View source for Covariant formulation of classical electromagnetism

{{Short description|Ways of writing certain laws of physics}}
{{broader2|Mathematical descriptions of the electromagnetic field|Classical electromagnetism and special relativity}}
{{Electromagnetism|Covariance}}

The '''[[covariance and contravariance of vectors|covariant]] formulation of [[classical electromagnetism]]''' refers to ways of writing the laws of classical electromagnetism (in particular, [[Maxwell's equations]] and the [[Lorentz force]]) in a form that is manifestly invariant under [[Lorentz transformation]]s, in the formalism of [[special relativity]] using rectilinear [[inertial frame|inertial coordinate system]]s. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as [[Maxwell's equations in curved spacetime]] or non-rectilinear coordinate systems.{{efn|
This article uses the [[classical treatment of tensors]] and [[Einstein summation convention]] throughout and the [[Minkowski metric]] has the form {{math|diag(+1, −1, −1, −1)}}. Where the equations are specified as holding in vacuum, one could instead regard them as the formulation of Maxwell's equations in terms of ''total'' charge and current.}}

== Covariant objects ==

=== Preliminary four-vectors ===
{{Main|Lorentz covariance}}
Lorentz tensors of the following kinds may be used in this article to describe bodies or particles:
* [[Displacement (vector)|four-displacement]]: <math display="block">x^\alpha = (ct, \mathbf{x}) = (ct, x, y, z) \,.</math>
* [[Four-velocity]]: <math display="block">u^\alpha = \gamma(c,\mathbf{u}) ,</math> where ''γ''('''u''') is the [[Lorentz factor]] at the 3-velocity '''u'''.
* [[Four-momentum]]: <math display="block">p^\alpha = ( E/c, \mathbf{p}) = m_0 u^{\alpha} </math> where <math>\mathbf{p}</math> is 3-momentum, <math>E</math> is the [[total energy]], and <math>m_0</math> is [[rest mass]].
* [[Four-gradient]]: <math display="block">\partial^{\nu} = \frac{\partial}{\partial x_{\nu}} = \left( \frac{1}{c} \frac{\partial}{\partial t}, - \mathbf{\nabla} \right) \,,</math>
* The [[d'Alembertian]] operator is denoted <math> {\partial}^2 </math>, <math display="block">\partial^2 = \frac{1}{c^2} {\partial^2 \over \partial t^2} - \nabla^2.</math>

The signs in the following tensor analysis depend on the [[sign convention#Metric signature|convention]] used for the [[metric tensor]]. The convention used here is {{nowrap|(+ − − −)}}, corresponding to the [[Minkowski space#Standard basis|Minkowski metric tensor]]:
<math display="block">\eta^{\mu \nu}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}</math>

=== Electromagnetic tensor ===
{{Main|Electromagnetic tensor}}
The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant [[antisymmetric tensor]] whose entries are '''B'''-field quantities.<ref name=Vanderlinde>
{{citation
  | last = Vanderlinde
  | first = Jack
  | title = classical electromagnetic theory
  | publisher = Springer
  | year = 2004
  | pages = 313–328
  | url = https://books.google.com/books?id=HWrMET9_VpUC&q=electromagnetic+field+tensor+vanderlinde&pg=PA316
  | isbn = 9781402026997
}}</ref>
<math display="block">F_{\alpha \beta} = \begin{pmatrix}
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{pmatrix}</math>
and the result of raising its indices is
<math display="block">F^{\mu \nu} \mathrel{\stackrel{\mathrm{def}}{=}} \eta^{\mu \alpha} \, F_{\alpha \beta} \, \eta^{\beta \nu} = \begin{pmatrix}
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{pmatrix} \,,</math>
where '''E''' is the [[electric field]], '''B''' the [[magnetic field]], and ''c'' the [[speed of light]].

