An exact solution of the Einstein field equations that is the metric outside a spinning sphere found by Kerr (1963). It was initially not possible to show that this solution fit smoothly with an exact interior solution. Cohen (1967) found a solution for a thin rotating spherical shell which is valid both inside and outside to lowest order in the rotation angular frequency and all orders in the shell mass and which satisfies the correct continuity conditions (Weinberg 1972, p. 241). However, the fact that Kerr's solution was unique and complete was subsequently demonstrated (Shapiro and Teukolsky 1983, p. 338).
Let the black hole have mass M and angular momentum J. The Kerr solution is then given by


where x is a quasi-Euclidean three-vector, a is a constant vector, scalar products , , and so on are defined in Euclidean geometry, is defined by

and (Weinberg 1972, p. 241). In Boyer-Lindquist coordinates, the metric becomes


where the black hole is rotating in the direction and



(Shapiro and Teukolsky 1983, p. 357). This metric is stationary (i.e., independent of t).