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General Relativity tensor proof

  1. Oct 19, 2004 #1
    Prove that if tauijkl is a tensor such that, in the {xi}-system, tauijkl=3tauiljk , then tauijkl=3 tauiljkin all coordinate systems.

    How would one go about proving this for all coordinate systems?
     
  2. jcsd
  3. Oct 19, 2004 #2
    Remember the similarity transformation of tensors. If I have a tensor that is given in the [tex]x_i[/tex] basis I can find what that tensor looks like in any other basis (say the [tex]x_j[/tex] basis) by the following

    [tex]T^{x_j} = U^\dagger T^{x_i} U[/tex] (1)

    Here, [tex]U[/tex] is a unitary tensor that relates the components of a vector described in the two different bases. The key property of [tex]U[/tex] is

    [tex]U^\dagger U = U U^\dagger = I[/tex] (2)

    Here, [tex]I[/tex] is the identity tensor. Anyway, play around with that. You won't need to actually calculate what [tex]U[/tex] is. You just have to know (1) and (2).
     
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