d) practical applications to the propagation of waves and
firefronts.Comment: 39 pages, 8 figure