The Schwarzschild solution describes the spacetime geometry outside a spherically symmetric massive object, characterised only by its mass M. In the previous chapter we derived further spherically symmetric solutions. Most real astrophysical objects, however, are rotating. In this case, a spherically symmetric solution cannot apply because the rotation axis of the object defines a special direction, so destroying the isotropy of the solution. For this reason, in general relativity it is not possible to find a coordinate system that reduces the spacetime geometry outside a rotating (uncharged) body to the Schwarzschild geometry. The non-linear field equations couple the source to the exterior geometry. Moreover, a rotating body is characterised not only by its mass M but also by its angular momentum J, and so we would expect the corresponding spacetime metric to depend upon these two parameters.

We now consider how to derive the metric describing the spacetime geometry outside a rotating body. Since the mathematical complexity in this case is far greater than that encountered in deriving the Schwarzschild metric (or the other spherically symmetric geometries discussed in the previous chapter), we shall content ourselves with just an outline of how the solution may be obtained.

The general stationary axisymmetric metric

In our derivation of the Schwarzschild solution, we began by constructing the general form of the static isotropic metric. We are now interested in deriving the spacetime geometry outside a steadily rotating massive body.