The exact vacuum solution for a rotating, uncharged black hole in general relativity is the Kerr metric.
The Kerr metric describes the geometry of empty spacetime around a stationary, rotating, axially symmetric mass. When the object is compact enough to form a black hole, the solution includes an event horizon and an ergosphere, a region outside the horizon where spacetime is dragged around by the black hole’s rotation. The solution is characterized primarily by mass and angular momentum because its electric charge is zero.
New Zealand mathematician Roy Kerr discovered the solution in 1963, nearly five decades after Karl Schwarzschild found the simpler non-rotating solution. Rotation makes Einstein’s nonlinear field equations substantially harder to solve, which is why Kerr’s result was a major advance. The Kerr metric is also used as an idealized model for the exterior spacetime of rotating astrophysical black holes.
The Reissner–Nordström metric is the common distractor: it describes a non-rotating, electrically charged black hole. Adding electric charge to the rotating case gives the Kerr–Newman metric. Setting the Kerr angular momentum to zero reduces the geometry to the Schwarzschild metric.