Calculate Riemann tensor according to veilbein

[oztopux]

Homework Statement

How to use veilbein to calculate Riemann tensor, Ricci tensor and Ricci scalar?
(please give me the details)
$de^a+\omega_{~b}^a\wedge e^b=0$,
$R_{~b}^a=d\omega_{~b}^a+\omega_{~c}^a\wedge\omega_{~b}^c$.
The metric is
$ds^2=-e^{2\alpha(r)}dt^2+e^{2\beta(r)}dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2$.
(Sean Carroll, page 195).

Homework Equations

the metric is:
$ds^2=-e^{2\alpha(r)}dt^2+e^{2\beta(r)}dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2$.
the veilbeins:
$e^0=e^\alpha dt,~~e^1=e^\beta dr,~~e^2=rd\theta,~~e^3=r\sin\theta d\phi$.

$de^a+\omega_{~b}^a\wedge e^b=0$.

1. $de^0+\omega_{~1}^0\wedge e^1=0$
$\alpha'e^\alpha dr\wedge dt+\omega_{~1}^0\wedge e^\beta dr=0$
$~~~\omega_{~1}^0=\alpha'e^\alpha e^{-\beta}dt$.

2. $de^2+\omega_{~1}^2\wedge e^1=0$
$dr\wedge d\theta+\omega_{~1}^2\wedge e^\beta dr=0$
$~~~\omega_{~2}^1=-e^{-\beta}d\theta$.

3. $de^3+\omega_{~1}^3\wedge e^1=0$
$\sin\theta dr\wedge d\phi+\omega_{~1}^3\wedge e^\beta dr=0$
$~~~\omega_{~3}^1=-e^{-\beta}\sin\theta d\phi$.

4. $de^3+\omega_{~2}^3\wedge e^2=0$
$r\cos\theta d\theta\wedge d\phi+\omega_{~2}^3\wedge rd\theta=0$
$~~~\omega_{~3}^2=-\cos\theta d\phi$.

I only calculate 4 spin connections, but how to calculate $\omega_{~2}^0,~\omega_{~3}^0$?

The Attempt at a Solution

 

[arkajad]

Why did you skip summations over index b in the first formula? Why do you have only one term there?