In the text I'm looking at, the Schwarzschild metric derivation, and it argues to the form ## ds^{2}= -e^{2\alpha(r)} dt^{2} + e^{\beta(r)}+r^{2}d\Omega^{2} ## [1]. Up to this point some of the ##R_{uv}=0## components have been used, not all.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Spherically symmetric vacuum solution static or stationary?

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