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sweetser is offline
Dec17-05, 05:10 PM
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P: 361
Hello Careful:

I understand that must not mix up their covariant and contravariant indices. Let me try and state this as a problem, and see if anyone can find a solution other than the one I wrote.

The asymmetric mixed second rank field strength tensor, [itex]\nabla_{\mu}A^{\nu}[/itex], like all asymmetric tensors, can be represented by a sum of symmetric tensor and an antisymmetric tensor. I don't know the theorem that says this, bue here is a reason why it can be done. The symmetric tensor is the average amount of change in the the potential, and the antisymmetric tensor is the deviation from the average amount of change tensor. Appropriately chosen averages and deviations can represent an arbitrary asymmetric tensor.

The exercise would be trivial without the word "mixed", like so:
[tex]\nabla^{\mu}A^{\nu}=\frac{1}{2}(\nabla^{\mu}A^{\nu}+\nabla^{\nu}A^{\mu} )+\frac{1}{2}(\nabla^{\mu}A^{\nu}-\nabla^{\nu}A^{\mu})[/tex]
The question can be made concrete: how would you write this tensor using standard indicies? (only showing 2 dimensions for clarity):

2 \nabla_0 b_0 & \nabla_0 b_1 - \nabla_1 b_0 \\
-\nabla_1 b_0 + \nabla_0 b_1 & 2 \nabla_1 b_1 \\

This looks like a reasonable matrix to represent with tensors. I am not sure how to write it in a proper way with indices.