=== Four-current ===
{{Main|Four-current}}
The four-current is the contravariant four-vector which combines [[electric charge density]] ''ρ'' and [[electric current density]] '''j''':
<math display="block">J^{\alpha} = ( c \rho, \mathbf{j} ) \,.</math>

=== Four-potential ===
{{Main|Four-potential}}
The electromagnetic four-potential is a covariant four-vector containing the [[electric potential]] (also called the [[scalar potential]]) ''ϕ'' and [[magnetic vector potential]] (or [[vector potential]]) '''A''', as follows:
<math display="block">A^{\alpha} = \left(\phi/c, \mathbf{A} \right)\,.</math>

The differential of the electromagnetic potential is
<math display="block">F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha} \,.</math>

In the language of [[differential forms]], which provides the generalisation to curved spacetimes, these are the components of a 1-form <math>A = A_{\alpha} dx^{\alpha}</math> and a 2-form <math display="inline">F = dA = \frac{1}{2} F_{\alpha \beta} dx^{\alpha} \wedge dx^{\beta}</math> respectively. Here, <math>d</math> is the [[exterior derivative]] and <math>\wedge</math> the [[wedge product]].

=== Electromagnetic stress–energy tensor ===
{{Main|Electromagnetic stress–energy tensor}}
The electromagnetic stress–energy tensor can be interpreted as the flux density of the momentum four-vector, and is a contravariant symmetric tensor that is the contribution of the electromagnetic fields to the overall [[stress–energy tensor]]:
<math display="block">T^{\alpha\beta} = \begin{pmatrix}
\varepsilon_{0} E^2/2 + B^2/2\mu_0 & S_x/c & S_y/c & S_z/c \\
S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\
S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\
S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz}
\end{pmatrix}\,,</math>
where <math> \varepsilon_0 </math> is the [[electric constant|electric permittivity of vacuum]], ''μ''<sub>0</sub> is the [[magnetic constant|magnetic permeability of vacuum]], the [[Poynting vector]] is
<math display="block">\mathbf{S} = \frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B} </math>
and the [[Maxwell stress tensor]] is given by
<math display="block">\sigma_{ij} = \varepsilon_0 E_i E_j + \frac{1}{\mu_0} B_i B_j - \left(\frac 1 2 \varepsilon_0 E^2 + \frac{1}{2\mu_0} B^2\right) \delta_{ij} \,.</math>

The electromagnetic field tensor ''F'' constructs the electromagnetic stress–energy tensor ''T'' by the equation:<ref>Classical Electrodynamics, Jackson, 3rd edition, page 609</ref>
<math display="block">T^{\alpha\beta} = \frac{1}{\mu_{0}} \left( \eta^{\alpha \nu}F_{\nu \gamma}F^{\beta \gamma} - \frac{1}{4} \eta^{\alpha\beta}F_{\gamma \nu}F^{\gamma \nu}\right)</math>
where ''η'' is the [[Minkowski metric]] tensor (with signature {{nowrap|(+ − − −)}}). Notice that we use the fact that
<math display="block">\varepsilon_{0} \mu_{0} c^2 = 1\,,</math>
which is predicted by Maxwell's equations.

== Maxwell's equations in vacuum ==
{{Main|Maxwell's equations}}
In vacuum (or for the microscopic equations, not including macroscopic material descriptions), Maxwell's equations can be written as two tensor equations.

The two inhomogeneous Maxwell's equations, [[Gauss's Law]] and [[Ampère's circuital law|Ampère's law]] (with Maxwell's correction) combine into (with {{nowrap|(+ − − −)}} metric):<ref>Classical Electrodynamics by Jackson, 3rd Edition, Chapter 11 Special Theory of Relativity</ref>

{{Equation box 1
|indent=:
|title='''[[Gauss's Law|Gauss]]&ndash;[[Ampère's circuital law|Ampère]] law'''
|equation=<math>\partial_{\alpha}F^{\alpha\beta} = \mu_{0} J^{\beta} </math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4
}}
The homogeneous equations – [[Faraday's law of induction]] and [[Gauss's law for magnetism]] combine to form <math>\partial^{\sigma} F^{\mu \nu} + \partial^{\mu} F^{\nu \sigma} + \partial^{\nu} F^{\sigma \mu} = 0</math>, which may be written using Levi-Civita duality as:
{{Equation box 1
|indent=:
|title='''[[Gauss's law for magnetism|Gauss]]&ndash;[[Faraday's law of induction|Faraday]] law'''
|equation=<math>\partial_{\alpha}\left(\varepsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}\right) = 0 </math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4
}}
where ''F''<sup>''αβ''</sup> is the [[electromagnetic tensor]], ''J''<sup>''α''</sup> is the [[four-current]], ''ε''<sup>''αβγδ''</sup> is the [[Levi-Civita symbol]], and the indices behave according to the [[Einstein summation convention]].

