Conformally invariant formalism for the electromagnetic field with currents in Robertson-Walker spaces

TL;DR

This paper develops a conformally invariant formalism for electromagnetic fields with currents in Robertson-Walker spaces, linking the Laplace-Beltrami equation in six dimensions to Maxwell's equations and gauge conditions.

Contribution

It introduces a geometric formalism that simplifies calculations and constructs an atlas for conformally flat spaces, enabling the use of Weyl rescaling to Minkowski space.

Findings

Derivation of Maxwell equations from the Laplace-Beltrami equation in six dimensions.

Construction of an atlas for conformally flat spaces including Robertson-Walker metrics.

Simplification of calculations through a geometric formalism.

Abstract

We show that the Laplace-Beltrami equation $\square_6 a =j$ in $(\setR^6,\eta)$, $\eta := \mathrm{diag}(+----+)$, leads under very moderate assumptions to both the Maxwell equations and the conformal Eastwood-Singer gauge condition on conformally flat spaces including the spaces with a Robertson-Walker metric. This result is obtained through a geometric formalism which gives, as byproduct, simplified calculations. In particular, we build an atlas for all the conformally flat spaces considered which allows us to fully exploit the Weyl rescalling to Minkowski space.