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Calculate Christoffel Symbols of 2D Metric

  1. May 28, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider metric ds2 = dx2 + x3 dy2 for 2D space.
    Calculate all non-zero christoffel symbols of metric.

    2. Relevant equations

    [itex]\Gamma[/itex]jik = [itex]\partial[/itex]ei / [itex]\partial[/itex] xk [itex]\times[/itex] ej


    3. The attempt at a solution

    Christoffel symbols, by definition, takes the partial of each vector basis with respect to component (in this case xk).

    So my instinct tells me to differentiate the metric with respect to x and y. Which would give me:

    metric differentiated with respect to (x) = 3x2 x1

    metric differentiated with respect to (y) = 0 x2

    Did I do this right or did I get this completely wrong? If I have done it wrong, please explain.

    Thank you very much in advance.
     
  2. jcsd
  3. May 28, 2013 #2
    Do you know the equation for calculating the christoffel symbols from the partial derivatives of the components of the metric tensor? Alternatively, for this problem,

    ex=ix

    ey=x(3/2)iy

    and

    ex=ix

    ey=x(-3/2)iy

    where ix and iy are unit vectors in the x- and y-directions, respectively.
     
    Last edited: May 28, 2013
  4. May 29, 2013 #3
    Hello! Thank you for the quick reply!

    The equation you have mentioned, the only equation I am aware of for relating christoffel symbols to the metric is:

    [itex]\Gamma[/itex]Lab = [itex]\frac{1}{2}[/itex] gLc (gac;b + gcb;a - gba;c)

    I'm having a hard time understanding the indices, thus just how to use this equation correctly. I'll give it a go and see if I can't arrive at the same answer.

    Thanks again!
     
  5. May 29, 2013 #4
    Those should be commas (indicating partial differentiation), not semicolons.
    For this problem, it would probably be easier to use your original formula. There are only two dimensions, so there are going to be only 8 gammas to evaluate. Either way, it shouldn't be much work.

    chet
     
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