Finslerian grounds for four--directional anisotropic kinematics

TL;DR

This paper extends special relativity to a four-directional anisotropic framework using Finsler geometry, revealing new invariance properties and a consistent relativistic structure that generalizes Lorentz transformations.

Contribution

It introduces a totally anisotropic kinematic transformation based on Finslerian metrics, expanding the geometric and group-theoretic foundations of relativistic kinematics.

Findings

Reveals group and invariance properties of anisotropic transformations

Establishes Finslerian metric for relativistic four-vectors

Maintains relativity principle and small-velocity correspondence

Abstract

Upon straightforward four--directional extension of the special--relativistic two--dimensional transformations to the four--dimensional case we lead to convenient totally anisotropic kinematic transformations, which prove to reveal many remarkable group and invariance properties. Such a promise is shown to ground the basic manifold with the Finslerian fourth-root metric function to measure length of relativistic four--vectors. Conversion to the framework of relativistic four--momentum is also elucidated. The relativity principle is strictly retained. An interesting particular algebra for subtraction and composition of three-dimensional relative velocities is arisen. The correspondence principle is operative in the sense that at small relative velocities the transformations introduced tend approximately to ordinary Lorentzian precursors. The transport synchronization remains valid. Abbreviation RF will be used for (inertial) reference frames. {\bf Keywords:} special relativity, invariance, Finsler geometry.