We consider three-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 [D. Dobkowski-Ryłko, J. Lewandowski, and T. Pawłowski, The Petrov type D isolated null surfaces, Classical Quantum Gravity 35, 175016 (2018); D. Dobkowski-Ryłko, J. Lewandowski, and T. Pawłowski, Local version of the no-hair theorem, Phys. Rev. D 98, 024008 (2018); J. Lewandowski and A. Szereszewski, The axial symmetry of Kerr without the rigidity theorem, Phys. Rev. D 97, 124067 (2018); D. Dobkowski-Ryłko, W. Kamiński, J. Lewandowski, and A. Szereszewski, The Petrov type D equation on genus $>0$ sections of isolated horizons, Phys. Lett. B 783, 415 (2018)]. From the four-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 two-dimensional manifolds diffeomorphic to 2-sphere. We have derived all the axisymmetric solutions to the Petrov type $D$ equation. They set a four-dimensional family of horizons and there is a four-dimensional family of the Kerr-NUT-dS (AdS) spacetimes in the literature. A surprising result is, that generically, our horizons do not correspond to those spacetimes. It means that among the exact type $D$ spacetimes there exists a new four-dimensional 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.

• Received 1 August 2019

DOI:https://doi.org/10.1103/PhysRevD.100.084058