[1] 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.
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.
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: If the said signature is instead (− + + +) then: This essentially defines the four-potential in terms of physically observable quantities, as well as reducing to the above definition.
in an inertial frame of reference is employed to simplify Maxwell's equations as:[2] where Jα are the components of the four-current, and 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:[3] where is the retarded time.
These homogeneous solutions in general represent waves propagating from sources outside the boundary.
[clarification needed] When flattened to a one-form (in tensor notation,
are, we are left with simply In infinite flat Minkowski space, every closed form is exact.