I'm kind of confused. I want to calculate the electromagnetic invariant I:=FμνFμν, but I'm not sure what is the easiest way to do so. So, I was trying to do it in matrix form, i.e. defining F:=(0−E1−E2−E3E10−B3B2E2B30−B1E3−B2B10)
and then calculate the quantity I, but I'm not sure how to obtain the matrix form for I (By "matrix form" I mean expressed in terms of the matrix F).
So, I have two questions, 1) what is the easiest way to calculate I?, and 2) how to obtain the matrix form for I starting with I:=FμνFμν?. I'm using the metric η:=diag(1,−1,−1,−1).
Answer
Notice that if we define matrices F=(Fμν) and η=(ημν), then notice that I=FμνFμν=ημαηνβFαβFμν=ημαFαβηνβFμν=−ημαFαβηβνFνμ=−tr(ηFηF)
where tr denotes the trace and I have used antisymmetry and symmetry of F and η respectively. I'm guessing this is the type of matrix expression you're after?
Cheers!
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