How do we solve for Einstein's equation in the vacuum with a cosmological constant, in the static spherically symmetric situation?
Attempt:
Following Sean Carroll's Spacetime and Geometry (p.195), I write the equations
\begin{align}R_{tt} - \frac 12 R g_{tt} + \Lambda g_{tt} &= 0 \tag{1}\\ R_{rr} - \frac 12 R g_{rr} + \Lambda g_{rr} &= 0\tag{2}\end{align}
And contract $R_{\mu\nu}-\frac 12 R g_{\mu\nu} + \Lambda g_{\mu\nu} = 0$ with the inverse metric to obtain $R = 4\Lambda$, which I plug back in $(1)$ and $(2)$.
Then, as in Carroll, I multiply $(1)$ by $\exp[-2(\alpha-\beta)]$, where $\alpha$ and $\beta$ are functions of $r$ and add it to $(2)$ to obtain:
$$\frac 2r \partial_r(\alpha +\beta) = \Lambda e^{-2\beta}\left(1-e^{-4(\alpha+\beta)}\right)\tag{3}$$
Can we solve this differential equation?
Addendum: I simplified $(3)$ to
$$\frac 2r \partial_r(\alpha +\beta) = \Lambda \left(1-e^{-4(\alpha+\beta)}\right)\tag{3*}$$
and obtained
$$\Lambda r^2 + 4C = \ln\left(e^{4(\alpha+\beta)}+1\right)$$
where $C$ is a constant of integration.
