Finding the Ricci tensor for the Schwarzschild metric

[saadhusayn]

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? Thank you!

 

[mark726]

Hello,

Thank you for sharing your work and question on using Cartan's method to find the Schwarzschild metric components. It looks like you have a good understanding of the method and have made some progress in calculating the Ricci tensor components. However, I would like to clarify a few things and offer some suggestions for obtaining the remaining components.

Firstly, it is important to note that the components $R_{abc}^{\space\space\space \space \space d}$ are not the same as the Ricci tensor components $R_{\mu \nu}$. The components $R_{abc}^{\space\space\space \space \space d}$ are the components of the Riemann tensor, which is a different tensor than the Ricci tensor. The Riemann tensor is a 4th order tensor, while the Ricci tensor is a 2nd order tensor.

To obtain the Ricci tensor components, you can use the following relation:

$$R_{\mu \nu} = R_{\mu \rho \nu}^{\space\space\space\space\space\space \rho} $$

where $R_{\mu \rho \nu}^{\space\space\space\space\space\space \rho}$ are the components of the Ricci tensor in the mixed form (one upper index and one lower index). These components can be calculated using the relation you have mentioned:

$$R_{\mu \rho \nu}^{\space\space\space\space\space\space \rho} = g^{\rho \sigma} R_{\mu \rho \nu \sigma} $$

where $R_{\mu \rho \nu \sigma}$ are the components of the Riemann tensor in the mixed form. So, for example, to obtain the component $R_{010}$, you can use the following relation:

$$R_{010} = R_{010}^{\space\space\space\space\space\space \rho} = g^{\rho \sigma} R_{010 \sigma} $$

where $g^{\rho \sigma}$ are the components of the inverse metric tensor, which you have already calculated. Similarly, you can obtain the other Ricci tensor components using the same method.

I hope this helps clarify the difference between the Riemann tensor and the Ricci tensor and