Erasmo Caponio

On the interplay between Lorentzian Causality and Finsler metrics of Randers type

We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on

On the energy functional on Finsler manifolds and applications to stationary spacetimes

M=\R\times S

Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric

and Randers metrics on

Convex domains of Finsler and Riemannian manifolds

S

Trajectories of charged particles in a region of a stationary spacetime

. In particular, for stationary spacetimes, we give a simple characterization of when they are causally continuous or globally hyperbolic (including in the latter case, when

Solutions to the Lorentz force equation with fixed charge-to-mass ratio in globally hyperbolic space–times

S

The Avez–Seifert theorem for the relativistic Lorentz force equation

is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. Causality suggests that the role of completeness in many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics, Bonnet-Myers, Synge theorems) is played, in Finslerian Geometry, by the compactness of symmetrized closed balls. Moreover, under this condition we show that for any Randers metric there exists another Randers metric with the same pregeodesics and geodesically complete. Even more, results on the differentiability of Cauchy horizons in spacetimes yield consequences for the differentiability of the Randers distance to a subset, and vice versa.

Convex Regions of Stationary Spacetimes and Randers Spaces. Applications to Lensing and Asymptotic Flatness

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 one to obtain existence and multiplicity results for light rays, using the Finsler theory. Finally the case of timelike geodesics with fixed energy is considered.

STANDARD STATIC FINSLER SPACETIMES

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.

Finsler geodesics in the presence of a convex function and their applications

A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop’s one in his classical result (Bishop, Indiana Univ Math J 24:169–172, 1974) for the Riemannian case. Ours not only can be extended to the Finsler setting but it also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface.