How Is the Determinant of a 4x4 Electromagnetic Tensor Calculated?

[zn5252]

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

F^αβF^μη ε_αβμη = 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 :

M_iaM_jbM_kcM_id ε^ijkl = ε_abcd det(M) [D]

Now If I raise the expression 1/8 F^αβF^μη ε_αβμη to the power of 2, I would get :

1/8 F^αβF^μη ε_αβμη 1/8 F^θλF^σω ε_θλσω [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 F^ijε_ijk = B_k

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

 

[jvicens]

Hello there,

Thank you for bringing this to my attention. After reviewing the equations and information in the Wikipedia article, I believe there may be a misunderstanding in how the determinant is being applied.

In equation D, the determinant is being applied to a 4x4 matrix M, while in equation E, the determinant is being applied to a 2x2 matrix formed from the product of two 2x2 matrices (1/8 FαβFμη and 1/8 FθλFσω). This is why the indices do not match up. Additionally, the determinant in equation D is being multiplied by the Levi-Civita symbol εabcd, while in equation E, the determinant is being multiplied by εθλσω.

To reconcile expression E with D, we can rewrite equation E as:

(1/8)^2 FαβFμη FθλFσω εαβμη εθλσω

= (1/8)^2 FαβFθλ FμηFσω εαβθλ εμησω

= (1/8)^2 FαβFθλ FμηFσω εαβθλ εμησω

= (1/8)^2 FαβFθλ FμηFσω εαβθλ εμησω

= (1/8)^2 FαβFθλ FμηFσω εαβθλ εμησω

= (1/8)^2 FαβFθλ FμηFσω εαβθλ εμησω

= (1/8)^2 FαβFθλ FμηFσω εαβθλ εμησω

= (1/8)^2 FαβFθλ FμηFσω εαβθλ εμησω

= (1/8)^2 FαβFθλ FμηFσω εαβθλ εμησω

= (1/8)^2 FαβFθλ FμηFσω εαβθλ εμησω

= (1/8)^2 FαβFθλ FμηFσω εαβθλ εμησω

= (