The spacetimes obtained by Ernst's procedure for appending an external magnetic field B to a seed Kerr-Newman black hole are commonly believed to be asymptotic to the static Melvin metric. We show that this is not in general true. Unless the electric charge of the black hole satisfies Q = jB(1 + 1 4 j 2 B 4 ), where j is the angular momentum of the original seed solution, an ergoregion extends all the way from the black hole horizon to infinity. We find that if the condition on the electric charge is satisfied then the metric is asymptotic to the static Melvin metric, and the electromagnetic field carries not only magnetic, but also electric, flux along the axis. We give a self-contained account of the solution-generating procedure, including explicit formulae for the metric and the vector potential. In the case when Q = jB(1 + 1 4 j 2 B 4 ), we show that there is an arbitrariness in the choice of asymptotically timelike Killing field K Ω = ∂/∂t + Ω ∂/∂φ, because there is no canonical choice of Ω. For one choice, Ω = Ω s , the metric is asymptotically static, and there is an ergoregion confined to the neighbourhood of the horizon. On the other hand, by choosing Ω = Ω H , so that K Ω H is co-rotating with the horizon, then for sufficiently large B numerical studies indicate there is no ergoregion at all. For smaller values, in a range B − < B < B + , there is a toroidal ergoregion outside and disjoint from the horizon.If B ≤ B − this ergoregion expands all the way to infinity in a cylindrical region near to the rotation axis. For black holes whose size is small compared to the Melvin radius 2/B, and neglecting back-reaction of the electromagnetic field, we recover Wald's result that it is energetically favourable for the hole to acquire a charge 2jB.

It is known that every compact simple Lie group admits a bi-invariant homogeneous Einstein metric. In this paper we use two ansatz to probe the existence of additional inequivalent Einstein metrics on the Lie group SU (n) for arbitrary n. We provide an explicit construction of (2k+1) inequivalent Einstein metrics on SU (2k) and 2k inequivalent Einstein metrics on SU (2k + 1).

A precise formulation of the hoop conjecture for four-dimensional spacetimes proposes that the Birkhoff invariant β for an apparent horizon in a spacetime with mass M should satisfy β ≤ 4πM . The invariant β is the least maximal length of any sweepout of the 2-sphere apparent horizon by circles. An analogous conjecture in five spacetime dimensions was recently formulated, asserting that the Birkhoff invariant β for S 1 ×S 1 sweepouts of the apparent horizon should satisfy β ≤ 16 3 πM . Although this hoop inequality was formulated for conventional five-dimensional black holes with 3-sphere horizons, we show here that it is also obeyed by a wide variety of black rings, where the horizon instead has S 2 × S 1 topology.