Thread: Relativistic Invariance View Single Post
P: 657
 Originally posted by turin Wouldn't this be just as complicated?
you be the judge

 Another thing: How do you know that this is going to give you the dot product? I mean, I can see that going throught the calculations would show this, but, how do you know to start with the Levi-Civita contraction in the first place? It seems kind of out-of-the-blue to me. Is there some theorem or something?
let s have a look:

the Levi-Civita tensor will select all the terms with even permutations of 1234 with a plus sign, and all the terms with odd permutations with a minus sign

$$\begin{multline*} F^{\mu\nu}F^{\rho\sigma}\epsilon_{\mu\nu\rho\sigma}\\ =4(F^{01}F^{23}+F^{02}F^{13}+F^{03}F^{12})=4\mathbf{E}\cdot\mathbf{B} \end{multline*}$$

and that s all i need to do, it is guaranteed to be lorentz invariant.