Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 23; Boxes 23.3 – 23.4.
The expressions for the components of the Ricci tensor for a spherically symmetric source look quite frightening as differential equations, and in the general case would be impossible to solve exactly. However, if we restrict ourselves to the vacuum, that is, to the region outside the source, things simplify a lot. In that case, because the stress-energy tensor , it follows from the Einstein equation that all components of the Ricci tensor must also be zero:
The metric has the form
and the Ricci components therefore give the PDEs:
The equation says
That is, can depend on only.
Next, notice that the terms in the brackets for and cancel in pairs except for a couple of terms, so we have
Plugging this into 5 we get
where is a constant of integration.
Now, from 11 and given that does not depend on , we must have independent of also. This can happen only if any dependence has on cancels out when we take the quotient , and this can happen only if for some functions and . In that case,
where we use total rather than partial derivatives in 18 because both and depend only on , and is another constant of integration.
The metric now looks like this:
In order for this metric to contain exactly one time coordinate, the coefficient of must be negative (giving the time coordinate), while the coefficients of the other three coordinates must be positive. Therefore and .
At this stage, we can transform the time coordinate so that
then replace by and drop the prime to get
We thus arrive (almost; we still have to find ) at the Schwarzschild metric. Note that in this form, the metric is independent of time, even though we haven’t assumed that the mass-energy of the source is independent of time, only that it is always spherically symmetric. Thus a star that expands or contracts while maintaining spherical symmetry would always give rise to the same metric. This is called Birkhoff’s theorem.
This choice of is the time measured by an observer at rest at infinity (), since to such an observer . This might look like a bit of a fudge, since we hid the time dependence of by sweeping it under the carpet with the rescaling of time in 23. However, on reflection, I think it does actually make sense, since in a more general case (if , say, or if the metric were non-diagonal), it wouldn’t be possible to find any time coordinate that gives a time-independent metric.