On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime

TL;DR

This paper develops a covariant method to construct symmetry operators for the Maxwell and Teukolsky equations on Kerr-NUT-(A)dS spacetime, enabling separation of variables by exploiting hidden symmetries.

Contribution

It reproduces and generalizes symmetry operators for Maxwell and Teukolsky equations using the Eisenhart-Duval lift, clarifying their geometric origin and relation to hidden symmetries.

Findings

Symmetry operators for Maxwell equations are constructed covariantly.

The symmetry operators match previous results up to first-order Killing vector operators.

The method is applied to the Teukolsky equation on Kerr-NUT-(A)dS spacetime.

Abstract

We focus on the method recently proposed by Lunin and Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k to solve the Maxwell equation on the Kerr-NUT-(A)dS spacetime by separation of variables. In their method, it is crucial that the background spacetime has hidden symmetries because they generate commuting symmetry operators with which the separation of variables can be achieved. In this work we reproduce these commuting symmetry operators in a covariant fashion. We first review the procedure known as the Eisenhart-Duval lift to construct commuting symmetry operators for given equations of motion. Then we apply this procedure to the Lunin-Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k (LFKK) equation. It is shown that the commuting symmetry operators obtained for the LFKK equation coincide with the ones previously obtained by Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k, up to first-order symmetry operators corresponding to Killing vector fields. We also address the Teukolsky equation on the Kerr-NUT-(A)dS spacetime and its symmetry operator is constructed.