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Homework Help: Contravariant components to covariant

  1. Jul 2, 2010 #1
    Hi, everyone

    I was playing with the coordinate transformations and metric tensors to get a feeling of how it all behaves, and got stuck with some basic problem I am hoping you can help me with.

    So, I have defined a coordinate system (s,t), with the s axis going along the x axis in the cartesian coordinates, and t axis going along the y=x line:
    s = x-y
    t = y*sqrt(2)

    with inverse transformation:

    x = s + t/sqrt(2)
    y = t/sqrt(2)

    If I am differentiating correctly, the metric tensor in these coordinates looks like:
    1 (2+sqrt(2))/2
    (2+sqrt(2))/2 1

    g11 = g22 = 1,
    g21=g12 = (2+sqrt(2))/2

    Now, I pick a point (3,1) in cartesian coordinates, and transform it to my new frame, and get the contravariant coordinates as (2, sqrt(2)).
    So far so good. What I am trying to do is find out what its covariant coordinates are going to be. I think, that covariant coordinates are supposed to be the lengths of orthogonal projections of the vector on the respective axes. From basic geometry, I get (3, 2*sqrt(2)).

    The problem is that when I try to multiply my metric tensor by the contravariant vector, I get a different answer - (3+sqrt(2), 2+2*sqrt(2))
    Clearly, there is something I am doing wrong here, but I can't figure out what it is :( Can somebody please help me spot the problem?

    Thanks a lot for your help!
  2. jcsd
  3. Jul 3, 2010 #2
    Your metric is wrong. The mixed components (g12 and g21) should be 1/sqrt(2).
    Because you did not show how you got your metric tensor, I can't say where you went wrong, but if your check your index dropping with the correct metric, you'll see that it fits.

    Hope, it helps ...
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