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It then says we have proven any spherically symmetric possess a time-like killing vector, and so is stationary. This is fine.

But , it then goes on to complete the derivtion of the Schwarzschild metric and explains that this is actually static.

QUESTION:

By Birkoff's theorem, this metric is the unique spherically symmetric vacuum solution, so haven't we proven that this solution is static?

If so, I don't understand the ordering of the text, or is it saying that form [1], were we have yet to use all ##R_{uv}=0##, at this point we can only conclude the metric to be stationary, but once we have used all ##R_{uv}=0## we see it is stationary,

**Thanks in advance.**