The dyonic Kerr-Schild ansatz

TL;DR

This work addresses the challenge of incorporating magnetic charge into Kerr–Schild spacetimes without relying on electric–magnetic duality rotations. It introduces a geometric dualization, constructing a one-form $l^{*}$ from the Kerr–Schild null congruence and forming a dyonic gauge potential $A = -S_{\text{e}}\,l + S_{\text{m}}\,l^{*}$, with the electrovac circularity theorem enforcing the electric and magnetic profiles. The approach yields a fully geometric derivation of the dyonic Kerr–Newman solution, and extends naturally to (A)dS backgrounds, producing the dyonic Kerr–Newman–(A)dS spacetime with charges $(q,p)$. This framework clarifies the geometry–gauge interplay in Kerr–Schild gravity and lays a foundation for generalizations to nonlinear theories and higher dimensions, where duality or additional fields may be incorporated within the same geometric scheme.

Abstract

We develop a geometric extension of the Kerr-Schild ansatz that incorporates both electric and magnetic sectors of the Maxwell field in a unified framework, without resorting to duality rotations. We start observing that the known purely electric solution satisfies Maxwell's equations due to a closedness condition obeyed by the Kerr-Schild null congruence. From the associated local exactness property, we construct a new one-form naturally linked to the congruence as a sort of Poincaré dualization. This leads us to propose a geometrically motivated dyonic vector potential within the Kerr-Schild ansatz, defined as a superposition of an electric contribution along the congruence and a magnetic one that aligns to the dualized one-form. We then show that for a stationary and axisymmetric Kerr-Schild ansatz, the electrovac circularity theorem uniquely constrains not only the scalar profile of the metric, but also those associated to the electric-magnetic splitting of the gauge field. The resulting formalism provides a transparent derivation of the dyonic Kerr-Newman solution and extends naturally to the (A)dS case, highlighting the intrinsic interplay between geometry and matter in a Kerr-Schild setting.