# Electromagnetic tensor

Electromagnetic tensor

The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field of a physical system in Maxwell's theory of electromagnetism. The field tensor was first used after the 4-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows some physical laws to be written in a very concise form.

## Definition

Mathematical note: In this article, the abstract index notation will be used.

The electromagnetic tensor starts with the Electromagnetic four-potential:

$A^{\mu} = \left( \frac{\phi}{c} , \vec A \right)$ and its covariant form is found by multiplying by the Minkowski metric η of signature (+,−,−,−) :
$A_{\mu} \, = \eta_{\mu\nu} A^{\nu} = \left( \frac{\phi}{c}, -\vec A \right)$

where

$\vec A \,$ is the vector potential and $\left(A_x, A_y, A_z \right)$ are its components
$\phi \,$ is the scalar potential and
$c \,$ is the speed of light.

Electric and magnetic fields are derived from the vector potentials and the scalar potential with two formulas:

$\vec{E} = -\frac{\partial \vec{A}}{\partial t} - \vec{\nabla} \phi \,$
$\vec{B} = \vec{\nabla} \times \vec{A} \,$

By definition, the electromagnetic tensor is the exterior derivative of the differential 1-form Aμ:

$F_{ \mu\nu } \ \stackrel{\mathrm{def}}{=}\ d A_\mu = \partial_\mu A_\nu-\partial_\nu A_\mu.$

F is therefore a differential 2-form on spacetime. In an inertial frame, the matrices of F read:

$F^{\mu\nu} = \begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{bmatrix}$

or

$F_{\mu\nu} = \begin{bmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end{bmatrix}$

### Properties

From the matrix form of the field tensor, it becomes clear that the electromagnetic tensor satisfies the following properties:

• antisymmetry: $F^{\mu\nu} \, = - F^{\nu\mu}$ (hence the name bivector).
• six independent components.

If one forms an inner product of the field strength tensor a Lorentz invariant is formed:

$F_{\mu\nu} F^{\mu\nu} = \ 2 \left( B^2 - \frac{E^2}{c^2} \right) = \mathrm{invariant}$

The product of the tensor $F^{\mu\nu} \,$ with its dual tensor gives the pseudoscalar invariant:

$\frac{1}{2}\epsilon_{\alpha\beta\gamma\delta}F^{\alpha\beta} F^{\gamma\delta} = \frac{4}{c} \left( \vec B \cdot \vec E \right) = \mathrm{invariant} \,$

where $\ \epsilon_{\alpha\beta\gamma\delta} \,$ is the completely antisymmetric unit pseudotensor of the fourth rank or Levi-Civita symbol. Caution: the sign for the above invariant depends on the convention used for the Levi-Civita symbol. The convention used here is $\ \epsilon_{0123} \,$ = +1.

Notice that:

$\det \left( F \right) = \frac{1}{c^2} \left( \vec B \cdot \vec E \right) ^{2}$

### Significance

Hidden beneath the surface of this complex mathematical equation is an ingenious unification of Maxwell's equations for electromagnetism. Consider the electrostatic equation

$\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0}$

which tells us that the divergence of the electric field vector is equal to the charge density, and the electrodynamic equation

$\vec{\nabla} \times \vec{B} - \frac{1}{c^2} \frac{ \partial \vec{E}}{\partial t} = \mu_0 \vec{J}$

that is the change of the electric field with respect to time, minus the curl of the magnetic field vector, is equal to negative 4π times the current density.

These two equations for electricity reduce to

$\partial_{\beta} F^{\alpha\beta} = \mu_0 J^{\alpha} \,$

where

$J^{\alpha} = ( c \, \rho , \vec{J} ) \,$ is the 4-current.

The same holds for magnetism. If we take the magnetostatic equation

$\vec{\nabla} \cdot \vec{B} = 0$

which tells us that there are no "true" magnetic charges, and the magnetodynamics equation

$\frac{ \partial \vec{B}}{ \partial t } + \vec{\nabla} \times \vec{E} = 0$

which tells us the change of the magnetic field with respect to time plus the curl of the electric field is equal to zero (or, alternatively, the curl of the electric field is equal to the negative change of the magnetic field with respect to time). With the electromagnetic tensor, the equations for magnetism reduce to

