Solution of Maxwell's equations on a de Sitter background

TL;DR

This paper provides an exact analytical solution to Maxwell's equations in de Sitter space-time using decoupling techniques, hypergeometric functions, and spherical harmonics, with implications for cosmology and gravitational wave research.

Contribution

It introduces a novel decoupling method and explicit solutions for Maxwell's equations in de Sitter space, advancing the understanding of electromagnetic fields in curved backgrounds.

Findings

Exact solutions for the vector potential components are expressed through hypergeometric functions.

The radial part of the solution is plotted for specific initial conditions.

The method paves the way for solving tensor wave equations in curved spacetime.

Abstract

The Maxwell equations for the electromagnetic potential, supplemented by the Lorenz gauge condition, are decoupled and solved exactly in de Sitter space-time studied in static spherical coordinates. There is no source besides the background. One component of the vector field is expressed, in its radial part, through the solution of a fourth-order ordinary differential equation obeying given initial conditions. The other components of the vector field are then found by acting with lower-order differential operators on the solution of the fourth-order equation (while the transverse part is decoupled and solved exactly from the beginning). The whole four-vector potential is eventually expressed through hypergeometric functions and spherical harmonics. Its radial part is plotted for given choices of initial conditions. We have thus completely succeeded in solving the homogeneous vector wave equation for Maxwell theory in the Lorenz gauge when a de Sitter spacetime is considered, which is relevant both for inflationary cosmology and gravitational wave theory. The decoupling technique and analytic formulae and plots are completely original. This is an important step towards solving exactly the tensor wave equation in de Sitter space-time, which has important applications to the theory of gravitational waves about curved backgrounds.