# General Relativity tensor proof

1. Oct 19, 2004

### 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?

2. Oct 19, 2004

### 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).