Stationary Vacuum Black Holes in 5 Dimensions

TL;DR

This paper classifies stationary vacuum black hole solutions in five dimensions with various horizon topologies, including new black lens solutions, by reducing Einstein's equations to a harmonic map problem and analyzing singularities.

Contribution

It provides the first explicit solutions for smooth vacuum black lenses in 5D and extends the analysis to spacetimes with orbifold singularities.

Findings

Classified all possible horizon topologies in 5D vacuum black holes.

Constructed the first candidates for smooth black lens solutions.

Derived conditions to avoid conical singularities for physical relevance.

Abstract

We study the problem of asymptotically flat bi-axially symmetric stationary solutions of the vacuum Einstein equations in $5$-dimensional spacetime. In this setting, the cross section of any connected component of the event horizon is a prime $3$-manifold of positive Yamabe type, namely the $3$-sphere $S^3$, the ring $S^1\times S^2$, or the lens space $L(p,q)$. The Einstein vacuum equations reduce to an axially symmetric harmonic map with prescribed singularities from $\mathbb{R}^3$ into the symmetric space $SL(3,\mathbb{R})/SO(3)$. In this paper, we solve the problem for all possible topologies, and in particular the first candidates for smooth vacuum non-degenerate black lenses are produced. In addition, a generalization of this result is given in which the spacetime is allowed to have orbifold singularities. We also formulate conditions for the absence of conical singularities which guarantee a physically relevant solution.