We now consider how to solve the Einstein field equations and so discover the metric functions gμν in any given physical situation. Clearly, the high degree of non-linearity in the field equations means that a general solution for an arbitrary matter distribution is analytically intractable. The problem becomes easier if we look for special solutions, for example those representing spacetimes possessing symmetries. The first exact solution to Einstein's equations was found by Karl Schwarzschild in 1916. As we shall see, the Schwarzschild solution represents the spacetime geometry outside a spherically symmetric matter distribution.

The general static isotropic metric

Schwarzschild sought the metric gµν representing the static spherically symmetric gravitational field in the empty space surrounding some massive spherical object such as a star. Thus, a good starting point for us is to construct the most general form of the metric for a static spatially isotropic spacetime.

A static spacetime is one for which some timelike coordinate x0 (say) with the following properties: (i) all the metric components gµν are independent of x0; and (ii) the line element ds2 is invariant under the transformation x0 → −x0. Note that (i) does not necessarily imply (ii), as is made clear by the example of a rotating star: time reversal changes the sense of rotation, but the metric components are constant in time. A spacetime that satisfies (i) but not (ii) is called stationary.