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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 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.

Like other potentials, many different electromagnetic four-potentials correspond to the same electromagnetic field, depending upon the choice of gauge.

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. Formulae are given in "SI units and "Gaussian-cgs units.

## Definition

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

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

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

In "special relativity, the electric and magnetic fields transform under "Lorentz transformations. This can be written in the form of a "tensor-the "electromagnetic tensor. This is written in terms of the electromagnetic four-potential and the "four-gradient as:

${\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}}$

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 ${\displaystyle \partial _{\alpha }A^{\alpha }=0}$ in an "inertial frame of reference is employed to simplify "Maxwell's equations as:[4]

SI units Gaussian units
${\displaystyle \Box A^{\alpha }=\mu _{0}J^{\alpha }}$ ${\displaystyle \Box A^{\alpha }={\frac {4\pi }{c}}J^{\alpha }}$

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

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

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

SI units Gaussian units
${\displaystyle \Box \phi ={\frac {\rho }{\epsilon _{0}}}}$ ${\displaystyle \Box \phi =4\pi \rho }$
${\displaystyle \Box \mathbf {A} =\mu _{0}\mathbf {j} }$ ${\displaystyle \Box \mathbf {A} ={\frac {4\pi }{c}}\mathbf {j} }$

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

${\displaystyle \phi (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int \mathrm {d} ^{3}x^{\prime }{\frac {\rho (\mathbf {r} ^{\prime },t_{r})}{\left|\mathbf {r} -\mathbf {r} ^{\prime }\right|}}}$
${\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int \mathrm {d} ^{3}x^{\prime }{\frac {\mathbf {j} (\mathbf {r} ^{\prime },t_{r})}{\left|\mathbf {r} -\mathbf {r} ^{\prime }\right|}},}$

where

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

is the "retarded time. This is sometimes also expressed with

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

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).["clarification needed]