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Christoffel Symbol

  1. May 20, 2014 #1
    Ansatz metric of the four dimensional spacetime:

    [itex]ds^2=a^2 g_{ab}dx^a dx^b - du^2[/itex]



    [itex]a(u)=[/itex]warped factor

    Christoffel symbol of a three dimensional AdS spacetime:

    [itex]\Gamma^{c}_{ab}= \frac{1}{2} g^{cd}(∂_b g_{da} + ∂_a g_{bd} - ∂_d g_{ba})[/itex]

    Now how to find [itex]\Gamma^{a}_{b}[/itex]?
  2. jcsd
  3. May 20, 2014 #2


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    First, what are the [itex]g_{ab}[/itex]? Are they functions of the "x"s only or also of u? Clearly the Christoffel symbols depend on exactly how they depend on the coordinates. Second, what do you mean by "a(u)"? Is it that the "a" in "[itex]a^2[/itex]"? You are already using "a" as an index. Surely "a(u)" is not an index so it would be better to use a different symbol.
  4. May 21, 2014 #3
    Ok I fix them:
  5. May 21, 2014 #4

    George Jones

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    What does [itex]\Gamma^{i}_{j}[/itex] mean, i.e., what is a [itex]\Gamma[/itex] with two indices?
  6. May 21, 2014 #5
    I don't know that's why I asked :confused:


    Well, one said that

    [itex]\Gamma^{\rho}_{\mu\nu} = ........\Gamma^{i}_{j}[/itex]

    [itex]R_{\mu\nu} = ........R_{ij}[/itex]


    [itex]\mu, \nu, \rho = (D + 1)[/itex] dimensional curved spacetime indices

    [itex]R_{ij} = \Lambda_{D} g_{ij}[/itex]

    [itex]\Lambda_{D} = [/itex] cosmological constant
    Last edited: May 21, 2014
  7. May 21, 2014 #6

    George Jones

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    Can you give specific references? In what articles or books have you seen this notation?
  8. May 21, 2014 #7
    It's not in any articles/books. I just met someone who told me that :(
  9. May 21, 2014 #8


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    Just giving my two cents, but I have never seen, in my study of differential geometry, a Christoffel symbol with 2 indices. The closest I can think of would be the the connection one-forms which are a set of 6 one-forms, and so they are sometimes written with two (anti-symmetric) indices...but...that's usually written in a way different notation.
  10. May 22, 2014 #9
    Oh okay
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