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arXiv · 2410.13410

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

Paul Tod
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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.

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Paul Tod. 2024-10-17. One-sided type-D metrics with aligned Einstein-Maxwell. https://arxiv.org/abs/2410.13410

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