Each of these tensor equations corresponds to four scalar equations, one for each value of ''β''.

Using the [[antisymmetric tensor]] notation and comma notation for the partial derivative (see [[Ricci calculus]]), the second equation can also be written more compactly as:
<math display="block"> F_{[\alpha \beta, \gamma]} = 0 .</math>

In the absence of sources, Maxwell's equations reduce to:
<math display="block">
  \partial^{\nu} \partial_\nu F^{\alpha\beta} \mathrel{\stackrel{\text{def}}{=}}
  \partial^2 F^{\alpha\beta} \mathrel{\stackrel{\text{def}}{=}}
  {1 \over c^2 }{\partial^2 F^{\alpha\beta} \over {\partial t}^2} - \nabla^2 F^{\alpha\beta} =
  0 \,,
</math>
which is an [[electromagnetic wave equation]] in the field strength tensor.

=== Maxwell's equations in the Lorenz gauge ===
<!--Lorenz is the correct name (not Lorentz). Ludvig Lorenz ≠ Hendrik Lorentz -->
{{Main|Lorenz gauge condition}}

The [[Lorenz gauge condition]] is a Lorentz-invariant gauge condition. (This can be contrasted with other [[gauge fixing|gauge conditions]] such as the [[Coulomb gauge]], which if it holds in one [[inertial frame]] will generally not hold in any other.) It is expressed in terms of the four-potential as follows:
<math display="block">\partial_{\alpha} A^{\alpha} = \partial^{\alpha} A_{\alpha} = 0 \,.</math>

In the Lorenz gauge, the microscopic Maxwell's equations can be written as:
<math display="block">{\partial}^2 A^{\sigma} = \mu_{0} \, J^{\sigma}\,.</math>

== Lorentz force ==

{{Main|Lorentz force}}

=== Charged particle ===
[[File:Lorentz force particle.svg|200px|thumb|[[Lorentz force]] '''f''' on a [[charged particle]] (of [[electric charge|charge]] ''q'') in motion (instantaneous velocity '''v'''). The [[electric field|'''E''' field]] and [[magnetic field|'''B''' field]] vary in space and time.]]

Electromagnetic (EM) fields affect the motion of [[electric charge|electrically charged]] matter: due to the [[Lorentz force]]. In this way, EM fields can be [[Particle detector|detected]] (with applications in [[particle physics]], and natural occurrences such as in [[Aurora (astronomy)|aurorae]]). In relativistic form, the Lorentz force uses the field strength tensor as follows.<ref>The assumption is made that no forces other than those originating in '''E''' and '''B''' are present, that is, no [[gravitation]]al, [[weak force|weak]] or [[strong force|strong]] forces.</ref>

Expressed in terms of [[coordinate time]] ''t'', it is:
<math display="block"> { d p_{\alpha} \over { d t } } =  q \, F_{\alpha \beta} \, \frac{d x^\beta}{d t} ,</math>
where ''p''<sub>''α''</sub> is the four-momentum, ''q'' is the [[Electric charge|charge]], and ''x''<sup>''β''</sup> is the position.

Expressed in frame-independent form, we have the four-force
<math display="block"> \frac{d p_{\alpha}}{d \tau} \, = q \, F_{\alpha \beta} \, u^\beta ,</math>
where ''u''<sup>''β''</sup> is the four-velocity, and ''τ'' is the particle's [[proper time]], which is related to coordinate time by {{nowrap|1=''dt'' = ''γdτ''}}.

=== Charge continuum ===
[[File:Lorentz force continuum.svg|200px|thumb|Lorentz force per spatial volume '''f''' on a continuous [[charge distribution]] ([[charge density]] ρ) in motion.]]

