Relativistic Finsler geometry

TL;DR

This paper explores the extension of general relativity into Finsler geometry, introducing concepts like color and anisotropy, and discusses how parallel displacement and geometrodynamics are affected by these generalizations.

Contribution

It provides a review of parallel displacement in Finsler geometry and discusses the implications of intrinsic anisotropy and direction dependence in physical interpretations.

Findings

Finsler geometry generalizes Riemannian structures by including direction dependence.

The concept of color introduces local anisotropy affecting geometrodynamics.

Parallel displacement in Finsler geometry can occur along directions other than the supporting direction.

Abstract

We briefly review some basic concepts of parallel displacement in Finsler geometry. In general relativity, the parallel translation of objects along the congruence of the fundamental observer corresponds to the evolution in time. By dropping the quadratic restriction on the measurement of an infinitesimal distance, the geometry is generalized to a Finsler structure. Apart from curvature a new property of the manifold complicates the geometrodynamics, the color. The color brings forth an intrinsic local anisotropy and many quantities depend on position and to a "supporting" direction. We discuss this direction dependence and some physical interpretations. Also, we highlight that in Finsler geometry the parallel displacement isn't necessarily always along the "supporting" direction. The latter is a fundamental congruence of the manifold and induces a natural 1+3 decomposition. Its internal deformation is given through the evolution of the irreducible components of vorticity, shear and expansion.