How to tell that the electromagnetic field tensor transforms as a tensor?

Question

Is any matrix a tensor in special relativity?

My question is inspired by the definition of the electromagnetic field tensor in Carroll's Spacetime and Geometry book. In equation (1.69), he defines a matrix which he says is the (0,2) electromagnetic field tensor.

In section 1.8, he states the Maxwell equations in tensor notation, and additionally states that in this form they manifestly transform as tensors. I assume that this is due to the tensorial nature of the electromagnetic field tensor; however, I don't understand why it is a tensor.

Would we have to prove that the electromagnetic field tensor transforms as a tensor by brute calculation? Alternatively, is there some clever and more obvious shorthand that you could use to come to the same realization?

Answer 1

Strictly speaking, a matrix is not a tensor, it is a representation of a tensor in a particular basis. You can't tell whether a given matrix is a tensor using only its components. You would have to know how it transforms to different reference frames.

For the electromagnetic field tensor, for example, you could write the equations for some physical configuration of electromagnetic fields, and then write the equations that describe the same physical configuration in a different reference frame, and show that applying the corresponding Lorentz transformation twice converts from one to the other.

$$F^{\mu\nu}_\text{frame 2} = [\Lambda(1,2)]_\alpha^\mu [\Lambda(1,2)]_\beta^\nu F^{\alpha\beta}_\text{frame 1}$$

where I've used $\Lambda(1,2)$ to denote the Lorentz transformation that transforms from frame 1 to frame 2. Doing this calculation out in full gets somewhat tedious, which is why many textbooks don't go through it in full detail, they just make it seem plausible. (But at least Einstein had to do it to show that the theory worked this way.)

Anyway, the point is that the tensorial nature of a quantity is really a consequence of the transformation law, not the representation (the matrix).

Answer 2

The answer to your question depends quite a bit on what you consider to be fundamental and what you consider to be derived.

A modern, manifestly relativistic treatment of E&M would define the electromagnetic field tensor as

$$ F^a = q\mathcal{F}^a_bv^b, $$

where $\mathcal{F}$ is the field tensor and $F$ is the four-force acting on a particle. By this definition, it's manifestly clear that $\mathcal{F}$ is a tensor; it's a linear operator that takes one vector and makes another. In this approach, the work you have left to do is to show how the components of $\mathcal{F}$ relate to the usual $\mathbf{E}$ and $\mathbf{B}$ three-vectors. For a treatment in this style, see my SR book, http://www.lightandmatter.com/sr/ , section 10.3.

A more lowbrow approach would be to define $\mathbf{E}$ and $\mathbf{B}$ as usual in a freshman physics class, then find their transformation laws. (This is done, e.g., in the classic 1965 book by Purcell, Electricity and Magnetism. The first edition can be found for free online because it was developed under an NSF grant. There is also a nice modern edition from Cambridge University Press.) Having done this, you then have to put them together as the components of $\mathcal{F}$ and show that these transformation laws make $\mathcal{F}$ transform as a tensor.

Answer 3

Would we have to prove that the electromagnetic field tensor transforms as a tensor by brute calculation?

Not if you accept that the derivative of a tensor is a tensor with one higher covariant rank.

In SR, the scalar electric and vector magnetic potential are components of a four-vector (a rank 1 tensor), the electromagnetic four-potential $A_{\mu}$.

Then, it is the case that the following is a rank 2 covariant anti-symmetric tensor, the electromagnetic tensor:

$$F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$$

Alternatively, is there some clever and more obvious shorthand that you could use to come to the same realization?

In component free notation, we have the tensor equation

$$\mathbf F \stackrel{\mathrm{def}}{=} \mathbf{dA}$$

which defines the electromagnetic tensor as the exterior derivative of the electromagnetic four-potential and leaves no possible doubt that $\mathbf F$ is a tensor.