View source for Electromagnetic four-potential

{{Use American English|date=March 2019}}{{Short description|Relativistic vector field}}
{{Electromagnetism|Covariance}}
An '''electromagnetic four-potential''' is a [[General relativity|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]].<ref>Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, {{ISBN|0-7167-0344-0}}</ref>

As measured in a given [[frame of reference]], and for a given [[Gauge theory|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 covariance|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]] {{nowrap|(+ − − −)}}.  See also [[covariance and contravariance of vectors]] and [[raising and lowering indices]] for more details on notation. Formulae are given in [[International System of Units|SI units]] and [[Gaussian units|Gaussian-cgs units]].

== Definition ==

The contravariant '''electromagnetic four-potential''' can be defined as:<ref name=Griffiths>{{cite book|title=Introduction to Electrodynamics|edition=3rd|author=D.J. Griffiths|publisher=Pearson Education, Dorling Kindersley|year=2007|isbn=978-81-7758-293-2}}</ref>

:{| class="wikitable"
|-
! SI units 
! Gaussian units
|-
| <math>A^\alpha = \left( \frac{1}{c}\phi, \mathbf{A} \right)\,\!</math> || <math>A^\alpha = (\phi, \mathbf{A})</math>
|}

in which ''ϕ'' is the [[electric potential]], and '''A''' is the [[magnetic vector potential|magnetic potential]] (a [[vector potential]]). The units of ''A<sup>α</sup>'' are [[volt|V]]·[[second|s]]·[[metre|m]]<sup>−1</sup> in SI, and [[maxwell (unit)|Mx]]·[[centimeter|cm]]<sup>−1</sup> in [[Gaussian units|Gaussian-cgs]].

The electric and magnetic fields associated with these four-potentials are:<ref name=grant />

:{| class="wikitable"
|-
! SI units 
! Gaussian units
|-
| <math>\mathbf{E} = -\mathbf{\nabla} \phi - \frac{\partial \mathbf{A}}{\partial t}</math> || <math>\mathbf{E} = -\mathbf{\nabla} \phi - \frac{1}{c} \frac{\partial \mathbf{A}}{\partial t} </math>
|-
| <math>\mathbf{B} = \mathbf{\nabla} \times \mathbf{A} </math> || <math>\mathbf{B} = \mathbf{\nabla} \times \mathbf{A} </math>
|}

In [[special relativity]], the electric and magnetic fields transform under [[Lorentz transformations]]. This can be written in the form of a rank two [[tensor]] - the [[electromagnetic tensor]]. 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:

:<math>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}
</math>

If the said signature is instead (− + + +) then:

:<math>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}
</math>

This essentially defines the four-potential in terms of physically observable quantities, as well as reducing to the above definition.

== In the Lorenz gauge ==
<!--- The correct name is Lorenz and NOT Lorentz (no t!!) Please do not change Lorenz to Lorentz!! Thanks ---> 
{{main|Mathematical descriptions of the electromagnetic field|Retarded potential}}
Often, the [[Lorenz gauge condition]] <math>\partial_{\alpha} A^{\alpha} = 0</math> in an [[inertial frame of reference]] is employed to simplify [[Maxwell's equations]] as:<ref name=Griffiths />

:{| class="wikitable"
|-
! SI units 
! Gaussian units
|-
| <math>\Box A^\alpha = \mu_0 J^\alpha</math> || <math> \Box A^\alpha = \frac{4 \pi}{c} J^\alpha </math>
|}

where ''J<sup>α</sup>'' are the components of the [[four-current]], and

:<math>\Box = \frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2 = \partial^\alpha \partial_\alpha</math>

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

:{| class="wikitable"
|-
! SI units 
! Gaussian units
|-
| <math>\Box \phi = \frac{\rho}{\epsilon_0}</math> || <math>\Box \phi = 4 \pi \rho</math>
|-
| <math>\Box \mathbf{A} = \mu_0 \mathbf{j}</math> || <math>\Box \mathbf{A} = \frac{4 \pi}{c} \mathbf{j}</math>
|}

