Uniqueness of stationary axisymmetric type D black holes with non-aligned electromagnetic field

TL;DR

This paper proves the uniqueness of certain stationary axisymmetric black hole spacetimes with specific geometric and electromagnetic properties, extending the understanding of solutions in Einstein-Maxwell theory and potential applications in alternative gravity theories.

Contribution

It establishes that the conformal-to-Carter metric ansatz is the only possible for a class of Weyl type D geometries and classifies the unique electrovacuum solutions with non-aligned electromagnetic fields.

Findings

The conformal-to-Carter metric is uniquely determined for the specified geometries.

The only non-trivial electrovacuum solution with non-aligned electromagnetic field is the Ovcharenko-Podolsky class.

The double-aligned electromagnetic field solution is the Plebański-Demiański class.

Abstract

We demonstrate the uniqueness of the spacetimes recently found by us in [H. Ovcharenko and J. Podolsky, Phys. Rev. D 112 (2025) 064076]. First, we prove that the conformal-to-Carter metric ansatz we used therein is the only possible for stationary axisymmetric geometries that are of Weyl type D, with geodesic and shear-free principal null directions (PNDs) which are orthogonal to polar directions, and whose specific 1-form $\theta$ is closed. Because this result is general, without employing any field equations, such conformal-to-Carter metric may find interesting applications also in various alternative theories of gravity. Then, we show that in the Einstein-Maxwell theory the only non-trivial electrovacuum solution for the conformal-to-Carter metric with the fully non-aligned and non-null electromagnetic field is the Ovcharenko-Podolsky class found in 2025. Complementarily, the only solution with the double-aligned and non-null electromagnetic field is the Pleba\'nski-Demia\'nski class found in 1976.