Electromagnetic tensor

The electromagnetic (field) tensor is the relativistic generalization of the electric and magnetic fields:

It is a second-rank antisymmetric tensor. It plays the part of the Lorentz transformation matrix in relativity. An alternative and equivalent formulation is the dual tensor

which is used in the same way and leads to the same transformations.

Derivation#

It's quite evident from the transformations of E and B that when you move from one frame of reference to another, the fields don't just change on their own: they mix. As such, we can't just describe them as (the spatial parts of) plain old four-vectors. We need some way of expressing them together, as a single joint object that encodes not just the fields, but the relationship between the fields. Looking back, we already did something similar in the past: when we wanted to represent the action of an electromagnetic field as one cohesive object, we made a tensor, specifically, the Maxwell stress tensor. Here, we follow in our own footsteps: the object we're looking for is an antisymmetric, second order tensor¹. But we are not working in 3D anymore, we're in spacetime! So this tensor is going to need to be $4\times4$.

Let's start from a four-vector in a frame $\mathcal{S}$. Such a vector, call it $a^{\mu}$, changes in relativity using a simple Lorentz transformation matrix $\Lambda$:

In $\Lambda$, superscript ($\mu$) is the row and subscript ($\nu$) is the column. We'll use the overbar now to denote the new frame $\bar{\mathcal{S}}$ instead of the prime $\mathcal{S}'$ (which gets in the way of the superscript). In case of constant-speed motion over the $x$ axis,

A second-rank tensor, call it $t$, has two indexes (it's a matrix) and will need two transformations, one for each index:

We also claimed we wanted our tensor to be antisymmetric. This just means $t^{\mu \nu}=-t^{\nu \mu}$, which is the same as saying that the diagonal is all zeros and that the off-diagonals are mirrored across the diagonal, with their sign flipped. In practice, a generic antisymmetric tensor reads

Count the unique elements: there's only six. Hopefully you start to see why we want an antisymmetric tensor and what these components are. But why not symmetric? Because a symmetric tensor would also have elements on the diagonal, for a total of ten. That's four too many, so that can't happen.

We, of course, want to see what the components are. To do so, let's look at what the transformation rule $(1)$ does to this tensor. $\bar{t}^{01}$ goes like

By definition of $\Lambda$, $\Lambda_{\lambda}^{0}$ (the first row) is nonzero only when $\lambda=0,1$. Similarly, $\Lambda_{\sigma}^{1}$ (the second row) is also nonzero only in $\sigma=0,1$. So, explicitly

Due to antisymmetry, $t^{00}=t^{11}=0$ and $t^{10}=-t^{01}$ so

You can repeat these procedure for every single one of the six unique elements:

The lettering might be a little different, but these are precisely the transformations of E and B:

If we compare the elements of the tensor directly with their corresponding electromagnetic transformation rule (first line of $(2)$ with first line of $(3)$, second line with second line), we can construct the tensor we were looking for²

This is known as the field tensor and it collectively describes the behavior of the entire electromagnetic field.

If you look closely, the field tensor is not the only tensor we can define. In fact, we if we compare the first line of $(3)$ with the second line of $(2)$ and the second with the first we get

This alternative tensor is known as the dual tensor and it describes much the same thing as field tensor. In fact, you can get $G$ from $F$ by substituting $\mathbf{E}/c\to \mathbf{B}$ and $\mathbf{B}\to-\mathbf{E}/c$. Which one you end up using is a matter of preference.

Thus, the field tensor packs up the entirety of the electromagnetic fields into a single object, which plays the part of a Lorentz transformation matrix.