Invarients from the Faraday tensor

[peterjaybee]

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?

 

[Cyosis]

Yes, by using the fact that the Faraday tensor is antisymmetric. This way you only have to calculate 6 terms.

 

[peterjaybee]

That is a good point. makes it much easier. Thanks

 

[Meir Achuz]

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.