# Covariant formulation of classical electromagnetism

The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

This article uses the classical treatment of tensors and Einstein summation convention throughout and the Minkowski metric has the form diag (+1, −1, −1, −1). Where the equations are specified as holding in a vacuum, one could instead regard them as the formulation of Maxwell's equations in terms of total charge and current.

For a more general overview of the relationships between classical electromagnetism and special relativity, including various conceptual implications of this picture, see Classical electromagnetism and special relativity.

## Covariant objects

### Preliminary 4-vectors

Lorentz tensors of the following kinds may be used in this article to describe bodies or particles:

${\displaystyle x^{\alpha }=(ct,{\mathbf {x} })=(ct,x,y,z)\,.}$
${\displaystyle u^{\alpha }=\gamma (c,{\mathbf {u}}),}$
where γ(u) is the Lorentz factor at the 3-velocity u.
${\displaystyle p^{\alpha }=(E/c,{\mathbf {p} })=m_{0}u^{\alpha }\,}$
where ${\displaystyle {\mathbf {p}}}$ is 3-momentum, ${\displaystyle E}$ is the total energy, and ${\displaystyle m_{0}}$ is rest mass.
${\displaystyle \partial ^{\nu }={\frac {\partial }{\partial x_{\nu }}}=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},-{\mathbf {\nabla } }\right)\,,}$
• The d'Alembertian operator is denoted ${\displaystyle {\partial }^{2}}$.

The signs in the following tensor analysis depend on the convention used for the metric tensor. The convention used here is +−−−, corresponding to the Minkowski metric tensor:

${\displaystyle \eta ^{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}\,}$

### Electromagnetic tensor

The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities. [1]

${\displaystyle F_{\alpha \beta }=\left({\begin{matrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{matrix}}\right)\,}$

and the result of raising its indices is

${\displaystyle F^{\mu \nu }\,{\stackrel {\mathrm {def} }{=}}\,\eta ^{\mu \alpha }\,F_{\alpha \beta }\,\eta ^{\beta \nu }=\left({\begin{matrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{matrix}}\right)\,.}$

where E is the electric field, B the magnetic field, and c the speed of light.

### Four-current

The four-current is the contravariant four-vector which combines electric charge density ρ and electric current density j:

${\displaystyle J^{\alpha }=(c\rho ,{\mathbf {j} })\,.}$

### Four-potential

The electromagnetic four-potential is a covariant four-vector containing the electric potential (also called the scalar potential) ϕ and magnetic vector potential (or vector potential) A, as follows:

${\displaystyle A^{\alpha }=\left(\phi /c,{\mathbf {A} }\right)\,.}$

The differential of the electromagnetic potential is

${\displaystyle F_{\alpha \beta }=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }\,.}$

### Electromagnetic stress–energy tensor

The electromagnetic stress–energy tensor can be interpreted as the flux density of the momentum 4-vector, and is a contravariant symmetric tensor that is the contribution of the electromagnetic fields to the overall stress–energy tensor:

${\displaystyle T^{\alpha \beta }={\begin{pmatrix}\epsilon _{0}E^{2}/2+B^{2}/2\mu _{0}&S_{x}/c&S_{y}/c&S_{z}/c\\S_{x}/c&-\sigma _{xx}&-\sigma _{xy}&-\sigma _{xz}\\S_{y}/c&-\sigma _{yx}&-\sigma _{yy}&-\sigma _{yz}\\S_{z}/c&-\sigma _{zx}&-\sigma _{zy}&-\sigma _{zz}\end{pmatrix}}\,,}$

where ε0 is the electric permittivity of vacuum, μ0 is the magnetic permeability of vacuum, the Poynting vector is

${\displaystyle {\mathbf {S} }={\frac {1}{\mu _{0}}}{\mathbf {E} }\times {\mathbf {B} }}$

and the Maxwell stress tensor is given by

${\displaystyle \sigma _{ij}=\epsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-\left({\frac {1}{2}}\epsilon _{0}E^{2}+{\frac {1}{2\mu _{0}}}B^{2}\right)\delta _{ij}\,.}$

The electromagnetic field tensor F constructs the electromagnetic stress–energy tensor T by the equation:

${\displaystyle T^{\alpha \beta }={\frac {1}{\mu _{0}}}\left(-\eta _{\gamma \nu }F^{\alpha \gamma }F^{\nu \beta }+{\frac {1}{4}}\eta ^{\alpha \beta }F_{\gamma \nu }F^{\gamma \nu }\right)}$

where η is the Minkowski metric tensor. Notice that we use the fact that

${\displaystyle \epsilon _{0}\mu _{0}c^{2}=1\,,}$

which is predicted by Maxwell's equations.

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