Relativistic particle in multiscale spacetimes

TL;DR

This paper investigates the behavior of relativistic particles in multiscale spacetimes, revealing that invariance principles lead to standard actions with multiscale features only in spatial directions, and explores the effects of relaxing these constraints.

Contribution

It demonstrates that invariance principles uniquely determine the relativistic particle action in multiscale spacetimes, showing multiscale effects are confined to spatial directions and analyzing consequences of relaxing these conditions.

Findings

Standard relativistic action is recovered with multiscale spatial geometry.

Allowing nontrivial time weights complicates charged particle dynamics.

Multiscale effects are restricted to spatial directions under symmetry constraints.

Abstract

We study the action and the dynamics of a relativistic particle, uncharged or charged, in multiscale spacetimes. Invariance under reparametrizations and Poincar\'e symmetries uniquely determine the action and the line element to be the usual ones, without the weight factors typical of particle mechanics in these geometries. The resulting spacetime is multiscale only along spatial directions. This version of the system is also dictated by recovery of the nonrelativistic limit together with compatibility with Maxwell and electrodynamics field theory. Giving up all these requirements and allowing for a nontrivial weight factor in the time direction produces a modified line element and considerably complicates the dynamics in the case of a charged particle.