Invarients from the Faraday tensor

peterjaybee

Hello,

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

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

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

clem

It also gets easier if you recognize that the first row is just [tex]-{\vec E}[/tex], and the 3X3 space-like part is just [tex]-{\vec B}[/tex], a bit mixed up.

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peterjaybee	Invarients from the Faraday tensor Tweet Hello, a full contraction of the faraday tensor with itself can be shown to be [tex]F_{\mu\nu}F^{\nu\mu}=2(E^{2}-c^{2}B^{2})[/tex] 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 Recognitions: Homework Help	Yes, by using the fact that the Faraday tensor is antisymmetric. This way you only have to calculate 6 terms.

clem Recognitions: Science Advisor	Invarients from the Faraday tensor It also gets easier if you recognize that the first row is just [tex]-{\vec E}[/tex], and the 3X3 space-like part is just [tex]-{\vec B}[/tex], a bit mixed up.