Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.13410 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909353343188992 |
|---|---|
| author | Tod, Paul |
| author_facet | Tod, Paul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13410 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | One-sided type-D metrics with aligned Einstein-Maxwell Tod, Paul General Relativity and Quantum Cosmology Differential Geometry 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. |
| title | One-sided type-D metrics with aligned Einstein-Maxwell |
| topic | General Relativity and Quantum Cosmology Differential Geometry |
| url | https://arxiv.org/abs/2410.13410 |