Electromagnetic four-potential

In the Lorenz gauge

Often, the Lorenz gauge condition ${\displaystyle {\mathrm{\partial }}_{\alpha }{A}^{\alpha }=0}$ in an inertial frame of reference is employed to simplify Maxwell's equations as:

where J^α are the components of the four-current, and

${\displaystyle ◻=\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2={\mathrm{\partial }}^{\alpha }{\mathrm{\partial }}_{\alpha }}$

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

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

${\displaystyle \begin{array}{cc}\varphi (\mathbf{r},t)& =\frac{1}{4\pi {ϵ}_0}\int {\mathrm{d}}^3{x}^{\mathrm{\prime }}\frac{\rho \left({\mathbf{r}}^{\mathrm{\prime }},t_r\right)}{\left|\mathbf{r}-{\mathbf{r}}^{\mathrm{\prime }}\right|}\\ \mathbf{A}(\mathbf{r},t)& =\frac{{\mu }_0}{4\pi }\int {\mathrm{d}}^3{x}^{\mathrm{\prime }}\frac{\mathbf{j}\left({\mathbf{r}}^{\mathrm{\prime }},t_r\right)}{\left|\mathbf{r}-{\mathbf{r}}^{\mathrm{\prime }}\right|},\end{array}}$

where

${\displaystyle t_r=t-\frac{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}{c}}$

is the retarded time. This is sometimes also expressed with

${\displaystyle \rho \left({\mathbf{r}}^{\prime },t_r\right)=\left[\rho \left({\mathbf{r}}^{\prime },t\right)\right],}$

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).

Gauge freedom

When flattened to a one-form (in tensor notation, ${\displaystyle A_{\mu }}$), the four-potential ${\displaystyle A}$ (normally written as a vector or, ${\displaystyle A^{\mu }}$ in tensor notation) can be decomposed via the Hodge decomposition theorem as the sum of an exact, a coexact, and a harmonic form,

${\displaystyle A=d\alpha +\delta \beta +\gamma }$.

There is gauge freedom in A in that of the three forms in this decomposition, only the coexact form has any effect on the electromagnetic tensor

${\displaystyle F=dA}$.

Exact forms are closed, as are harmonic forms over an appropriate domain, so ${\displaystyle dd\alpha =0}$ and ${\displaystyle d\gamma =0}$, always. So regardless of what ${\displaystyle \alpha }$ and ${\displaystyle \gamma }$ are, we are left with simply

${\displaystyle F=d\delta \beta }$.

In infinite flat Minkowski space, every closed form is exact. Therefore the ${\displaystyle \gamma }$ term vanishes. Every gauge transform of ${\displaystyle A}$ can thus be written as

${\displaystyle A\Rightarrow A+d\alpha }$.