- #1
binbagsss
- 1,266
- 11
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 possesses 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.
It then says we have proven any spherically symmetric possesses 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.