Which exact solution describes a black hole that has both electric charge and rotation?

Which exact solution describes a black hole that has both electric charge and rotation? The answer is the Kerr–Newman metric.

The Kerr–Newman metric is an exact solution of the Einstein–Maxwell equations for a stationary, rotating, electrically charged black hole. It combines mass, angular momentum and electric charge in one relativistic description, making it the charged rotating member of the classic black-hole solution family.

Ezra Newman and collaborators obtained the solution in 1965, extending Roy Kerr’s 1963 solution for an uncharged rotating black hole. The metric includes an outer event horizon and, under suitable parameter limits, an inner horizon and an ergosphere where frame dragging occurs.

The related names are easy to confuse. Setting charge to zero yields the Kerr metric; removing rotation gives the Reissner–Nordström solution; removing both produces the Schwarzschild metric. Kerr–Newman is mainly theoretical because astrophysical black holes are expected to carry very little net electric charge.