Spherically symmetric vacuum solution static or stationary?

binbagsss
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.
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,

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

Yes. (Technically, that's not exactly what is proven, because the proof of Birkhoff's theorem does not require the additional KVF to be timelike everywhere. But I don't think we need to go into that here.)

If so, I don't understand the ordering of the text

Which text are you looking at?

binbagsss
Yes. (Technically, that's not exactly what is proven, because the proof of Birkhoff's theorem does not require the additional KVF to be timelike everywhere. But I don't think we need to go into that here.)

Which text are you looking at?

Lecture Notes on General Relatvitiy, 1997, Sean Carroll, page 169.

Mentor
Lecture Notes on General Relatvitiy, 1997, Sean Carroll, page 169.

Ok. The equation you labeled [1] in the OP is equation (7.20) in the text. In the paragraph following that equation, Carroll notes that the metric described in that equation is actually static, not just stationary. He doesn't show that explicitly, and the rest of the derivation is not intended to show that; it's just intended to show that, when you take the rest of the Ricci tensor components into account, the metric turns out to be the Schwarzschild metric--i.e., that there must be a particular relationship between the functions ##\alpha## and ##\beta## in equation (7.20). But none of that is necessary to show that the metric in (7.20) is static; that can be done just from the form of (7.20).

A static metric, as Carroll notes in the paragraph following equation (7.20), is stationary and hypersurface orthogonal; i.e., the timelike KVF is orthogonal to some family of hypersurfaces. In the coordinate chart in which equation (7.20) is expressed, the vector field ##\partial / \partial_t## is the KVF; and it is easy to show that this vector field is orthogonal to surfaces of constant ##t##, i.e., constant coordinate time: this is obvious from the fact that there are no "cross terms" in the metric (terms of the form ##dt dr##, ##dt d\theta##, or ##dt d\phi##). This shows that the metric of equation (7.20) is static, regardless of any other considerations.