A Green's function for the source-free Maxwell equations on $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$

TL;DR

This paper constructs an explicit Green's function for the source-free Maxwell equations on a complex product spacetime involving anti-de Sitter space and spheres, using Fourier analysis and differential geometry techniques.

Contribution

It introduces a novel method to solve Maxwell's equations on $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$ by formulating the problem via the Hodge Laplacian and employing Fourier expansion in eigenforms.

Findings

Explicit Green's function for Maxwell equations on the given spacetime.

Solution expressed as a Fourier series of eigenforms on $\mathbb{S}^3$.

Method applicable to similar problems in complex geometric backgrounds.

Abstract

We compute a Green's function giving rise to the solution of the Cauchy problem for the source-free Maxwell's equations on a causal domain $\mathcal{D}$ contained in a geodesically normal domain of the Lorentzian manifold $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$, where $AdS^5$ denotes the simply connected $5$-dimensional anti-de-Sitter space-time. Our approach is to formulate the original Cauchy problem as an equivalent Cauchy problem for the Hodge Laplacian on $\mathcal{D}$ and to seek a solution in the form of a Fourier expansion in terms of the eigenforms of the Hodge Laplacian on $\mathbb{S}^3$. This gives rise to a sequence of inhomogeneous Cauchy problems governing the form-valued Fourier coefficients corresponding to the Fourier modes and involving operators related to the Hodge Laplacian on $AdS^5 \times \mathbb{S}^2$, which we solve explicitly by using Riesz distributions and the method of spherical means for differential forms. Finally we put together into the Fourier expansion on $\mathbb{S}^3$ the modes obtained by this procedure, producing a $2$-form on $\mathcal{D}\subset AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$ which we show to be a solution of the original Cauchy problem for Maxwell's equations.