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I'm confused by a very basic property of electric fields. The electric field is a vector field. Vectors are tensors. Wikipedia has the following statement in the article about the electromagnetic tensor:

The electromagnetic tensor is completely isomorphic to the electric and magnetic fields, though the electric and magnetic fields change with the choice of the reference frame, while the electromagnetic tensor does not.

This suggests, to me, that electric and magnetic fields are not tensors even though they are vector fields. What am I not understanding?

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    $\begingroup$ Electric and magnetic fields are vectors under the group of rotations ($SO(3)$) but combine to form a second-rank antisymmetric tensor under the Lorentz group. $\endgroup$
    – suresh
    Oct 2 '14 at 12:26
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The statement from the Wikipedia articles is, as written, wrong. The EM field tensor - as a tensor - does change under change of reference frames. It is covariant, but not invariant under the Lorentz group, while the electric and magnetic field are neither, but they are covariant under the rotation group.

The electric and magnetic fields are ordinary, non-relativistic 3-vectors, i.e. they transform as elements of the fundamental representation of the rotation group $\mathrm{SO}(3)$ as $E_i \mapsto \sum_j R_{ij}E_j$ for every matrix $R \in \mathrm{SO}(3)$. One can call them tensors of rank 1 under this group, living on Euclidean space.1

The EM field tensor is a rank 2 relativistic tensor, i.e. it is an object with two indices transforming under the Lorentz group $\mathrm{SO}(1,3)$ as $F_{\mu\nu} \mapsto \Lambda^\sigma_\mu\Lambda^\rho_\nu F_{\rho\sigma}$ for any $\Lambda \in \mathrm{SO}(1,3)$. Since it is antisymmetric, $F$ has 6 independent entries - which map one-to-one to the three entries of the electric field and the three entries of the magnetic field. The electromagnetic tensor lives, in contrast to the electric and magnetic fields, on Minkowski space, and is thus more natural to consider when doing relativity.


1If we are precise, the magnetic field is a pseudovector $B_{ij}$, which is, on 3D Euclidean space, an antisymmetric tensor of rank 2 (see here for more on pseudovectors vs. vectors) which maps isomorphically onto its normal vector $\tilde{B}_k = \epsilon^{ijk}B_{ij}$ as long as we consider $\mathrm{SO}(3)$ and not $\mathrm{O}(3)$. The $F_{\mu\nu}$ is constructed as $F_{ij} = B_{ij}$ and $F_{0i} = E_i$, where $i,j$ are spatial indices from $1$ to $3$.

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    $\begingroup$ Except the electric field is a vector while the magnetic field is a pseudovector! $\endgroup$
    – Slereah
    Oct 2 '14 at 12:58
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    $\begingroup$ The EM field tensor - as a tensor - does change under change of reference frames. Many people do prefer to describe tensors as invariant. When people adopt that convention, they say that the tensor's components in a particular basis are basis-dependent, but they consider the tensor itself to be invariant. $\endgroup$
    – user4552
    Oct 2 '14 at 14:51

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