{{see also|continuum mechanics}}
The density of force due to electromagnetism, whose spatial part is the Lorentz force, is given by
<math display="block">f_{\alpha} = F_{\alpha\beta}J^{\beta} .</math>
and is related to the electromagnetic stress–energy tensor by
<math display="block">f^{\alpha} = - {T^{\alpha\beta}}_{,\beta} \equiv  - \frac{\partial T^{\alpha\beta}}{\partial x^\beta}.</math>

== Conservation laws ==

=== Electric charge ===
The [[continuity equation]]:
<math display="block">
  {J^\beta}_{,\beta} \mathrel\overset{\text{def}}{\mathop{=}}
  \partial_\beta J^\beta =
  \partial_\beta \partial_\alpha F^{\alpha \beta}/\mu_0 =
  0.
</math>
expresses [[charge conservation]].

=== Electromagnetic energy–momentum ===

Using the Maxwell equations, one can see that the [[electromagnetic stress–energy tensor]] (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current four-vector
<math display="block">{T^{\alpha\beta}}_{,\beta} + F^{\alpha\beta} J_\beta = 0</math>
or
<math display="block">\eta_{\alpha \nu} { T^{\nu \beta } }_{,\beta} + F_{\alpha \beta} J^\beta = 0,</math>
which expresses the conservation of linear momentum and energy by electromagnetic interactions.

== Covariant objects in matter ==

=== Free and bound four-currents ===

In order to solve the equations of electromagnetism given here, it is necessary to add information about how to calculate the electric current, ''J''<sup>''ν''</sup>. Frequently, it is convenient to separate the current into two parts, the free current and the bound current, which are modeled by different equations;
<math display="block">J^{\nu} = {J^{\nu}}_{\text{free}} + {J^{\nu}}_{\text{bound}} \,,</math>
where
<math display="block">\begin{align}
     {J^{\nu}}_{\text{free}} = \begin{pmatrix} c\rho_{\text{free}}, & \mathbf{J}_{\text{free}} \end{pmatrix} &= \begin{pmatrix} c \nabla \cdot \mathbf{D}, & - \frac{\partial \mathbf{D}}{\partial t} + \nabla\times\mathbf{H}\end{pmatrix} \,, \\
  {J^{\nu}}_{\text{bound}} = \begin{pmatrix} c\rho_{\text{bound}}, & \mathbf{J}_{\text{bound}} \end{pmatrix} &= \begin{pmatrix} - c \nabla \cdot \mathbf{P}, & \frac{\partial \mathbf{P}}{\partial t} + \nabla\times\mathbf{M} \end{pmatrix} \,.
\end{align}</math>

[[Maxwell's equations#Maxwell's "macroscopic" equations|Maxwell's macroscopic equations]] have been used, in addition the definitions of the [[electric displacement]] '''D''' and the [[magnetic field|magnetic intensity]] '''H''':
<math display="block">\begin{align}
  \mathbf{D} &= \varepsilon_0 \mathbf{E} + \mathbf{P}, \\
  \mathbf{H} &= \frac{1}{\mu_0} \mathbf{B} - \mathbf{M} \,.
\end{align}</math>
where '''M''' is the [[magnetization]] and '''P''' the [[electric polarization]].

=== Magnetization–polarization tensor ===

The bound current is derived from the '''P''' and '''M''' fields which form an antisymmetric contravariant magnetization-polarization tensor 
<ref name=Vanderlinde />
<ref>However, the  assumption that <math>M^{\mu\nu}</math>,  <math>D^{\mu\nu}</math>,  and even <math>F^{\mu\nu}</math>, are relativistic tensors in a polarizable medium, is without foundation. The quantity<math display="block">A^{\alpha} = \left(\phi/c, \mathbf{A} \right)\,</math>is not a four vector in a polarizable medium, so <math display="block">F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha} \,</math> does not produce a tensor.</ref>
<ref>
{{citation
 | last = Franklin
 | first = Jerrold
 | title = Can electromagnetic fields form tensors in a polarizable medium?
 | url =https://www.academia.edu/104227822}}</ref><ref>{{Citation 
 | last = Gonano
 | first = Carlo
 | title = Definition for Polarization P and Magnetization M Fully Consistent with Maxwell's Equations
 | url = https://www.jpier.org/PIERB/pier.php?paper=15100606
}}</ref>
<math display="block">\mathcal{M}^{\mu \nu} = \begin{pmatrix}
      0  & P_x c & P_y c & P_z c \\
 - P_x c &    0 & -M_z   &  M_y \\
 - P_y c &  M_z &    0   & -M_x \\
 - P_z c & -M_y &  M_x   &    0
\end{pmatrix},</math>
which determines the bound current
<math display="block">{J^\nu}_\text{bound} = \partial_\mu \mathcal{M}^{\mu \nu} \,.</math>

