Berwald spacetimes and very special relativity

Abstract

In this work we study Berwald spacetimes and their vacuum dynamics, where the latter are based on a Finsler generalization of the Einstein's equations derived from an action on the unit tangent bundle. In particular, we consider a specific class of spacetimes which are non-flat generalizations of the very special relativity (VSR) line element, to which we refer as very general relativity (VGR). We derive necessary and sufficient conditions for the VGR line element to be of Berwald type. We present two novel examples with the corresponding vacuum field equations: a Finslerian generalization of vanishing scalar invariant (VSI) spacetimes in Einstein's gravity as well as the most general homogeneous and isotropic VGR spacetime.

Foundations of Finsler Spacetimes from the Observers’ Viewpoint

“…(i) The axioms D 1 -D 4 , which allow one to define radar coordinates, should apply to particles P, Q which do not intersect. Otherwise, spurious differential issues might appear even in the case of Lorentz-Minkowski spacetime 24 .…”

“…However, for most choices of b, Bogoslovsky metric (12) is an example which does not lie under PW definition, in spite of having a regular cone and a connection extendible to it (indeed, both of them the same as Lorentz-Minkowski space). Next, let us analyze a generalization of Bogoslovsky metrics from norms to arbitrary manifolds considered in [24]. Let L Bog = g(·, ·) (1−b) (β ⊗ β) b , where g is a (time-oriented) Lorentzian metric and β a 1-form in a manifold M ; notice that, whenever β remains g-causal, the future cone C of g agrees with the lightlike vectors for L Bog and this metric is well-defined on all the g-causal vectors.…”

“…Many of them were motivated by the search of a unified theory which solved disturbing issues of compatibility with Quantum Mechanics (Kaluza-Klein, M-theory, quantum field gravity...) while, since the 90's, unexpected cosmological measurements led to further alternatives (cosmological constant, quintaessence, theories with varying speed of light...). However, the possibility to consider a Finslerian modification of GR has not settled in the mainstream of research and it has been scarcely considered in the literature until recent times (some examples are references [1,13,23,24,26,32,36,39,43,50,60,64,67,73]). Certainly, the generality of Finsler Geometry in comparison with the Riemannian setup (namely, analogous to the generality of the convex open subsets of an affine space in comparison with the ellipsoids) is a big drawback, as the number of new variables and parameters would seem immeasurable.…”

Redshift in Finsler spacetimes

“…Girelli et al [11]. Moreover, Finsler geometry comes up naturally also in curved versions of Very Special Relativity, see Gibbons et al [12] and, for the more special case where the resulting Finsler spacetime is of Berwald type, Fuster et al [13], and in the Standard Model Extension, see e.g. Kostelecký [14].…”

Energy-momentum and angular-momentum of a gyratonic pp -waves spacetime

“…The ppwaves space-time were investigated in the 1950's and 1960's specially by Peres, Pirani and Bondi [2,3], and most recently in the Refs. [4][5][6][7][8][9][10][11].…”