## Covariant formulation of classical electromagnetism |

Part of a series of articles about |

The ** covariant formulation of classical electromagnetism** refers to ways of writing the laws of classical electromagnetism (in particular,

This article uses the *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

- covariant objects
- maxwell's equations in vacuum
- lorentz force
- conservation laws
- covariant objects in matter
- maxwell's equations in matter
- lagrangian for classical electrodynamics
- see also
- notes and references
- further reading

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

- where
*γ*(**u**) is theLorentz factor at the 3-velocity**u**.

- where is 3-momentum, is the
total energy , and isrest mass .

- The
d'Alembertian operator is denoted .

The signs in the following tensor analysis depend on the `+−−−`, corresponding to the

The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant ^{[1]}

and the result of raising its indices is

where **E** is the **B** the *c* the

The four-current is the contravariant four-vector which combines *ρ* and **j**:

The electromagnetic four-potential is a covariant four-vector containing the *ϕ* and **A**, as follows:

The differential of the electromagnetic potential is

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

where *ε*_{0} is the *μ*_{0} is the

and the

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

where *η* is the

which is predicted by Maxwell's equations.