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?

  • $\begingroup$ 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. $\endgroup$ – A.V.S. Dec 13 '18 at 9:07
  • $\begingroup$ right, thank you, I've been taking square roots. How stupid of me! $\endgroup$ – saad Dec 13 '18 at 9:23
  • $\begingroup$ Do you mean Riemann tensor components? Ricci tensor is 0 for the Schwarzschild solution. $\endgroup$ – enumaris Dec 13 '18 at 19:20
  • $\begingroup$ @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. $\endgroup$ – saad Dec 14 '18 at 2:03

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