The Kerr solution

Most known stars are rotating relative to their local inertial system (relative to the stars) and are therefore not spherically symmetric; their gravitational field is not described by the Schwarzschild solution. In Newtonian gravitational theory, although the field certainly changes because of the rotational flattening of the star, it still remains static, while in the Einstein theory, on the other hand, the flow of matter acts to produce fields. The metric will still be time-independent (for a timeindependent rotation of the star), but not invariant under time reversal. We therefore expect that the gravitational field of a rotating star will be described by an axisymmetric stationary vacuum solution which goes over to a flat space at great distance from the source. Depending on the distribution of matter within the star there will be different types of vacuum fields which, in the language of the Newtonian gravitational theory, differ, for example, in the multipole moments of the matter distribution. One of these solutions is the Kerr (1963) solution, found almost fifty years after the discovery of the Schwarzschild metric. It proves to be especially important for understanding the gravitational collapse of a rotating star. To avoid misunderstanding we emphasize that the Kerr solution is not the gravitational field of an arbitrary axisymmetric rotating star, but rather only the exterior field of a very special source.

We shall now discuss the Kerr solution and its properties. Since its mathematical structure is rather complicated, we shall not construct a derivation from the Einstein field equations.