Electromagnetic tensor

Definition

The electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic four-potential, A, a differential 1-form:

$${\displaystyle F\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\mathrm{d}A.}$$

Therefore, F is a differential 2-form— an antisymmetric rank-2 tensor field—on Minkowski space. In component form,

$${\displaystyle F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }.}$$

where ${\displaystyle \mathrm{\partial }}$ is the four-gradient and ${\displaystyle A}$ 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

$${\displaystyle \begin{array}{cc}F=& (E_x/c)dx\wedge dt+(E_y/c)dy\wedge dt+(E_z/c)dz\wedge dt\\ & +B_{x}dy\wedge dz+B_{y}dz\wedge dx+B_{z}dx\wedge dy,\end{array}}$$

where ${\displaystyle dt}$ is the time element times the speed of light ${\displaystyle c}$.

This is the exterior derivative of its 1-form antiderivative, the covariant form of the four-potential, is

$${\displaystyle A=(\varphi /c)dt-A_{x}dx-A_{y}dy-A_{z}dz,}$$

where ${\displaystyle \varphi (\mathbf{x},t)}$ has ${\displaystyle -\mathbf{\nabla }\varphi =\mathbf{E}}$ (${\displaystyle \varphi }$ is a scalar potential for the irrotational/conservative vector field ${\displaystyle \mathbf{E}}$) and ${\displaystyle \mathbf{A}(\mathbf{x},t)}$ has ${\displaystyle \mathbf{\nabla }\times \mathbf{A}=\mathbf{B}}$ (${\displaystyle \mathbf{A}}$ is a vector potential for the solenoidal vector field ${\displaystyle \mathbf{B}}$).

The electric and magnetic fields can be obtained from the components of the electromagnetic tensor. The relationship is simplest in Cartesian coordinates:

$${\displaystyle E_i=c{F}_{0i},}$$ where c is the speed of light, and $${\displaystyle B_i=-\frac{1}{2}{\epsilon }_{ijk}{F}^{jk},}$$ where ${\displaystyle {\epsilon }_{ijk}}$ is the Levi-Civita tensor. This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor will transform covariantly, and the fields in the new frame will be given by the new components.

In contravariant matrix form with metric signature (+,−,−,−), $${\displaystyle F^{\mu \nu }=\left[\begin{array}{cccc}0& -E_x/c& -E_y/c& -E_z/c\\ E_x/c& 0& -B_z& B_y\\ E_y/c& B_z& 0& -B_x\\ E_z/c& -B_y& B_x& 0\end{array}\right].}$$

The covariant form is given by index lowering,

$${\displaystyle \begin{array}{cc}{F}_{\mu \nu }& ={\eta }_{\alpha \nu }{F}^{\beta \alpha }{\eta }_{\mu \beta }\\ & =\left[\begin{array}{cccc}0& E_x/c& E_y/c& E_z/c\\ -E_x/c& 0& -B_z& B_y\\ -E_y/c& B_z& 0& -B_x\\ -E_z/c& -B_y& B_x& 0\end{array}\right].\end{array}}$$ The Faraday tensor's Hodge dual is $${\displaystyle \begin{array}{cc}{G}^{\alpha \beta }& =\frac{1}{2}{\epsilon }^{\alpha \beta \gamma \delta }{F}_{\gamma \delta }\\ & =\left[\begin{array}{cccc}0& -B_x& -B_y& -B_z\\ B_x& 0& E_z/c& -E_y/c\\ B_y& -E_z/c& 0& E_x/c\\ B_z& E_y/c& -E_x/c& 0\end{array}\right]\end{array}}$$

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:

1. Antisymmetry: $${\displaystyle F^{\mu \nu }=-F^{\nu \mu }}$$
2. Six independent components: In Cartesian coordinates, these are simply the three spatial components of the electric field (E_x, E_y, E_z) and magnetic field (B_x, B_y, B_z).
3. Inner product: If one forms an inner product of the field strength tensor a Lorentz invariant is formed $${\displaystyle F_{\mu \nu }{F}^{\mu \nu }=2\left(B^2-\frac{E^2}{c^2}\right)}$$ meaning this number does not change from one frame of reference to another.
4. Pseudoscalar invariant: The product of the tensor ${\displaystyle F^{\mu \nu }}$ with its Hodge dual ${\displaystyle G^{\mu \nu }}$ gives a Lorentz invariant: $${\displaystyle G_{\gamma \delta }{F}^{\gamma \delta }=\frac{1}{2}{\epsilon }_{\alpha \beta \gamma \delta }{F}^{\alpha \beta }{F}^{\gamma \delta }=-\frac{4}{c}\mathbf{B}\cdot \mathbf{E}}$$ where ${\displaystyle {\epsilon }_{\alpha \beta \gamma \delta }}$ is the rank-4 Levi-Civita symbol. The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here is ${\displaystyle {\epsilon }_{0123}=-1}$. This and the previous Lorentz invariant vanish in the crossed field case.
5. Determinant: $${\displaystyle detF=\frac{1}{c^2}{\left(\mathbf{B}\cdot \mathbf{E}\right)}^2}$$ which is proportional to the square of the above invariant.
6. Trace: $${\displaystyle \mathrm{tr}F={F^{\mu }}_{\mu }=0}$$

Significance

This tensor simplifies and reduces Maxwell's equations as four vector calculus equations into two tensor field equations. In electrostatics and electrodynamics, Gauss's law and Ampère's circuital law are respectively:

$${\displaystyle \begin{array}{cccc}\mathrm{\nabla }\cdot \mathbf{E}& =\frac{\rho }{{\epsilon }_0},& \mathrm{\nabla }\times \mathbf{B}& =\frac{1}{c^2}\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}+{\mu }_0\mathbf{J}\end{array}}$$

and reduce to the inhomogeneous Maxwell equation:

$${\displaystyle {\mathrm{\partial }}_{\alpha }{F}^{\beta \alpha }=-{\mu }_0{J}^{\beta },}$$ where ${\displaystyle J^{\alpha }=(c\rho ,\mathbf{J})}$ is the four-current.

In magnetostatics and magnetodynamics, Gauss's law for magnetism and Maxwell–Faraday equation are respectively:

$${\displaystyle \begin{array}{cccc}\mathrm{\nabla }\cdot \mathbf{B}& =0,& \mathrm{\nabla }\times \mathbf{E}& =-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\end{array}}$$

which reduce to the Bianchi identity:

$${\displaystyle {\mathrm{\partial }}_{\gamma }{F}_{\alpha \beta }+{\mathrm{\partial }}_{\alpha }{F}_{\beta \gamma }+{\mathrm{\partial }}_{\beta }{F}_{\gamma \alpha }=0}$$

or using the index notation with square brackets^{[note 1]} for the antisymmetric part of the tensor:

$${\displaystyle {\mathrm{\partial }}_{[\alpha }{F}_{\beta \gamma ]}=0}$$ Using the expression relating the Faraday tensor to the four-potential, one can prove that the above antisymmetric quantity turns to zero identically (${\displaystyle \equiv 0}$). This tensor equation reproduces the homogeneous Maxwell's equations.