Here we will only consider the first half of the question(v2).
The Birkhoff's Theorem is e.g. proven (at a physics level of rigor) in Ref. 1 and Ref. 2. Imagine that we have managed to argue that the metric is of the form of eq. (5.38) in Ref. 1 or eq. (7.13) in Ref. 2:
ds2 = −e2α(r,t)dt2+e2β(r,t)dr2+r2dΩ2.(A)
It is a straightforward exercise to calculate the corresponding Ricci tensor Rμν, see eq. (5.41) in Ref. 1 or eq. (7.16) in Ref. 2. The notation is here x0≡t, x1≡r, x2≡θ, and x3≡ϕ. Assuming a vanishing cosmological constant Λ=0, the Einstein's equations in vacuum read
Rμν = 0 .
The argument is now as follows.
From Rtr=0 follows that β is independent of t.
From e2(β−α)Rtt+Rrr=0 follows that ∂r(α+β)=0. In other words, the function f(t):=α+β is independent of r.
Define a new coordinate variable T:=∫tdt′ ef(t′). Then the metric (A) becomes
ds2 = −e−2βdT2+e2βdr2+r2dΩ2.(B)
Rename the new coordinate variable T→t. Then eq. (B) corresponds to setting α=−β in eq. (A).
From Rθθ=0 follows that
1=e−2β(1−2r∂rβ)≡∂r(re−2β),
so that re−2β=r−R for some real integration constant R. In other words, we have derived the Schwarzschild solution,
e2α = e−2β = 1−Rr.
Finally, if we switch back to the original t coordinate variable, the metric (A) becomes
ds2 = −(1−Rr)e2f(t)dt2+(1−Rr)−1dr2+r2dΩ2.(C)
It is interesting that the metric (C) is the most general metric of the form (A) that satisfies Einstein's vacuum equations with Λ=0. The only freedom is the function f=f(t), which reflects the freedom to reparametrize the t coordinate variable.
References:
Sean Carroll, Spacetime and Geometry: An Introduction to General Relativity, 2003.
Sean Carroll, Lecture Notes on General Relativity, Chapter 7. The pdf file is available here.
This post imported from StackExchange Physics at 2014-05-01 12:16 (UCT), posted by SE-user Qmechanic