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Stationary property of geodesics - Dirac

  1. Jan 23, 2015 #1
    In Dirac's book on GRT, top of page 17, he has this: (I'll use letters instead of Greeks)

    gcdgac(dva/ds) becomes (dvd/ds)

    I seems to me that that only works if the metric matrix is diagonal.
    (1) Is that correct?
    (2) If so, that doesn't seem to be a legitimate limitation on the property of geodesics??
  2. jcsd
  3. Jan 23, 2015 #2


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    No. ##g^{cd}g_{ac} = \delta^{d}{}{}_{a}## by definition of the inverse of any metric tensor, hence the desired result.
  4. Jan 24, 2015 #3
    Ah yes, I see that now. Thanks.

    PS. Must the metric matrix always be symmetric? For example, a skew-symmetric matrix can produce the same ds^2 interval as a diagonal. Just curious.
  5. Jan 24, 2015 #4


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    It gives you the same ##ds^2## as some diagonal matrix, but it won't work for calculating the inner product of two different vectors.
  6. Jan 24, 2015 #5


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    A metric is, by definition, symmetric. :)
  7. Jan 29, 2015 #6


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    To expand on this just a bit, the metric must be symmetric because the inner product is commutative; ##g(a, b) = g(b, a)## for any two vectors ##a## and ##b##. If you write this out in components, you get that ##g## must be a symmetric matrix.
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