Easy way of calculating Riemann tensor?

[dingo_d]

Homework Statement

Is there any painless way of calculating the Riemann tensor?

I have the metric, and finding the Christoffel symbols isn't that hard, especially if I'm given a diagonal metric.

Out of 40 components, most will be zero. But how do I know how to pick the indices of Riemann tensor, given the non vanishing Christoffel symbols?

Because I can't just go and put all the possible combinations :\

I have the metric:

$ds^2= -B(r)\text{dt}^2 +A(r)\text{dr}^2+r^2 \left(d \theta<br /> ^2+\sin^2\theta d\phi^2\right)$

And I have 13 Christoffel symbols that are different from zero. Mathematica says that there will be 6 nonzero components.

I also know that there are possible 20 linearly independent components of Riemann tensor, but how do I figure out which combinations of $t,r,\theta,\phi$ will give me the correct answer?

 

[member 553083]

Homework Equations The Riemann tensor is given by:R^{\alpha\beta}_{\gamma\delta}= \frac{\partial \Gamma^{\alpha\beta}_\delta}{\partial x^\gamma} - \frac{\partial \Gamma^{\alpha\beta}_\gamma}{\partial x^\delta} + \Gamma^{\alpha\sigma}_\gamma \Gamma^{\beta\delta}_\sigma - \Gamma^{\alpha\sigma}_\delta \Gamma^{\beta\gamma}_\sigmaThe Attempt at a Solution I have found the Christoffel symbols and I have written them in terms of the metric components.I am trying to figure out how to calculate the Riemann tensor components but I am stuck. I would appreciate if someone could explain to me how to pick the correct indices for the Riemann tensor given the non-zero Christoffel symbols.