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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2411.02096 |
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| _version_ | 1866917282364522496 |
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| author | Tod, Paul |
| author_facet | Tod, Paul |
| contents | I review the twistor theory construction of stationary and axisymmetric, Lorentzian signature solutions of the Einstein vacuum equations and the related toric Ricci-flat metrics of Riemannian signature, \cite{W,MW,F,FW}. The construction arises from the Ward construction \cite{W2} of anti-self-dual Yang-Mills fields as holomorphic vector bundles on twistor space, with the observation of Witten \cite{LW} that the Einstein equations for these metrics include the anti-self-dual Yang-Mills equations. The principal datum for a solution is the holomorphic patching matrix $P$ for a holomorphic vector bundle on a reduced twistor space, and $P$ is typically simpler than the corresponding metric to write down.
I give a catalogue of examples, building on earlier collections \cite{F,AG}, and consider the inverse problem: how far does the rod structure of such a metric, together with its asymptotics, determine $P$? |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02096 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rod Structures and Patching Matrices: a review Tod, Paul Differential Geometry General Relativity and Quantum Cosmology 83C60 I review the twistor theory construction of stationary and axisymmetric, Lorentzian signature solutions of the Einstein vacuum equations and the related toric Ricci-flat metrics of Riemannian signature, \cite{W,MW,F,FW}. The construction arises from the Ward construction \cite{W2} of anti-self-dual Yang-Mills fields as holomorphic vector bundles on twistor space, with the observation of Witten \cite{LW} that the Einstein equations for these metrics include the anti-self-dual Yang-Mills equations. The principal datum for a solution is the holomorphic patching matrix $P$ for a holomorphic vector bundle on a reduced twistor space, and $P$ is typically simpler than the corresponding metric to write down. I give a catalogue of examples, building on earlier collections \cite{F,AG}, and consider the inverse problem: how far does the rod structure of such a metric, together with its asymptotics, determine $P$? |
| title | Rod Structures and Patching Matrices: a review |
| topic | Differential Geometry General Relativity and Quantum Cosmology 83C60 |
| url | https://arxiv.org/abs/2411.02096 |