Incidence Relations for Self-Dual Black Holes

Joon-Hwi Kim

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arXiv · 2608.07774

Incidence Relations for Self-Dual Black Holes

Joon-Hwi Kim

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Abstract

In Penrose's nonlinear graviton construction, a self-dual spacetime arises as the moduli space of holomorphic twistor lines in curved twistor space, whose explicit parametrization is concretely given by the incidence relation. We demonstrate a concrete local construction of the incidence relations for self-dual black hole spacetimes in Kerr-Schild coordinates: Eguchi-Hanson, self-dual Taub-NUT, and self-dual Pleba\'nski-Demia\'nski. This is facilitated by implementing the Dunajski-Mason recursion in the framework of Pleba\'nski's second heavenly equation, for which we explicitly find the Pleba\'nski scalar. Our results describe closed-form formulae. Notably, for the self-dual Taub-NUT solution, the exact incidence relation exhibits linear dependence on the gravitational coupling. The construction of zero-rest-mass fields on self-dual black hole backgrounds is briefly sketched as an application.

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Joon-Hwi Kim. 2026-08-07. Incidence Relations for Self-Dual Black Holes. https://arxiv.org/abs/2608.07774

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