Ehlers as EM duality in the double copy

TL;DR

This paper explores the connection between Ehlers transformations, electromagnetic duality, and the double copy framework in gravity and gauge theories, revealing how solution-generating symmetries translate across these contexts.

Contribution

It explicitly maps EM duality to a U(1) subgroup of SL(2,R) in the double copy, extending the understanding of solution symmetries in gravity and gauge theories.

Findings

Identifies a map between Maxwell fields and target space scalars in the double copy.

Clarifies the role of Ehlers-Harrison transformations in Einstein-Maxwell solutions.

Interprets charged black hole equations as a truncation of Einstein-Yang-Mills double copy.

Abstract

Given a solution to 4D Einstein gravity with an isometry direction, it is known that the equations of motion are identical to those of a 3D $\sigma$-model with target space geometry $SU(1,1)/U(1)$. Thus, any transformation by $SU(1, 1) \cong SL(2,\mathbb{R})$ is a symmetry for the action and allows one to generate new solutions in 4D. Here we clarify and extend recent work on electromagnetic (EM) duality in the context of the classical double copy. In particular, for pure gravity, we identify an explicit map between the Maxwell field of the single copy and the scalars in the target space, allowing us to identify the $U(1) \subset SL(2, \mathbb{R})$ symmetry dual to EM duality in the single copy. Moreover, we extend the analysis to Einstein-Maxwell theory, where we highlight the role of Ehlers-Harrison transformations and, for spherically symmetric charged black hole solutions, we interpret the equations of motion as a truncation of the putative single copy for Einstein-Yang-Mills theory.