Editing Electromagnetic four-potential (section)

== 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 unit of ''A<sup>α</sup>'' is [[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 {{nowrap|(+ − − −)}}, 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 {{nowrap|(− + + +)}} 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.