I am calculating the Riemann tensor for the Schwarzschild solution. I've calculated all 9 non-vanishing Christoffel symbols already. Now I need to evaluate the Riemann tensor and I find no easy way to do it. I have
$$ R^\alpha{}_{\beta\gamma\delta} = \partial_\gamma\Gamma^\alpha_{\beta\delta} – \partial_\delta\Gamma^\alpha_{\beta\gamma} + \Gamma^\mu_{\beta\delta} \Gamma^\alpha_{\mu\gamma} – \Gamma^\mu_{\beta\gamma} \Gamma^\alpha_{\mu\delta} $$
and I only have the following connections different from zero:
$$ \Gamma^r_{tt}, \Gamma^r_{\theta\theta},\Gamma^r_{\phi\phi},\Gamma^r_{rr}, \Gamma^\theta_{r\theta},\Gamma^\phi_{r\phi}, \Gamma^\theta_{\phi\phi}, \Gamma^\phi_{\theta\phi}, \Gamma^t_{tr} $$
I'm thinking of putting in the first partial derivative $\partial_\gamma\Gamma^\alpha_{\beta\delta}$ only one of those symbols I already have, however I'm afraid it won't make me go trough every possible non-vanishing Riemann tensor component I need.
Will I get all the components? Are there other easy ways to do it?
PS1: Yes, I know the symmetries and that there are only 20 independent components
PS2: I also know that I have the answer in the book, but I want to do it myself to practice
PS3: I don't want a specifically method for Schwarzschild solution only, a more "general easy way out" of it.
Best Answer
There is a relatively fast approach to computing the Riemann tensor, Ricci tensor and Ricci scalar given a metric tensor known as the Cartan method or method of moving frames. Given a line element,
$$ds^2 = g_{\mu\nu}dx^\mu dx^\nu$$
you pick an orthonormal basis $e^a = e^a_\mu dx^\mu$ such that $ds^2 = \eta_{ab}e^a e^b$. The first Cartan structure equation,
$$de^a + \omega^a_b \wedge e^b = 0$$
allows one to solve for the spin connection components $\omega^a_b$ from which one can compute the Ricci tensor in the orthonormal basis:
$$R^a_b = d\omega^a_b + \omega^a_c \wedge \omega^c_b.$$
The entire process simply requires exterior differentiation of the basis and spin connection. The Riemann components may be deduced from the relation,
$$R^a_b = R^a_{bcd} \, e^c \wedge e^d$$
possibly with a factor of $\frac12$ depending on your conventions. To convert back to the coordinate basis, one must simply contract with the basis back:
$$R^\mu_{\nu \lambda \kappa} = (e^{-1})^\mu_a \, R^a_{bcd}\, e^b_\nu \, e^c_\lambda \, e^d_\kappa.$$
For an explicit calculation see my previous answers here, here and here. The gravitational physics lectures at pirsa.org also provide explicit examples. As for using computer algebra systems, if all you're looking to do is compute curvature tensors, Hartle's textbook for Mathematica is your best option or the GREAT package. If you'd like to do more advanced stuff like perturbation theory, then xAct is required.