=== Electric displacement tensor ===

If this is combined with ''F''<sup>''μν''</sup> we get the antisymmetric contravariant  electromagnetic displacement tensor which combines the '''D''' and '''H''' fields as follows:
<math display="block">
  \mathcal{D}^{\mu \nu} =
  \begin{pmatrix}
      0 & -D_xc & -D_yc & -D_zc \\
   D_xc &     0 & - H_z &   H_y \\
   D_yc &   H_z &     0 &  -H_x \\
   D_zc &  -H_y & H_x   &     0
  \end{pmatrix}.
</math>

The three field tensors are related by:
<math display="block">\mathcal{D}^{\mu \nu} = \frac{1}{\mu_{0}} F^{\mu \nu} - \mathcal{M}^{\mu \nu} </math>
which is equivalent to the definitions of the '''D''' and '''H''' fields given above.

== Maxwell's equations in matter ==

The result is that [[Ampère's circuital law|Ampère's law]],
<math display="block">\mathbf{\nabla} \times \mathbf{H} - \frac{\partial \mathbf{D}} {\partial t} = \mathbf{J}_{\text{free}},</math>
and [[Gauss's law]],
<math display="block">\mathbf{\nabla} \cdot \mathbf{D} = \rho_{\text{free}},</math>
combine into one equation:
{{Equation box 1
|indent=:
|title='''[[Gauss's Law|Gauss]]&ndash;[[Ampère's circuital law|Ampère]] law''' ''(matter)''
|equation=<math>{J^{\nu}}_{\text{free}} = \partial_{\mu} \mathcal{D}^{\mu \nu} </math>
|cellpadding
|border
|border colour = #0073CF
}}

The bound current and free current as defined above are automatically and separately conserved
<math display="block">\begin{align}
  \partial_\nu {J^\nu}_\text{bound} &= 0 \, \\
   \partial_\nu {J^\nu}_\text{free} &= 0 \,.
\end{align}</math>

=== Constitutive equations ===

{{main|Constitutive equation}}

==== Vacuum ====

In vacuum, the constitutive relations between the field tensor and displacement tensor are:
<math display="block">\mu_0 \mathcal{D}^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha \beta} \eta^{\beta \nu} \,.</math>

Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define ''F''<sup>''μν''</sup> by
<math display="block">F^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha \beta} \eta^{\beta \nu} ,</math>
the constitutive equations may, in ''vacuum'', be combined with the Gauss–Ampère law to get:
<math display="block">\partial_\beta F^{\alpha \beta} = \mu_0 J^{\alpha}. </math>

The electromagnetic stress–energy tensor in terms of the displacement is:
<math display="block">T_\alpha{}^\pi = F_{\alpha\beta} \mathcal{D}^{\pi\beta} - \frac{1}{4} \delta_\alpha^\pi F_{\mu\nu} \mathcal{D}^{\mu\nu} ,</math>
where ''δ<sub>α</sub><sup>π</sup>'' is the [[Kronecker delta]]. When the upper index is lowered with ''η'', it becomes symmetric and is part of the source of the gravitational field.

