Rotating charged black strings and three-dimensional black holes

Einstein-Maxwell equations with a cosmological constant are analyzed in a four-dimensional stationary spacetime admitting in addition a two-dimensional group ${G}_{2}$ of spatial isometries. We find charged rotating black string solutions. For open black strings the mass ($M$), angular momentum ($J$), and charge ($Q$) line densities can be defined using the Hamiltonian formalism of Brown and York. It is shown through dimensional reduction that $M$, $J$, and $Q$ are, respectively, the mass, angular momentum, and charge of a related three-dimensional black hole. For closed black strings one can define the total mass, charge, and angular momentum of the solution. These closed black string solutions have a flat torus topology. The black string solutions are classified according to the mass, charge, and angular momentum parameters. The causal structure is studied and some Penrose diagrams are shown. There are similarities between the charged rotating black string and the Kerr-Newman spacetime. The solution has Cauchy and event horizons, ergosphere, timelike singularities, closed timelike curves, and extremal cases. Both the similarities and differences of these black strings and Kerr-Newman black holes are explored. We comment on the implications these solutions might have on the hoop conjecture.