Electrodynamics/Electromagnetic Field Tensors

Before discussing Maxwell's Equations, it's useful to define two tensors for the Electromagnetic field. These two tensors will be used in the remainder of the book for several applications, including Maxwell's Equations.

Field Tensors

We will define two field tensors, F and G, and we will define another tensor, σ, that we can use to generate one field tensor from the other.

$𝔽=\left[\begin{array}{cccc}0& B_z& -B_y& -i{E}_x\\ -B_z& 0& B_x& -i{E}_y\\ B_y& -B_x& 0& -i{E}_z\\ i{E}_x& i{E}_y& i{E}_z& 0\end{array}\right]$

$𝔾=\left[\begin{array}{cccc}0& E_z& -E_y& -i{B}_x\\ -E_z& 0& E_x& -i{B}_y\\ E_y& -E_x& 0& -i{B}_z\\ i{B}_x& i{B}_y& i{B}_z& 0\end{array}\right]$

Completely Antisymmetrical Tensor

We can define a completely antisymmetrical tensor of the fourth rank (4 dimensional, so we cannot visualize it) as follows:

${\sigma }_{\mu \nu \lambda \rho }=0$ if any two indices are the same.

${\sigma }_{1234}=1$

${\sigma }_{\mu \nu \lambda \rho }=1$if the indices are all different, and if the total difference in the indices from 1234 is even.

${\sigma }_{\mu \nu \lambda \rho }=-1$if the indices are all different and if the total difference in the indices from 1234 is odd.

We can define the two field tensors, F and G in terms of this new tensor as:

$𝔾=\frac{i}{2}\sum {\sigma }_{\mu \nu \lambda \rho }{𝔽}_{\lambda \rho }$