Einstein equations in the spherically symmetric, static case

Question

This question is not about the solutions but much rather about the equations we write in GR for a spherically symmetric, static vacuum 4D spacetime.

The Einstein equations are $$G_{\mu\nu}=0\;\;\;\Rightarrow\;\;\; g^{\mu\nu}G_{\mu\nu}=0\;\;\;\Rightarrow \;\;\;R=0$$ so that in conclusion we can solve $$R_{\mu\nu}=0 \;\;\;\text{or}\;\;\;R^{\mu}_{\nu}=0$$ Moreover, we can choose the coordinates so that the metric takes the form $$ds^2=-f(r)\,dt^2+h^{-1}(r)\,dr^2+r^2\!\left(d\theta^2+\sin^2\!\theta \,d\varphi^2\right)$$ In this way when we write explicitly the equations $R^{\mu}_{\nu}=0$ we get three non trivial equations, namely: $$R^{0}_0=0\;\;,\qquad R^r_r=0\;\;,\qquad R^{\theta}_{\theta}\equiv R^{\varphi}_{\varphi}=0$$

Using the first two equations and the formal identity $R^{\theta}_{\theta}\equiv R^{\varphi}_{\varphi}$ , we can write the third simply as $R=0$.

My question is what is the meaning of this third equation, when the variables are just the two functions $f(r)\;$ and $\;h(r)$? Does it act like a constraint or is there any way to get around it? and what does it tell us about GR in spherically symmetric and static space-times?

I would guess that it corresponds to a relation we impose between $f$, $h$ (and the coordinate $r$) when we write the solid angle part as in flat space, but I don't really understand its role.

Answer 1

The metric in a vacuum around a spherically symmetric and static massive body is in principle written as
$ds^2 = -U(r) dt^2 + V(r) dr^2 + W(r) r^2 (d\theta^2 + \sin^2 \theta d\phi^2)$.
The three independent equations $R_{t t} = R_{r r} = R_{\theta \theta} = 0$, ($R_{\theta \theta}$ and $R_{\phi \phi}$ linearly dependent), allow for the determination of the three radial functions $U(r)$, $V(r)$, $W(r)$.

However the three radial functions, being $r$ simply a radial parameter and not an actual distance, can be reduced to two funtions. Put $W(r) r^2 = \hat r^2$, express $U(r) = \hat U(\hat r)$ and $V(r) = \hat V(\hat r)$. Then removing the hats and putting $U(r) = e^{2 \nu(r)}$, $V(r) = e^{2 \lambda(r)}$ the line element becomes
$ds^2 = -e^{2 \nu(r)} dt^2 + e^{2 \lambda(r)} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)$
If you start from this metric assumption, you have three independent equations for two functions $\nu(r)$ and $\lambda(r)$. One equation is redundant, however to determine the expression of the radial functions you may choose which equations are more convenient for an easy computation.

Note: There is not a specific meaning of the third equation, as each of the three equations can be taken as redundant compared to the other two equations.

Answer 2

$\let\a=\alpha \let\b=\beta \let\th=\vartheta$

My question is what is the meaning of this third equation

I think the answer is the following. You know that Einstein's tensor is divergenceless. If you expand $G^\a_{\b;\a}$ you'll find that three of the four equations are trivial identities. The only one surviving is the $r$-component and you'll be able to see that it simply says $$G^r_{r,r} = 0$$ (no sum).

So $G^r_r$ is a constant and if the spacetime is asymptotically flat the constant is zero. Then one of Einstein's equations is automatically satisfied and only the remaining two $$G^t_t = 0 \qquad G^\th_\th = 0$$ are real equations for $f(r)$, $h(r)$.