For a given charge and current distribution, {{nowrap|''ρ''('''r''', ''t'')}} and {{nowrap|'''j'''('''r''', ''t'')}}, the solutions to these equations in SI units are:<ref name=grant>{{cite book|title=Electromagnetism|url=https://archive.org/details/electromagnetism0000gran|url-access=registration|edition=2nd|author=I.S. Grant, W.R. Phillips|publisher=Manchester Physics, John Wiley & Sons|year=2008|isbn=978-0-471-92712-9}}</ref>

:<math>\begin{align}
       \phi (\mathbf{r}, t) &= \frac{1}{4 \pi \epsilon_0} \int \mathrm{d}^3 x^\prime \frac{\rho\left( \mathbf{r}^\prime, t_r\right)}{ \left| \mathbf{r} - \mathbf{r}^\prime \right|} \\
  \mathbf A (\mathbf{r}, t) &= \frac{\mu_0}{4 \pi} \int \mathrm{d}^3 x^\prime \frac{\mathbf{j}\left( \mathbf{r}^\prime, t_r\right)}{ \left| \mathbf{r} - \mathbf{r}^\prime \right|},
\end{align}</math>

where 
:<math>t_r = t - \frac{\left|\mathbf{r} - \mathbf{r}'\right|}{c}</math> 

is the [[retarded time]].  This is sometimes also expressed with 
:<math>\rho\left(\mathbf{r}', t_r\right) = \left[\rho\left(\mathbf{r}', t\right)\right],</math> 

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 [[Homogeneous differential equation|inhomogeneous]] [[differential equation]], any solution to the homogeneous equation can be added to these to satisfy the [[boundary condition]]s. 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''{{i sup|−2}} (the [[induction field]]) and a component decreasing as ''r''{{i sup|−1}}  (the [[radiation field]]).{{clarify|date=March 2016}}

== Gauge freedom ==

When [[Musical isomorphism|flattened]] to a [[one-form]] (in tensor notation, <math>A_\mu</math>), the four-potential <math>A</math> (normally written as a vector or, <math>A^\mu</math> in tensor notation) can be decomposed{{clarify|date=November 2022|reason=What are the operators? d is the four gradient, but what is delta. Which differential complex?}} via the [[Hodge theory|Hodge decomposition theorem]] as the sum of an [[Closed and exact differential forms|exact]], a coexact, and a harmonic form,

:<math>A = d \alpha + \delta \beta + \gamma</math>.

There is [[gauge freedom]] in {{math|''A''}} in that of the three forms in this decomposition, only the coexact form has any effect on the [[electromagnetic tensor]]

:<math>F = d A</math>.

Exact forms are closed, as are harmonic forms over an appropriate domain, so <math>d d \alpha = 0</math> and <math>d\gamma = 0</math>, always. So regardless of what <math>\alpha</math> and <math>\gamma</math> are, we are left with simply

:<math>F = d \delta \beta</math>.

== See also ==
* [[Four-vector]]
* [[Covariant formulation of classical electromagnetism]]
* [[Jefimenko's equations]]
* [[Gluon field]]
* [[Aharonov–Bohm effect]]

== References ==
{{reflist}}
* {{cite book| author=Rindler, Wolfgang| title=Introduction to Special Relativity (2nd)| location=Oxford| publisher=Oxford University Press| year=1991| isbn=0-19-853952-5| url-access=registration| url=https://archive.org/details/introductiontosp0000rind}}
* {{cite book | author = Jackson, J D | title=Classical Electrodynamics (3rd) | location =New York | publisher=Wiley | year = 1999 | isbn=0-471-30932-X}}

[[Category:Theory of relativity]]
[[Category:Electromagnetism]]
[[Category:Four-vectors]]