Invarients from the Faraday tensor

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Hello,

a full contraction of the faraday tensor with itself can be shown to be

F_{\mu\nu}F^{\nu\mu}=2(E^{2}-c^{2}B^{2})

I have done this by calculating 16 terms in the sum i.e. F11F11 + F12F21, and get this answer, but this is very tedious.

Is there a faster way to show this that I am missing?
 
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Yes, by using the fact that the Faraday tensor is antisymmetric. This way you only have to calculate 6 terms.
 
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That is a good point. makes it much easier. Thanks
 
It also gets easier if you recognize that the first row is just -{\vec E}, and the 3X3 space-like part is just -{\vec B}, a bit mixed up.
 

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