Dual Faraday Tensor - Levi Civita Problem

[MidnightR]

Ok I'm trying to calculate the Dual Faraday Tensor, but I'm having trouble with the notation I think...

$*F^{ab} = \frac{1}{2}E^{abcd}F_{cd}$ where E is the levi-civita symbol.

I'm correct in that

E^(1234) =

1 for (1234,3412,2341,4123)
-1 for (4321,3214,2143,1432)
0 otherwise

But this only gives me 8 terms in my 4x4 matrix? Which is incorrect?

F_cd is the Faraday tensor, which is also a 4x4 matrix.

 

[fzero]

I'm correct in that

E^(1234) =

1 for (1234,3412,2341,4123)
-1 for (4321,3214,2143,1432)
0 otherwise

But this only gives me 8 terms in my 4x4 matrix? Which is incorrect?

You are both missing permutations like 1324 and you have the wrong sign on others. For example, 2341 and 4123 are both odd, while 2143 and 4321 are even.

 

[MidnightR]

You're right, thanks. It works now.

 

[Edigam]

So 1324 is odd and 3214 is even. Correct?

 

[dextercioby]

No, because $\epsilon^{1324} = - \epsilon^{3124} = - \left( - \epsilon^{3214}\right)$, so that when the first is odd (and it is), the last is odd as well.

 

[Edigam]

Thank's.