Electromagnetic four-potential

An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.

As measured in a given frame of reference, and for a given gauge, the first component of the electromagnetic four-potential is conventionally taken to be the electric scalar potential, and the other three components make up the magnetic vector potential. While both the scalar and vector potential depend upon the frame, the electromagnetic four-potential is Lorentz covariant.

Like other potentials, many different electromagnetic four-potentials correspond to the same electromagnetic field, depending upon the choice of gauge.

This article uses tensor index notation and the Minkowski metric sign convention (+ − − −). See also covariance and contravariance of vectors and raising and lowering indices for more details on notation. Formulae are given in SI units and Gaussian-cgs units.

The contravariant electromagnetic four-potential can be defined as:

in which ϕ is the electric potential, and A is the magnetic potential (a vector potential). The unit of A^α is V·s·m⁻¹ in SI, and Mx·cm⁻¹ in Gaussian-CGS.

The electric and magnetic fields associated with these four-potentials are:

In special relativity, the electric and magnetic fields transform under Lorentz transformations. This can be written in the form of a rank two tensor – the electromagnetic tensor. The 16 contravariant components of the electromagnetic tensor, using Minkowski metric convention (+ − − −), are written in terms of the electromagnetic four-potential and the four-gradient as: