One-sided type-D metrics with aligned Einstein-Maxwell

TL;DR

This paper studies four-dimensional Einstein-Maxwell metrics with aligned type-D Weyl tensors, showing they can be described via Toda and Laplace equations, revealing linearization and symmetries in these solutions.

Contribution

It demonstrates that such Einstein-Maxwell metrics with aligned type-D Weyl tensors can be expressed through solutions of the Toda and Laplace equations, highlighting their linear structure and symmetries.

Findings

Metrics can be derived from solutions of the SU(∞)-Toda equation.

When a second Killing vector exists, metrics relate to axisymmetric Laplace solutions.

The field equations linearize under certain symmetry conditions.

Abstract

We consider four-dimensional, Riemannian metrics for which one or other of the self-dual or anti-self-dual Weyl tensors is type-D and which satisfy the Einstein-Maxwell equations with the corresponding Maxwell field aligned with the type-D Weyl spinor, in the sense of sharing the same Principal Null Directions (or PNDs). Such metrics always have a valence-2 Killing spinor, and therefore a Hermitian structure and at least one Killing vector. We rederive the results of Araneda (\cite{ba}), that these metrics can all be given in terms of a solution of the $SU(\infty)$-Toda field equation, and show that, when there is a second Killing vector commuting with the first, the method of Ward can be applied to show that the metrics can also be given in terms of a pair of axisymmetric solutions of the flat three-dimensional Laplacian. Thus in particular the field equations linearise. Some examples of the constructions are given.