# Determinant of a 4x4 tensor

1. Apr 16, 2013

### zn5252

hi there,
In this wikipedia article https://en.wikipedia.org/wiki/Electromagnetic_tensor
we have the following invariant :

FαβFμη εαβμη = 8 E*B

However the determinant is the square of this quantity divided by 8, i.e. ( E*B )2 .

Now from the definition of the determinant for a 4x4 matrix , we have :

MiaMjbMkcMid εijkl = εabcd det(M) [D]

Now If I raise the expression 1/8 FαβFμη εαβμη to the power of 2, I would get :

1/8 FαβFμη εαβμη 1/8 FθλFσω εθλσω [E]

Now If I compare this expression with equation D above, I see that some indices do not fall into the right place and also in Equation D, we have the expression for the determinant , however in expression E, we see that we have the matrix F multiplied 4 times much like in expression D (or is it rows or columns that get multiplied).

We should also bear in mind that the magnetic field is the spatial part of F and that the electric field is the time part :

Fi0 = Ei and Fijεijk = Bk

How can we reconcile expression E with D then ? or is this an error perhaps ?