Lorentz tensors of the following kinds may be used in this article to describe bodies or particles:
- where γ(u) is the Lorentz factor at the 3-velocity u.
- where is 3-momentum, is the total energy, and is rest mass.
- The d'Alembertian operator is denoted .
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:
The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities.
and the result of raising its indices is
where E is the electric field, B the magnetic field, and c the speed of light.
The four-current is the contravariant four-vector which combines electric charge density ρ and electric current density j:
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:
The differential of the electromagnetic potential is
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:
where ε0 is the electric permittivity of vacuum, μ0 is the magnetic permeability of vacuum, the Poynting vector is
and the Maxwell stress tensor is given by
The electromagnetic field tensor F constructs the electromagnetic stress–energy tensor T by the equation:
where η is the Minkowski metric tensor. Notice that we use the fact that
which is predicted by Maxwell's equations.