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. [1]

As measured in a given frame of reference, and for a given gauge, the first component of the electromagnetic four-potential is 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.

In this article, index notation and the Minkowski metric (+−−−) will be used, see also Ricci calculus, 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.

Definition

The electromagnetic four-potential can be defined as:[2]

SI units cgs units
$A^\alpha = \left( \phi / c , \mathbf{A} \right)\,\!$ $A^\alpha = (\phi, \mathbf{A})$

in which ϕ is the electric potential, and A is the magnetic potential (a vector potential). The units of Aα are V·s·m−1 in SI, and Mx·cm−1 in Gaussian-cgs.

The electric and magnetic fields associated with these four-potentials are:[3]

SI units cgs units
$\mathbf{E} = -\mathbf{\nabla} \phi - \frac{\partial \mathbf{A}}{\partial t}$ $\mathbf{E} = -\mathbf{\nabla} \phi - \frac{1}{c} \frac{\partial \mathbf{A}}{\partial t}$
$\mathbf{B} = \mathbf{\nabla} \times \mathbf{A}.$ $\mathbf{B} = \mathbf{\nabla} \times \mathbf{A}.$

In special relativity, the electric and magnetic fields must be written in the form of a tensor so they transform correctly under Lorentz transformations - achieved by the electromagnetic tensor. This is written in terms of the electromagnetic four-potential as:

$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.$

This essentially defines the four-potential in terms of physically observable quantities, as well as reducing to the above definition.

In the Lorenz gauge

Often, the Lorenz gauge condition $\partial_{\alpha} A^{\alpha} = 0$ in an inertial frame of reference is employed to simplify Maxwell's equations as:[4]

SI units cgs units
$\Box A^\alpha = \mu_0 J^\alpha$ $\Box A^\alpha = \frac{4 \pi}{c} J^\alpha$

where Jα are the components of the four-current, and

$\Box = \frac{1}{c^2} \frac{\partial^2} {\partial t^2}-\nabla^2$

is the d'Alembertian operator. In terms of the scalar and vector potentials, this last equation becomes:

SI units cgs units
$\Box \phi =- \frac{\rho}{\epsilon_0}$ $\Box \phi = 4 \pi \rho$
$\Box \mathbf{A} =- \mu_0 \mathbf{j}$ $\Box \mathbf{A} = \frac{4 \pi}{c} \mathbf{j}$

For a given charge and current distribution, ρ(r, t) and j(r, t), the solutions to these equations in SI units are:[5]

$\phi (\mathbf{r}, t) = \frac{1}{4 \pi \epsilon_0} \int \mathrm{d}^3 x^\prime \frac{\rho( \mathbf{r}^\prime, t_r)}{ \left| \mathbf{r} - \mathbf{r}^\prime \right|}$
$\mathbf A (\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \mathrm{d}^3 x^\prime \frac{\mathbf{j}( \mathbf{r}^\prime, t_r)}{ \left| \mathbf{r} - \mathbf{r}^\prime \right|},$

where

$t_r = t - \frac{\left|\mathbf{r}-\mathbf{r}'\right|}{c}$

is the retarded time. This is sometimes also expressed with

$\rho(\mathbf{r}',t_r)=[\rho(\mathbf{r}',t)],$

where the square brackets are meant to indicate that the time should be evaluated at the retarded time. Of course, since the above equations are simply the solution to an inhomogeneous differential equation, any solution to the homogeneous equation can be added to these to satisfy the boundary conditions. These homogeneous solutions in general represent waves propagating from sources outside the boundary.

When the integrals above are evaluated for typical cases, e.g. of an oscillating current (or charge), they are found to give both a magnetic field component varying according to r −2 (the induction field) and a component decreasing as r −1 (the radiation field).