In a exterior region without matter to a stationary black hole, spherical symmetric, where the cosmological constant is not zero. From the Cartan's structural equations for space without torsion, we get the following non zero Ricci's tensor:
$$ R_{r0} = 0 \\ R_{00} =e^{-(U + V)}[\partial_r ( e^{(-V + U)} \partial_rU)] + \frac{2}{r} e^{-2V} \partial_rU \\ R_{rr} = -e^{-(U+V)}[\partial_r(e^{(-V + U)} \partial_rU] + \frac{2}{r} e^{-2V} \partial_rV \\ R_{\theta \theta} = R_{\phi \phi} =\frac{e^{-2V}}{r}(\partial_rV - \partial_rU) + \frac{1}{r^2} (1 - e^{-2V}) $$
Then I am asked to find components of the metric resolving Einstein's equations and right here I'm kinda stuck, I can write the general Einstein's equation:
$$ G_{\mu \nu} + \Lambda g_{\mu\nu} = R_{\mu \nu} - \frac{R}{2}g_{\mu \nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu \nu}$$
and then how can find the components of the metric using this?
Edit:
Corrected Einstein's Equation
Edit2:
Rereading the exercise, I noticed I had more information that I didn't wrote on this post.
The metric components are given by:
$$ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu} = \eta_{\mu \nu} \omega^{\mu} \otimes \omega^{\nu} $$
and $$ \begin{align} \omega^0 = e^{U(r)dt} \\ \omega^1 = e^{V(r)dt} \\ \omega^{\theta} = rd\theta \\ \omega^{\phi} = r\sin\theta d\phi \end{align} $$
From this I can compute the metric and write:
$$ g_{\mu \nu} = diag(-e^{2U(r)}, e^{2V(r)}, r^2 , r^2 \sin^2 \theta) $$
So for the component $00$, we write:
$$R_{00} - \frac{R}{2}g_{00} + \Lambda g_{00} = 0 \\ \Leftrightarrow R_{00} - \frac{g^{\alpha \beta} R_{\alpha \beta}}{2}g_{00} + \Lambda g_{00} = 0 $$