# Petrov type D equation on horizons of nontrivial bundle topology

## TL;DR

This paper investigates Petrov type D equations on horizons with nontrivial U(1) bundle topology, deriving axisymmetric solutions and revealing a new family of spacetimes that extend known black hole solutions.

## Contribution

It derives all axisymmetric solutions to the Petrov type D equation on nontrivial U(1) bundles over spheres, uncovering new spacetimes beyond known Einstein solutions.

## Key findings

- Derived all axisymmetric solutions for fixed cosmological constant.
- Found horizons are generally not embeddable in known exact solutions.
- Identified a new family of spacetimes extending Kerr-(A)dS and Taub-NUT.

## Abstract

We consider $3$-dimensional isolated horizons (IHs) generated by null curves that form nontrivial $U(1)$ bundles. We find a natural interplay between the IH geometry and the $U(1)$-bundle geometry. In this context we consider the Petrov type D equation introduced and studied in previous works \cite{DLP1,DLP2,LS,DKLS1}. From the $4$-dimensional spacetime point of view, solutions to that equation define isolated horizons embeddable in vacuum spacetimes (with cosmological constant) as Killing horizons to the second order such that the spacetime Weyl tensor at the horizon is of the Petrov type D. From the point of view of the $U(1)$-bundle structure, the equation couples a $U(1)$-connection, a metric tensor defined on the base manifold and the surface gravity in a very nontrivial way. We focus on the $U(1)$-bundles over $2$-dimensional manifolds diffeomorphic to $2$-sphere. We have derived all the axisymmetric solutions to the Petrov type D equation. For a fixed value of the cosmological constant they set a $3$-dimensional family as one could expect. A surprising result is, that generically our horizons are not embeddable in the known exact solutions to Einstein's equations. It means that among the exact type D spacetimes there exists a new family of spacetimes that generalize the properties of the Kerr- (anti) de Sitter black holes on one hand and the Taub-NUT spacetimes on the other hand.