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Matrix Formalism of GR

  1. Nov 13, 2009 #1
    Are there any good references out there for writing the equations of GR in matrix format? For example:

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

    where the matrix version of g_mn (g) would be hermitian, dx+ is the conjugate...

    covariant derivative:

    Y_n||m = dY_n/dx_m - {n, km} Y_k -> Y||m = dY/dx_m - G_m Y

    in matrix format:

    dg/dx_m + G_m g + g G+_m = 0 is the vanishing covariant derivative of the metric.

    This is just a different way to write the same mathematics. It seams it would be easier to work with, but I can't find any good references for it. Hasn't someone else done this already? I'm just looking for something that gives the original GR back, no new theories, just new notation...
     
  2. jcsd
  3. Nov 13, 2009 #2

    pervect

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    I've seen one, but I can't locate it. I think it was aimed at electrical engineers, if I recall correctly. "Exploring black holes" might work, I don't know, I haven't read the whole thing, just some of the free downloads. Several chapters of this book are available on the internet. Much of the approach is very reminiscent of MTW's "Gravitation", without the high-level math. So it might not be quite what you asked for, but parts of it are free - and I'd check it out. Take a look at

    http://www.eftaylor.com/general.html

    However, there are good reasons for the serious student to not use matrix notation and to learn tensors. Rank 2 tensors can easily be represented in matrix form, so they aren't the issue. The problem is representing rank 4 tensors, such as the Riemann curvature tensor. I suppose you *could* think of a rank 4 as a general linear map from one matrix to another. This requires not a 2-d array, but a 4-d data structure. For space-time that's 4x4x4x4 = 256 numbers, which however are not all independent in the case of the Riemann, which must obey some identies (the Bianchi identites).
     
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