Kahler decoupling for Kerr perturbations
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| Format: | Preprint |
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2026
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| author | Green, Stephen R. Krasnov, Kirill Shaw, Adam |
| author_facet | Green, Stephen R. Krasnov, Kirill Shaw, Adam |
| contents | The Euclidean Kerr metric is conformal, in two distinct ways, to a Kahler metric, with conformal factors determined by the repeated eigenvalue of the two chiral halves of the Weyl curvature. A Lorentzian analogue holds, where the conformally related metric is complex but retains key features of Kahler geometry. We show that this hidden Kahler structure provides a geometric explanation for the existence of decoupled equations for curvature scalars, such as the Teukolsky equations. The essential mechanism is that, on a Kahler background, self-dual 2-forms are parallel with respect to a natural covariant derivative, so differential operators acting on them preserve their decomposition and do not mix components. In this way, decoupling is seen to be a direct consequence of Kahler geometry. We make this mechanism explicit in two ways. First, we show that the spin-k Teukolsky operator can be obtained from a Laplace-type operator associated with the Kahler metric by a similarity transformation. Second, for electromagnetic perturbations, we use the conformal invariance of Maxwell's equations delta F = 0 to show that they imply d delta F = 0, where delta is the co-differential of the Kahler metric. This operator automatically decouples, and the resulting equations for the extremal components coincide with the spin-one Teukolsky equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_22424 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kahler decoupling for Kerr perturbations Green, Stephen R. Krasnov, Kirill Shaw, Adam General Relativity and Quantum Cosmology High Energy Physics - Theory Differential Geometry The Euclidean Kerr metric is conformal, in two distinct ways, to a Kahler metric, with conformal factors determined by the repeated eigenvalue of the two chiral halves of the Weyl curvature. A Lorentzian analogue holds, where the conformally related metric is complex but retains key features of Kahler geometry. We show that this hidden Kahler structure provides a geometric explanation for the existence of decoupled equations for curvature scalars, such as the Teukolsky equations. The essential mechanism is that, on a Kahler background, self-dual 2-forms are parallel with respect to a natural covariant derivative, so differential operators acting on them preserve their decomposition and do not mix components. In this way, decoupling is seen to be a direct consequence of Kahler geometry. We make this mechanism explicit in two ways. First, we show that the spin-k Teukolsky operator can be obtained from a Laplace-type operator associated with the Kahler metric by a similarity transformation. Second, for electromagnetic perturbations, we use the conformal invariance of Maxwell's equations delta F = 0 to show that they imply d delta F = 0, where delta is the co-differential of the Kahler metric. This operator automatically decouples, and the resulting equations for the extremal components coincide with the spin-one Teukolsky equations. |
| title | Kahler decoupling for Kerr perturbations |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Differential Geometry |
| url | https://arxiv.org/abs/2604.22424 |