Komar fluxes of circularly polarized light beams and cylindrical metrics

TL;DR

This paper derives the metric for a circularly polarized electromagnetic beam in cylindrical spacetime, analyzing energy, momentum, and angular momentum fluxes using Komar integrals, and clarifies the factor of 2 in the Tolman-Komar formula.

Contribution

It provides a new derivation of the cylindrical metric for polarized light beams and clarifies the Komar fluxes and the factor 2 issue in the Tolman-Komar energy formula.

Findings

Derived the metric from Einstein-Maxwell equations.

Identified the missing angular momentum in plane wave solutions.

Clarified the factor 2 in the Tolman-Komar energy expression.

Abstract

The mass per unit length of a cylindrical system can be found from its external metric as can its angular momentum. Can the fluxes of energy, momentum and angular momentum along the cylinder also be so found? We derive the metric of a beam of circularly polarized electromagnetic radiation from the Einstein-Maxwell equations. We show how the uniform plane wave solutions miss the angular momentum carried by the wave. We study the energy, momentum, angular momentum and their fluxes along the cylinder both for this beam and in general. The three Killing vectors of any stationary cylindrical system give three Komar flux vectors which in turn give six conserved fluxes. We elucidate Komar's mysterious factor 2 by evaluating Komar integrals for systems that have no trace to their stress tensors. The Tolman-Komar formula gives twice the energy for such systems which also have twice the gravity. For other cylindrical systems their formula gives correct results.