Anisotropic conformal invariance of lightlike geodesics in pseudo-Finsler manifolds

Abstract

In this paper, we prove that lightlike geodesics of a pseudo-Finsler manifold and its focal points are preserved up to reparametrization by anisotropic conformal changes, using the Chern connection and the anisotropic calculus developed in [, ] and the fact that geodesics are critical points of the energy functional and Jacobi fields, the kernel of its index form. This result has applications to the study of Finsler spacetimes.

Anisotropic connections and parallel transport in Finsler spacetimes

“…Recently, one of the authors has developed systematically the anisotropic calculus [10,11], namely, how to make computations with an anisotropic connection, which can be seen as a natural and intuitive generalization of the usual Koszul connections. Some applications have been obtained in [9,16]. In the present article, we revisit this notion, showing precisely its relations with the other elements of the standard setting and providing a further insight on its associated parallel transport.…”

“…(2) The cocycle (12) for N a i (x, y) implies the cocycle (5) for Γ a ij . Then, the homogeneity of ν implies (14) and thus (16).…”

Lightlike hypersurfaces and time-minimizing geodesics in cone structures

“…Conversely, each cone structure C uniquely determines a (nonempty) class of anisotropically equivalent Lorentz-Finsler metrics [12, Rem. 5.9], any of which will be called compatible with C. All these metrics share the same lightlike pregeodesics (see, e.g., [13,Prop. 3.4]), 5 which coincide with the cone geodesics of C, as shown by the following result [12,Thm.…”

“…Unlike the first case in which the indicatrix is transversal to the position vector, the null cone does not determine the pseudo-Finsler metric in some open subset of the tangent bundle, and it turns out that the lightlike geodesics are determined only up to reparametrization. Nevertheless, using a certain quotient space, it is possible to define some kind of curvature invariants (see [14]), and on the other hand, the focal points of these lightlike geodesics do not depend on the pseudo-Finsler metric used to compute them [13]. Additionally, if these null cones enclose a convex subset in every tangent space, then it is possible to study their causal relations, namely, the connections between points by means of curves whose tangent vectors lie always inside the null cones.…”

“…Anyway, the smoothness on all A * will not be strictly needed here, just on a neighbourhood of C, as we will only focus on lightlike directions 5. The geodesics of a Lorentz-Finsler metric L are the (smooth) autoparallel curves of its Chern connection (see[9,13] for background).…”