Foundations of anisotropic relativistic mechanics

TL;DR

This paper establishes the theoretical framework for anisotropic relativistic mechanics, incorporating both conventional and real anisotropies, and provides methods to derive fundamental dynamical quantities and dispersion relations within this context.

Contribution

It introduces a general method to determine dynamical quantities in anisotropic relativistic mechanics and formulates these theories using pseudo-Finslerian spacetime geometry.

Findings

Derived explicit expressions for Lagrangian, energy, and momentum in anisotropic relativistic kinematics.

Presented dispersion relations for these anisotropic theories.

Outlined a potential extension to three spatial dimensions.

Abstract

We lay down the foundations of particle dynamics in mechanical theories that satisfy the relativity principle and whose kinematics can be formulated employing reference frames of the type usually adopted in special relativity. Such mechanics allow for the presence of anisotropy, both conventional (due to non-standard synchronisation protocols) and real (leading to detectable chronogeometrical effects, independent of the choice of synchronisation). We give a general method for finding the fundamental dynamical quantities (Lagrangian, energy and momentum), and write their explicit expression in all the kinematics compatible with the basic requirements. We also write the corresponding dispersion relations and outline a formulation of these theories in terms of a pseudo-Finslerian spacetime geometry. Although the treatment is restricted to the case of one spatial dimension, an extension to three dimensions is almost straightforward.