Separation of Maxwell equations in Kerr-NUT-(A)dS spacetimes

TL;DR

This paper proves the separability of Maxwell equations in a broad class of higher-dimensional Kerr-NUT-(A)dS spacetimes using a specialized ansatz and principal tensor, leading to explicit solutions via separated ordinary differential equations.

Contribution

It demonstrates the explicit separability of Maxwell equations in higher-dimensional spacetimes with principal tensors, generalizing previous approaches with a new ansatz and eigenfunction method.

Findings

Maxwell equations are separable in Kerr-NUT-(A)dS spacetimes.

Solutions can be expressed as eigenfunctions of commuting operators.

Separated equations reduce to ordinary differential equations with clear eigenvalues.

Abstract

In this paper we explicitly demonstrate separability of the Maxwell equations in a wide class of higher-dimensional metrics which include the Kerr-NUT-(A)dS solution as a special case. Namely, we prove such separability for the most general metric admitting the principal tensor (a non-degenerate closed conformal Killing-Yano 2-form). To this purpose we use a special ansatz for the electromagnetic potential, which we represent as a product of a (rank 2) polarization tensor with the gradient of a potential function, generalizing the ansatz recently proposed by Lunin. We show that for a special choice of the polarization tensor written in terms of the principal tensor, both the Lorenz gauge condition and the Maxwell equations reduce to a composition of mutually commuting operators acting on the potential function. A solution to both these equations can be written in terms of an eigenfunction of these commuting operators. When incorporating a multiplicative separability ansatz, it turns out that the eigenvalue equations reduce to a set of separated ordinary differential equations with the eigenvalues playing a role of separability constants. The remaining ambiguity in the separated equations is related to an identification of D-2 polarizations of the electromagnetic field. We thus obtained a sufficiently rich set of solutions for the Maxwell equations in these spacetimes.