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Anelectromagnetic four-potential is arelativisticvector function from which theelectromagnetic field can be derived. It combines both anelectric scalar potential and amagnetic vector potential into a singlefour-vector.^[1]

As measured in a givenframe of reference, and for a givengauge, 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 isLorentz covariant.

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

This article usestensor index notation and theMinkowski metricsign convention(+ − − −). See alsocovariance and contravariance of vectors andraising and lowering indices for more details on notation. Formulae are given inSI units andGaussian-cgs units.

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

SI units	Gaussian units
${\displaystyle A^{\alpha }=\left(\frac{1}{c}\varphi ,\mathbf{A}\right)}$	${\displaystyle A^{\alpha }=(\varphi ,\mathbf{A})}$

in whichϕ is theelectric potential, andA is themagnetic potential (avector potential). The unit ofA^α isV·s·m⁻¹ in SI, andMx·cm⁻¹ inGaussian-CGS.

The electric and magnetic fields associated with these four-potentials are:^[3]

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

Inspecial relativity, the electric and magnetic fields transform underLorentz transformations. This can be written in the form of a rank twotensor – theelectromagnetic tensor. The 16 contravariant components of the electromagnetic tensor, usingMinkowski metric convention(+ − − −), are written in terms of the electromagnetic four-potential and thefour-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.

Often, theLorenz gauge condition${\displaystyle {\mathrm{\partial }}_{\alpha }{A}^{\alpha }=0}$ in aninertial frame of reference is employed to simplifyMaxwell's equations as:^[2]

SI units	Gaussian units
${\displaystyle ◻A^{\alpha }={\mu }_0{J}^{\alpha }}$	${\displaystyle ◻A^{\alpha }=\frac{4\pi }{c}{J}^{\alpha }}$

whereJ^α are the components of thefour-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 thed'Alembertian operator. In terms of the scalar and vector potentials, this last equation becomes:

SI units	Gaussian units
${\displaystyle ◻\varphi =-\frac{\rho }{{ϵ}_0}}$	${\displaystyle ◻\varphi =4\pi \rho }$
${\displaystyle ◻\mathbf{A}=-{\mu }_0\mathbf{j}}$	${\displaystyle ◻\mathbf{A}=\frac{4\pi }{c}\mathbf{j}}$

For a given charge and current distribution,ρ(r,t) andj(r,t), the solutions to these equations in SI units are:^[3]

where

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

is theretarded 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 aninhomogeneousdifferential equation, any solution to the homogeneous equation can be added to these to satisfy theboundary 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 tor⁻² (theinduction field) and a component decreasing asr⁻¹ (theradiation field).^{[clarification needed]}

Whenflattened to aone-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^{[clarification needed]} via theHodge decomposition theorem as the sum of anexact, a coexact, and a harmonic form,

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

There isgauge freedom inA in that of the three forms in this decomposition, only the coexact form has any effect on theelectromagnetic 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

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

• Four-vector
• Covariant formulation of classical electromagnetism
• Jefimenko's equations
• Gluon field
• Aharonov–Bohm effect

• Rindler, Wolfgang (1991).Introduction to Special Relativity (2nd). Oxford: Oxford University Press.ISBN 0-19-853952-5.
• Jackson, J D (1999).Classical Electrodynamics (3rd). New York: Wiley.ISBN 0-471-30932-X.

• Theory of relativity
• Electromagnetism
• Four-vectors

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