The electromagnetic fourpotential is a covariant fourvector consisting of the electric potential and magnetic vector potential. All formulas in this article are given in SI units, and in parentheses in Gaussiancgs units. The definition of the electromagnetic fourpotential is:
in which φ is the electrical potential, and is the magnetic potential, a vector potential.
The units of A^{α} are volt·seconds/meter in SI, and maxwell/centimeter in Gaussiancgs.
The electric and magnetic fields associated with these fourpotentials are:
It is useful to group the potentials together in this form because A^{α} is a contravariant vector, hence A_{α} = g_{αβ}A^{β}, is a covariant vector. This means that A_{α} transforms in the same way as the gradient of a scalarvalued function, e.g. under arbitrary curvilinear coordinate transformations. So, for example, the inner product
is the same in every inertial frame of reference.
Often, physicists employ the Lorenz gauge condition in an inertial frame of reference to simplify Maxwell's equations as:
where are the components of the fourcurrent,
and
In terms of the scalar and vector potentials, this last equation becomes:
For a given charge and current distribution, and , the solutions to these equations in SI units are
where is the retarded time. This is sometimes also expressed with , 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).
