# Finding the Ricci tensor components for the Schwarzschild metric

I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the components of the Ricci tensor.

The given distance element is

$$ds^2 = e^{2 \lambda} dt^2 - e^{2 \mu} dr^2 - r^2({d \theta^2}+ \sin^2 \theta d\phi^2)$$

Thus $$g_{\mu \nu} = \text{diag}(e^{2 \lambda}, -e^{2\mu}, -r^2, -r^2 \sin^2\theta)$$ and $$g^{\mu \nu} = \text{diag}(e^{-2\lambda}, -e^{-2\mu}, -\frac{1}{r^2}, -\frac{1}{r^2 \sin^2 \theta})$$

Using Cartan's structure equations, I have worked out the Ricci tensor components $$R_{011}^{\space\space\space\space\space\space 0}$$, $$R_{022}^{\space\space\space\space\space\space 0}$$, $$R_{033}^{\space\space\space\space\space 0}$$, $$R_{122}^{\space\space\space\space\space\space 1}$$,$$R_{133}^{\space\space\space\space\space\space 1}$$ and $$R_{233}^{\space\space\space\space\space\space2}$$. Then $$R_{00} = R_{0\rho0}^{\space\space\space\space\space\space \rho} = R_{000}^{\space\space\space\space\space\space 0}+ R_{010}^{\space\space\space\space\space\space 1} + R_{020}^{\space\space\space\space\space\space 2} + R_{030}^{\space\space\space\space\space\space 3}$$

Now $$R_{000}^{\space\space\space \space \space 0} = 0$$ by antisymmetry of the first two indices. The last 3 terms look like the known components $$R_{011}^{\space\space\space\space\space\space 0}$$, $$R_{022}^{\space\space\space\space\space\space 0}$$, $$R_{033}^{\space\space\space\space\space 0}$$ except with adjacent upstairs and downstairs indices switched. Is the following how we obtain them?

$$R_{abc}^{\space\space\space \space \space d } = g^{de} R_{abce} = -g^{de} R_{abec} = -g_{cf}g^{de} R_{abe}^{\space\space\space \space \space f}$$. Now $$\because$$ the metric is a diagonal one, the only surviving term is the one for which $$f = c$$ and $$d = e$$. Therefore,

$$R_{abc}^{\space\space\space \space \space d} = - g_{cc}g^{dd}R_{abd}^{\space\space\space \space \space c}$$ (no summation)

So, for example,

$$R_{010}^{\space \space \space \space \space 1} = -g_{00}g^{11} R_{011}^{\space \space \space \space \space 0} = -(e^{2 \lambda}) (-e^{-2 \mu}) R_{011}^{\space \space \space \space \space 0}$$

...and so on and so forth for the other Riemann tensor components.

If it isn't, how would we get these components from the known components of the Riemann tensor?

• Why would you take square root of $e^{2\lambda}$ and other components in your $\mathrm{diag}$ expression? For that matter taking square root of $-r^2$ should have produced an imaginary number, which would have been a hint that something is wrong. – A.V.S. Dec 13 '18 at 9:07
• right, thank you, I've been taking square roots. How stupid of me! – saad Dec 13 '18 at 9:23
• Do you mean Riemann tensor components? Ricci tensor is 0 for the Schwarzschild solution. – enumaris Dec 13 '18 at 19:20
• @enumaris Yes, the Riemann tensor components to find the traces (i.e. the Ricci tensor components) and then set them equal to zero so that we can find $\lambda$ and $\mu$ and thus the Schwarzschild solution. – saad Dec 14 '18 at 2:03