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Easy way of calculating Riemann tensor?

  1. Jan 10, 2012 #1
    1. The problem statement, all variables and given/known data

    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:

    [itex]ds^2= -B(r)\text{dt}^2 +A(r)\text{dr}^2+r^2 \left(d \theta
    ^2+\sin^2\theta d\phi^2\right)[/itex]

    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 [itex]t,r,\theta,\phi[/itex] will give me the correct answer?
  2. jcsd
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