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:

$$ds^2~=~-e^{2\alpha(r,t)}dt^2 + e^{2\beta(r,t)}dr^2 +r^2 d\Omega^2.
\qquad\qquad(A) $$

It is a straightforward exercise to calculate the corresponding Ricci tensor $R_{\mu\nu}$, see eq. (5.41) in Ref. 1 or eq. (7.16) in Ref. 2. The notation is here $x^0\equiv t$, $x^1\equiv r$, $x^2\equiv\theta$, and $x^3\equiv\phi$. Assuming a vanishing cosmological constant $\Lambda=0$, the Einstein's equations in vacuum read

$$R_{\mu\nu}~=~0~. $$

The argument is now as follows.

From $R_{tr}=0$ follows that $\beta$ is independent of $t$.

From $e^{2(\beta-\alpha)} R_{tt}+R_{rr}=0$ follows that $\partial_r(\alpha+\beta)=0$. In other words, the function $f(t):=\alpha+\beta $ is independent of $r$.

Define a new coordinate variable $T:=\int^t dt'~e^{f(t')}$. Then the metric $(A)$ becomes
$$ds^2~=~-e^{-2\beta}dT^2 + e^{2\beta}dr^2 +r^2 d\Omega^2.\qquad\qquad(B)$$

Rename the new coordinate variable $T\to t$. Then eq. $(B)$ corresponds to setting $\alpha=-\beta$ in eq. $(A)$.

From $R_{\theta\theta}=0$ follows that
$$1=e^{-2\beta}(1-2r\partial_r\beta)\equiv\partial_r(re^{-2\beta}),$$
so that $re^{-2\beta}=r-R$ for some real integration constant $R$. In other words, we have derived the Schwarzschild solution,
$$e^{2\alpha}~=~e^{-2\beta}~=~1-\frac{R}{r}.$$

Finally, if we switch back to the original $t$ coordinate variable, the metric $(A)$ becomes

$$ds^2~=~-\left(1-\frac{R}{r}\right)e^{2f(t)}dt^2
+ \left(1-\frac{R}{r}\right)^{-1}dr^2 +r^2 d\Omega^2.\qquad\qquad(C)$$

It is interesting that the metric $(C)$ is the most general metric of the form $(A)$ that satisfies Einstein's vacuum equations with $\Lambda=0$. 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