In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a tensor that describes the electromagnetic field in spacetime. The EM tensor field was developed by Arnold Sommerfeld after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The EM tensor allows related physical laws to be written concisely, and allows for the quantization of the electromagnetic field by a Lagrangian formulation.
Definition
The electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic four-potential, A, a differential 1-form:
<math display="block">F \ \stackrel{\mathrm{def{=}\ \mathrm{d}A.</math>
Therefore, F is a differential 2-form— an antisymmetric rank-2 tensor field—on Minkowski space. In component form,
<math display="block">F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.</math>
where <math>\partial</math> is the four-gradient and <math>A</math> is the four-potential.
SI units for Maxwell's equations and the particle physicist's sign convention for the signature of Minkowski space , will be used throughout this article.
Relationship with the classical fields
The Faraday differential 2-form is given by
<math display="block">
\begin{align}
F ={}&(E_x/c)\ dx \wedge dt + (E_y/c)\ dy \wedge dt + (E_z/c)\ dz \wedge dt \\
&+ B_x\ dy \wedge dz + B_y\ dz \wedge dx + B_z\ dx \wedge dy,
\end{align}
</math>
where <math> dt </math> is the time element times the speed of light <math> c </math>.
This is the exterior derivative of its 1-form antiderivative, the covariant form of the four-potential, is
<math display="block">
F^{\mu\nu} = \begin{bmatrix}
0 & -E_x/c & -E_y/c & -E_z/c \\
E_x/c & 0 & -B_z & B_y \\
E_y/c & B_z & 0 & -B_x \\
E_z/c & -B_y & B_x & 0
\end{bmatrix}.
</math>
The covariant form is given by index lowering,
<math display="block">\begin{align}
F_{\mu\nu} &= \eta_{\alpha\nu} F^{\beta\alpha} \eta_{\mu\beta} \\[1ex]
&= \begin{bmatrix}
0 & E_x/c & E_y/c & E_z/c \\
-E_x/c & 0 & -B_z & B_y \\
-E_y/c & B_z & 0 & -B_x \\
-E_z/c & -B_y & B_x & 0
\end{bmatrix}.
\end{align}</math>
The Faraday tensor's Hodge dual is
<math display="block">
\begin{align}
G^{\alpha\beta} &= \tfrac{1}{2} \varepsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta} \\[1ex]
&= \begin{bmatrix}
0 & -B_x & -B_y & -B_z \\
B_x & 0 & E_z/c & -E_y/c \\
B_y & -E_z/c & 0 & E_x/c \\
B_z & E_y/c & -E_x/c & 0
\end{bmatrix}
\end{align}
</math>
From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is assumed, and the electric and magnetic fields are with respect to the coordinate system's reference frame, as in the equations above.
Properties
The matrix form of the field tensor yields the following properties: