Topological Censorship and Higher Genus Black Holes

Motivated by recent interest in black holes whose asymptotic geometry approaches that of anti-de Sitter spacetime, we give a proof of topological censorship applicable to spacetimes with such asymptotic behavior. Employing a useful rephrasing of topological censorship as a property of homotopies of arbitrary loops, we then explore the consequences of topological censorship for horizon topology of black holes. We find that the genera of horizons are controlled by the genus of the space at infinity. Our results make it clear that there is no conflict between topological censorship and the non-spherical horizon topologies of locally anti-de Sitter black holes. More specifically, let D be the domain of outer communications of a boundary at infinity “scri.” We show that the Principle of Topological Censorship (PTC), that every causal curve in D having endpoints on scri can be deformed to scri, holds under reasonable conditions for timelike scri, as it is known to do for a simply connected null scri. We then show that the PTC implies that the fundamental group of scri maps, via inclusion, onto the fundamental group ofD, i.e., every loop inD is homotopic to a loop in scri. We use this to determine the integral homology of preferred spacelike hypersurfaces (Cauchy surfaces or analogues thereof) in the domain of outer communications of any 4-dimensional spacetime obeying the PTC. From this, we establish that the sum of the 1 Dept. of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124, USA. e-mail:[email protected] 2 Dept. of Physics and Astronomy, University of British Columbia, 6224 Agriculture Road, Vancouver, BC, Canada V6T 1Z1. e-mail: [email protected] 3 Dept. of Physics and Astronomy, University of British Columbia, 6224 Agriculture Road, Vancouver, BC, Canada V6T 1Z1. e-mail: [email protected] 4 Dept. of Mathematical Sciences and Theoretical Physics Institute, University of Alberta, Edmonton, AB, Canada T6G 2G1. e-mail: [email protected]