Klein-Gordon equation for a particle in brane model

TL;DR

This paper derives and solves the Klein-Gordon equation for a free particle in a brane universe, incorporating brane radius variations and establishing related conservation laws and dispersion relations.

Contribution

It introduces a novel derivation of the Klein-Gordon equation in a brane context, accounting for dynamic brane radius effects.

Findings

Derived conservation laws on the brane based on symmetry

Obtained approximate solutions to the equation of motion

Derived dispersion relations for particles in the brane model

Abstract

Brane model of universe is considered for free particle. Conservation laws on the brane are obtained using the symmetry properties of the brane. Equation of motion is derived for a particle using variation principle from these conservation laws. This equation includes terms accounting the variation of brane radius. Its solution is obtained at some approximations and dispersion relation for a particle is derived.