General Relativity tensor proof

[regretfuljones]

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?

[BLaH!]

Remember the similarity transformation of tensors. If I have a tensor that is given in the x_i basis I can find what that tensor looks like in any other basis (say the x_j basis) by the following

T^{x_j} = U^\dagger T^{x_i} U (1)

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

U^\dagger U = U U^\dagger = I (2)

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