Vacuum Petrov type D horizons of non-trivial $U(1)$ bundle structure over Riemann surfaces with genus $>0$

Abstract

We consider isolated horizons (Killing horizons up to the second order) whose null flow has the structure of a U(1) principal fiber bundle over a compact Riemann surface. We impose the vacuum Einstein equations (with the cosmological constant) and the condition that the spacetime Weyl tensor is of Petrov D type on the geometry of the horizons. We derive all the solutions in the case when the genus of the surface is $>1$. By doing so for all the non-trivial bundles, we complete the classification. We construct the embedding spacetimes and show that they are locally isometric to the toroidal or hyperbolic generalization of the Taub-NUT-(anti-) de Sitter spacetimes for horizons of genus $1$ or $>1$ respectively, after performing Misner's identification of the spacetime. The horizon bundle structure can be naturally extended to bundle structure defined on the entire spacetime.

24 pages, minor typos corrected, version published in PRD

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• Symmetric non-expanding horizons
• Horizons and Null Infinity: A Fugue in 4 voices
• Non-singular extension of the Kerr-NUT-(anti) de Sitter spacetimes
• Extremal isolated horizons with $\Lambda$ and the related unique type D black holes