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Riemann-Christoffel tensor

  1. May 1, 2010 #1
    I'm trying to work through getting the Riemann-Christoffel tensor using covariant differentiation and I don't see where two terms cancel. I have the correct result, plus these two terms:

    d/dx^(sigma) *{alpha nu, tau}*A^(alpha)
    d/dx^(nu) *{alpha sigma, tau}*A^(alpha)

    Sorry, I couldn't figure out how to do this with LaTeX. The A^(alpha) is just an arbitrary contravariant vector, and the {a n, t} and {a sigma, t} are Christoffel symbols.

    Somehow these two are supposed to be equal (in order to cancel). I know the Christoffel symbols are symmetric in the lower indices, but that doesn't help me much. Can anyone shed some light on why the two are the same?
  2. jcsd
  3. May 1, 2010 #2
    Or more simply put:

    {alpha nu, tau}*d/dx^sigma - {alpha sigma, tau}*d/dx^nu = 0

    How can I show this is true? Is there some way of writing this with the nu and sigma switched in one of the terms?

  4. May 2, 2010 #3


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    Science Advisor

    I'm afraid you have to be a little more specific. I don't see why an expression like

    \Gamma^{\tau}_{\alpha\nu} \frac{\partial}{\partial x^{\sigma}} -
    \Gamma^{\tau}_{\alpha\sigma} \frac{\partial}{\partial x^{\nu}}

    would disappear. On what does it act? Maybe you can rewrite the partial derivatives in terms of covariant ones?

    And for future questions: learn how to use latex. You can look at the code I've written down. It's a matter of hours to get the basics, and eventually you will need it anyway if you study physics or math ;)
  5. May 2, 2010 #4
    Thanks for the reply (and the latex sample!). It is acting on a vector A:

    \Gamma^{\tau}_{\alpha\nu} \frac{\partial A^{\alpha}}{\partial x^{\sigma}} -
    \Gamma^{\tau}_{\alpha\sigma} \frac{\partial A^{\alpha}}{\partial x^{\nu}}

    I've tried rewriting the partials as

    A^{\alpha}_{\sigma} - \Gamma^{\alpha}_{\mu\sigma} A^{\mu}


    A^{\alpha}_{\nu} - \Gamma^{\alpha}_{\mu\nu} A^{\mu}

    which would give me

    \Gamma^{\tau}_{\alpha\nu} A^{\alpha}_{\sigma} - \Gamma^{\tau}_{\alpha\nu} \Gamma^{\alpha}_{\mu\sigma} A^{\mu} - \Gamma^{\tau}_{\alpha\sigma} A^{\alpha}_{\nu} + \Gamma^{\tau}_{\alpha\sigma} \Gamma^{\alpha}_{\mu\nu} A^{\mu}

    but it didn't get me anywhere. Any ideas?
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