On Fermat's principle for causal curves in time oriented Finsler spacetimes

Abstract

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 second variation of the 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. C 2012 American Institute of Physics.This notion of reversibility is different from the one considered by Beem 3 and also different than the one considered in Ref. 25. Our definition of reversible metric is stronger than the corresponding notions of Beem and that the one considered in the theory of Pfeifer-Wohlfart. B. Elementary causality notions for Finsler spacetimesThe fundamental causal notions of a Finsler spacetime (M, L) is a natural generalization of the Lorentzian causal framework. 4 A vector field X ∈ TM is said to be timelike if L(x, X(x)) < 0 at all points x ∈ M and a curve λ : I −→ M is timelike if the tangent vector field is timelike L(λ(s),λ(s)) < 0. A vector field X ∈ is lightlike if L(x, X(x)) = 0, ∀x ∈ M; a curve is lightlike if its tangent vector field is lightlike. Similar notions hold for spacelike vector and curves. A curve is causal if it is either timelike and has constant speed g˙λ(λ,λ) := L(λ,λ) = g i j (λ, T )λ iλ j or if it is lightlike.The following facts can be proved from the definition of Finsler spacetime.

Bounce Cosmology in Generalized Modified Gravities

“…From the form of the two Friedmann equations (41), (42) we can see that we obtain extra contributions that reflect the Finsler-like structure of the tangent bundle. In particular, these induce an effective energy density and pressure of geometrical origin as…”

“…These are natural extensions of Riemannian geometry in which the physical quantities may directly depend on observer 4-velocity, and this velocitydependence reflects the Lorentz-violating character of the kinematics. Such a property is called dynamic anisotropy [35][36][37][38][39][40][41][42][43][44][45][46]. Additionally, Finsler and Finsler-like geometries are strongly connected to the effective geometry within anisotropic media [47,48] and naturally enter the analogue gravity program [49].…”

On the Analyticity of Static Solutions of a Field Equation in Finsler Gravity

“…This fact had led several authors to assume that only one of these connected components should be considered as a privileged one by the point of view of causality. The choice can be done, e.g., by prescribing a timelike, globally defined, vector field Y and taking at each x ∈M the connected component which is the boundary of the set of timelike vectors containing Y(x) (such as, for example, in [16,17]) or by a priori restricting L to a cone sub-bundle A of TM, like in Asanov's definition of a Finsler norm F (such as, for example, in [18][19][20][21]) or by looking only at the cone structure, without considering as fundamental the function L (see [22][23][24][25]). In some physical models, anyway, indefinite Finsler metrics L arise as the metrics invariant under the action of the symmetry group considered and, in general, they are defined and smooth only on an open cone sub-bundle of TM.…”

Schwarzschild–Finsler–Randers spacetime: geodesics, dynamical analysis and deflection angle

“…In the framework of applications of Finsler geometry, many works in different directions of geometrical and physical structures have contributed to the extension of research for theoretical and observational approaches during the last years. We cite some works from the literature of the applications of Finsler geometry [7,8,13,[21][22][23][24][25][26][27][28][29][30][31][32].…”