Ergoregions in Magnetised Black Hole Spacetimes

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 solution. 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 give a self-contained account of the solution-generating procedure, including 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 $K_\Omega= \partial/\partial t+ \Omega \partial/\partial \phi$, because there is no canonical choice of $\Omega$. For one choice, $\Omega=\Omega_s$, the metric is asymptotically static, and there is an ergoregion confined to the neighbourhood of the horizon. On the other hand, by choosing $\Omega=\Omega_H$, so that $K_{\Omega_H}$ 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 $B\le B_-$ 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, we recover Wald's result that it is energetically favourable for the hole to acquire a charge $2jB$.