Notes on maximal slices of five-dimensional black holes

Abstract

We consider maximal slices of the Myers-Perry black hole, the doubly spinning black ring, and the Black Saturn solution. These slices are complete, asymptotically flat Riemannian manifolds with inner boundaries corresponding to black hole horizons. Although these spaces are simply connected as a consequence of topological censorship, they have non-trivial topology. In this note we investigate the question of whether the topology of spatial sections of the horizon uniquely determines the topology of the maximal slices. We show that the horizon determines the homological invariants of the slice under certain conditions. The homological analysis is extended to black holes for which explicit geometries are not yet known. We believe that these results could provide insights in the context of proving existence of deformations of this initial data. For the topological slices of the doubly spinning black ring and the Black Saturn we compute the homotopy groups up to dimension 3 and show that their 4-dimensional homotopy group is not trivial.

Soliton mechanics

“…As a last example, we consider asymptotically flat dipole black rings [15] where the horizon topology is S 1 ×S 2 and Σ ∼ = R 4 #(S 2 ×D 2 ) [20,21] .The rings are a solution to five dimensional Einstein-Maxwell theory (and also the minimal supergravity theory because the Chern-Simons term is of no consequence to the solutions). For convenience to match with the conventions used in [15], in this section we take g IJ = 1/2 in the general formalism of [13].…”

Mass–Angular-Momentum Inequality For Black Ring Spacetimes

“…According to [22,23], the maximal constant time slices in the domain of outer communication of the stationary black ring family are diffeomorphic to…”

Plumbing constructions and the domain of outer communication for 5-dimensional stationary black holes

“…Recall that according to the discussion in Section 2 the boundary ∂P(ξ 1 , ξ 2 ) = S 3 . This entails replacing the horizon rod with the sequence of rod structures (0, 1), (1,0), to obtain the expanded or enhanced rod structure {(1, 0), (0, 1), (1, 0), (0, 1)}. The compactified Cauchy surface of the DOC is thenM 4 = S 2 × S 2 , which may be computed from the chart in [29, pg.…”

“…By filling in the two ring horizons with the trivial disc bundle over S 2 the resulting extended rod structure sequence takes the form Observe that both sets of rod structures begin and end with (1, 0), (0, 1) indicating that the asymptotic end is of the form R + × S 3 , and both horizon rods are bounded between the axis rods (1, 0), (−1, 2) signifying that the horizon topology is the lens space L(2, 1) = RP 3 . The only difference between the two sequences is that addition of two axis rods in (4.11) having rod structures (0, 1), (1,0). This adds two additional corners and changes the topology of the corresponding DOCs.…”

“…A basic question posed in the literature [1] is the following. Does the topology of horizon crosssections and the asymptotic end uniquely determine the topology of the domain of outer communication for stationary (vacuum) black holes in 5-dimensions?…”