Observable Effects in a Class of Spherically Symmetric Static Finsler Spacetimes

Abstract

After some introductory discussion of the definition of Finsler spacetimes and their symmetries, we consider a class of spherically symmetric and static Finsler spacetimes which are small perturbations of the Schwarzschild spacetime. The deviations from the Schwarzschild spacetime are encoded in three perturbation functions φ0(r), φ1(r) and φ2(r) which have the following interpretations: φ0 perturbs the time function, φ1 perturbs the radial length measurement and φ2 introduces a spatial anisotropy which is a genuine Finsler feature. We work out the equations of motion for freely falling particles and for light rays, i.e. the timelike and lightlike geodesics, in this class of spacetimes, and we discuss the bounds placed on the perturbation functions by observations in the Solar system.

Finsler spacetime geometry in physics

“…However most of the existing definitions so far turned out to be still too restrictive, since different communities formulated them having in mind different examples they wanted to cover. For example the definition by the author and collaborator in [51] excludes Randers type metrics, since there exists no power of a Randers Finsler Lagrangian that is smooth on T M \ {0}, while nth root Finsler spacetimes (14) are excluded by the definition of Lämmerzahl, Perlick and Hasse [37] and by Minguzzi [46], since the signature of the L-metric changes on T M \ {0}, as well as by the definition of Javaloyes and Sánchez [32], since their square is not smooth where they vanish.…”

“…The following definition of Finsler spacetimes, formulated in [30], summarizes conditions on L and g L such that freely falling causal curves exist and the geometry is well defined along nearly all of these causal curves. It is distilled from three existing definitions, see [32,37,51], in such a way that huge classes of Finsler spacetimes according to the earlier definitions are Finsler spacetimes according to this latest definition, even if not all. Moreover, it includes most examples encountered in Section II.…”

Foundations of Finsler Spacetimes from the Observers’ Viewpoint

“…(b) Product metrics −dt 2 + F 2 or, with more generality, the rough Lorentz-Finsler version of static spacetimes −Λ(x)dt 2 + F 2 (x, y), with natural coordinates (x, y) at T M , are never smooth at ∂ t whenever F is Finsler but not Riemannian. Consequently, some authors have included the possible existence of non-smooth directions as a fundamental ingredient of Lorentz-Finsler metrics (see for example [13,51]). Nevertheless, as explained in Rem.…”

Finsler metrics and relativistic spacetimes

“…4.1]. On the other, physically, there are situations where it is natural to consider non-differentiable directions, so that one may allow this possibility explicitly [31]. However, we will not go through these questions in the remainder (essentially, "smooth" might include some residual non-differentiable points with no big harm).…”

“…In [35,31], the definition is essentially as in [5] with the opposite sign of L (and an increasing attention to relax differentiability). The definition in [36] is somewhat more involved: they introduce an r-homogeneous function L, with r ≥ 2 endowed with a fundamental tensor of signature one and, then, they associate a homogeneous Finsler function F = r √ L. This allows one to deal with non-differentiability in cases that extend the lightlike vectors in our definition above.…”