Electromagnetic four-potential

The electromagnetic four-potential is a four-vector defined in SI units (and gaussian units in parentheses) as


 * $$A^{\alpha}=\left(\frac{\phi}{c}, \vec A \right) \qquad \left(A^a=(\phi, \vec A)\right)$$

in which $$\phi$$ is the electrical potential, and $$\vec A$$ is the magnetic potential, a vector potential.

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


 * $$\vec{E} = -\vec{\nabla} \phi - \frac{\partial \vec{A}}{\partial t}  \qquad   \left( -\vec{\nabla} \phi - \frac{1}{c} \frac{\partial \vec{A}}{\partial t} \right) $$
 * $$\vec{B} = \vec{\nabla} \times \vec{A} $$

It is useful to group the potentials together in this form because $$A_{\alpha}$$ is a Lorentz covariant vector, meaning that it transforms in the same way as the spacetime coordinates (t, x) under transformations in the Lorentz group: rotations and Lorentz boosts. As a result, the inner product


 * $$A^{\alpha} A_{\alpha} = |\vec{A}|^2 -\frac{\phi^2}{c^2} \qquad \left(A^a A_a \, = |\vec{A}|^2 - \phi^2 \right)$$

is the same in every inertial reference frame.

Often, physicists employ the Lorenz gauge condition $$\partial_{\alpha} A^{\alpha} = 0$$ to simplify Maxwell's equations as:


 * $$\Box^2 A_{\alpha} = -\mu_0 J_{\alpha}  \qquad   \left( \Box^2 A_{\alpha} = -\frac{4 \pi}{c} J_{\alpha} \right)$$

where $$J_\alpha$$ are the components of the four-current,

and


 * $$\Box^2 = \nabla^2 -\frac{1}{c^2} \frac{\partial^2} {\partial t^2}$$ is the d'Alembertian operator.

In terms of the scalar and vector potentials, this last equation becomes:


 * $$\Box^2 \phi = -\frac{\rho}{\epsilon_0}   \qquad   \left(\Box^2 \phi = -4 \pi \rho \right)$$


 * $$\Box^2 \vec{A} = -\mu_0 \vec{j}  \qquad   \left( \Box^2 \vec{A} = -\frac{4 \pi}{c} \vec{j} \right) $$

For a given charge and current distribution, $$\rho(\vec{x},t)$$ and $$\vec{j}(\vec{x},t)$$, the solutions to these equations in SI units are


 * $$\phi (\vec{x}, t) = \frac{1}{4 \pi \epsilon_0} \int \mathrm{d}^3 x^\prime \frac{\rho( \vec{x}^\prime, \tau)}{ \left| \vec{x} - \vec{x}^\prime \right|}$$


 * $$\vec A (\vec{x}, t) = \frac{\mu_0}{4 \pi} \int \mathrm{d}^3 x^\prime \frac{\vec{j}( \vec{x}^\prime, \tau)}{ \left| \vec{x} - \vec{x}^\prime \right|}$$,

where $$\tau = t - \frac{\left|\vec{x}-\vec{x}'\right|}{c}$$ is the retarded time. This is sometimes also expressed with $$\rho(\vec{x}',\tau)=[\rho(\vec{x}',t)]$$, 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 as $$ r^{-2} $$ (the induction field) and a component decreasing as $$r^{-1} $$  (the radiation field).