$\partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0$

## The field tensor and relativity

The field tensor derives its name from the fact that the electromagnetic field is found to obey the tensor transformation law, this general property of (non-gravitational) physical laws being recognised after the advent of special relativity. This theory stipulated that all the (non-gravitational) laws of physics should take the same form in all coordinate systems - this led to the introduction of tensors. The tensor formalism also leads to a mathematically elegant presentation of physical laws. For example, Maxwell's equations of electromagnetism may be written using the field tensor as:

$F_{[\alpha\beta,\gamma]} \, = 0$ and $F^{\alpha\beta}{}_{,\beta} \, = \mu_0 J^{\alpha}$

The second equation implies conservation of charge:

$J^\alpha{}_{,\alpha} \, = 0$

These laws can be generalised to curved spacetime by simply replacing partial with covariant derivatives:

$F_{[\alpha\beta;\gamma]} \, = 0$ and $F^{\alpha\beta}{}_{;\beta} \, = \mu_0 J^{\alpha}$

where the semi-colon represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the curved space Maxwell equations. Again, the second equation implies charge conservation (in curved spacetime):

$J^\alpha{}_{;\alpha} \, = 0$

## Lagrangian formulation of classical electromagnetism without charges and currents

When there are no electric charges (ρ=0) and no electric currents (j=0), Classical electromagnetism and Maxwell's equations can be derived from the action defined:

$\mathcal{S} = \int \left( -\begin{matrix} \frac{1}{4 \mu_0} \end{matrix} F_{\mu\nu} F^{\mu\nu} \right) \mathrm{d}^4 x \,$

where

$\mathrm{d}^4 x \;$   is over space and time.

This means the Lagrangian density is

 $\mathcal{L} \,$ $= -\begin{matrix} \frac{1}{4\mu_0} \end{matrix} F_{\mu\nu} F^{\mu\nu} \,$ $= -\begin{matrix} \frac{1}{4\mu_0} \end{matrix} \left( \partial_\mu A_\nu - \partial_\nu A_\mu \right) \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right). \,$ $= -\begin{matrix} \frac{1}{4\mu_0} \end{matrix} \left( \partial_\mu A_\nu \partial^\mu A^\nu - \partial_\nu A_\mu \partial^\mu A^\nu - \partial_\mu A_\nu \partial^\nu A^\mu + \partial_\nu A_\mu \partial^\nu A^\mu \right).$

The far left and far right terms are the same because μ and ν are just dummy indices after all. The two middle terms are also the same, so the Lagrangian density is

 $\mathcal{L} \,$ $= -\begin{matrix} \frac{1}{2\mu_0} \end{matrix} \left( \partial_\mu A_\nu \partial^\mu A^\nu - \partial_\nu A_\mu \partial^\mu A^\nu \right).$

We can then plug this into the Euler-Lagrange equation of motion for a field:

$\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0 . \,$

The second term is zero because the Lagrangian in this case only contains derivatives. So the Euler-Lagrange equation becomes:

$\partial_\nu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) = 0. \,$

The quantity in parentheses above is just the field tensor, so this finally simplifies to

 $\partial_\nu F^{\mu \nu} = 0. \,$

That equation is just another way of writing the two inhomogeneous Maxwell's equations as long as you make the substitutions:

$~E^i /c \ \ = -F^{0 i} \,$
$\epsilon^{ijk} B^k = -F^{ij} \,$

where $i \,$ and $j \,$ take on the values of 1, 2, and 3.

When there are charges or currents, the Lagrangian needs an extra term to account for the coupling between them and the electromagnetic field. In that case $\partial_\nu F^{\mu \nu}$ is equal to the 4-current instead of zero.

### Role in quantum electrodynamics and field theory

The Lagrangian of quantum electrodynamics extends beyond the classical Lagrangian established in relativity, from $\mathcal{L}=\bar\psi(i\hbar c \, \gamma^\alpha D_\alpha - mc^2)\psi -\frac{1}{4 \mu_0}F_{\alpha\beta}F^{\alpha\beta},$  to incorporate the creation and annihilation of photons (and electrons).

In quantum field theory it is used as the template for the gauge field strength tensor. By being employed in addition to the local interaction Lagrangian it reprises its usual role in QED.

## References

• Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 0-19-514665-4.
• Jackson, John D. (1999). Classical Electrodynamics. John Wiley & Sons, Inc.. ISBN 0-471-30932-X.
• Peskin, Michael E.; Schroeder, Daniel V. (1995). An Introduction to Quantum Field Theory. Perseus Publishing. ISBN 0-201-50397-2.

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