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GR in Matrix format

  1. Nov 18, 2009 #1
    So I've been working on writing these equations in matrix format. I have encountered a question / issue and was hoping someone on this forum could check my math and point out what I've done wrong (I can't see it):

    ds^2 = g_mn dx_m dx_n -> ds^2 = dx+ g dx

    where dx is a 4 by 1 column matrix, dx+ is its conjugate transpose, and g is the 4x4 matrix version of the metric tensor. The symmetry of g in matrix form is the equivalent of saying that g is a hermitian matrix (operator... distance is a real number).

    The covariant derivative becomes:

    DY_m = dY/dx_m - G_m Y

    Where G_m are 4 (4x4) matrices. They are the matrix equivalents to the Christoffel symbols. The covariant derivative of the matrix form of g is:

    dg / dx_m + (G_m) g + g (G+_m) which we let go to 0.

    If we say under coordinate transform: Y' = SY where S is a 4x4 transformation matrix, then:

    g' = S+-1 g S-1, G_m' = dx_n/dx'_m (S G_n S-1 - dS/dx_n S-1)

    Now, if we take S = e^ik (a phase shift), we see that G shifts by the needed amount for our gauge condition.

    The question I have is this:

    What is wrong with the formula for the covariant derivative of the matrix g? The solution to that equation as I have it is:

    G_m = -1/2 dg/dx_m g-1 + iA

    A is a real constant value, unknown, meaning the solution isn't unique. Also it doesn't match what we would get from tensors.
     
  2. jcsd
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