==== Linear, nondispersive matter ====

Thus we have reduced the problem of modeling the current, ''J''<sup>''ν''</sup> to two (hopefully) easier problems &mdash; modeling the free current, ''J''<sup>''ν''</sup><sub>free</sub> and modeling the magnetization and polarization, <math> \mathcal{M}^{\mu\nu}</math>. For example, in the simplest materials at low frequencies, one has
<math display="block">\begin{align}
  \mathbf{J}_\text{free} &= \sigma \mathbf{E} \, \\
              \mathbf{P} &= \varepsilon_0 \chi_e \mathbf{E} \, \\
              \mathbf{M} &= \chi_m \mathbf{H} \,
\end{align}</math>
where one is in the instantaneously comoving inertial frame of the material, ''σ'' is its [[electrical conductivity]], ''χ''<sub>e</sub> is its [[electric susceptibility]], and ''χ''<sub>m</sub> is its [[magnetic susceptibility]].

The constitutive relations between the <math>\mathcal{D}</math> and ''F'' tensors, proposed by [[Hermann Minkowski|Minkowski]] for a linear materials (that is, '''E''' is [[Proportionality (mathematics)|proportional]] to '''D''' and '''B''' proportional to '''H'''), are:
<math display="block">\begin{align}
         \mathcal{D}^{\mu\nu}u_\nu &= c^2\varepsilon F^{\mu\nu} u_\nu \\
  {\star\mathcal{D}^{\mu\nu}}u_\nu &= \frac{1}{\mu}{\star F^{\mu\nu}} u_\nu
\end{align}</math>
where ''u'' is the four-velocity of material, ''ε'' and ''μ'' are respectively the proper [[permittivity]] and [[Permeability (electromagnetism)|permeability]] of the material (i.e. in rest frame of material), <math>\star</math> and denotes the [[Hodge star operator]].

== Lagrangian for classical electrodynamics ==

=== Vacuum ===
The [[Lagrangian (field theory)|Lagrangian]] density for classical electrodynamics is composed by two components: a field component and a source component:
<math display="block"> \mathcal{L} \, = \, \mathcal{L}_\text{field} + \mathcal{L}_\text{int} = - \frac{1}{4 \mu_0} F^{\alpha \beta} F_{\alpha \beta} - A_{\alpha} J^{\alpha} \,.</math>

In the interaction term, the four-current should be understood as an abbreviation of many terms expressing the electric currents of other charged fields in terms of their variables; the four-current is not itself a fundamental field.

The [[Euler–Lagrange equation|Lagrange equations]] for the electromagnetic lagrangian density <math> \mathcal{L}\mathord\left(A_{\alpha}, \partial_{\beta} A_{\alpha}\right)</math> can be stated as follows:
<math display="block"> \partial_{\beta}\left[\frac{\partial \mathcal{L}}{\partial (\partial_{\beta}A_{\alpha})}\right]  - \frac{\partial \mathcal{L}}{\partial A_{\alpha}}=0 \,.</math>

Noting
<math display="block">\begin{align}
  F^{\lambda \sigma} &= F_{\mu \nu}\eta^{\mu\lambda}\eta^{\nu\sigma}, \\
         F_{\mu \nu} &= \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}\, \\
  {\partial \left(\partial_{\mu} A_{\nu}\right) \over \partial\left(\partial_{\rho} A_{\sigma}\right)}
                     &= \delta_{\mu}^{\rho} \delta_{\nu}^{\sigma}
\end{align}</math>
the expression inside the square bracket is
<math display="block">\begin{align}
  \frac{\partial \mathcal{L}}{\partial (\partial_\beta A_\alpha)}
    & = - \ \frac{1}{4 \mu_0}\ \frac{\partial \left(F_{\mu \nu}\eta^{\mu\lambda}\eta^{\nu\sigma}F_{\lambda \sigma}\right)}{\partial \left(\partial_\beta A_\alpha\right)} \\
    & = - \ \frac{1}{4 \mu_0}\ \eta^{\mu\lambda}\eta^{\nu\sigma}
\left(F_{\lambda\sigma}\left(\delta^\beta_\mu \delta^\alpha_\nu - \delta^\beta_\nu \delta^\alpha_\mu\right) + F_{\mu\nu}\left(\delta^\beta_\lambda \delta^\alpha_\sigma - \delta^\beta_\sigma \delta^\alpha_\lambda\right)
\right) \\
    & = - \ \frac{F^{\beta\alpha}}{\mu_0}\,.
\end{align}</math>

The second term is
<math display="block">\frac{\partial \mathcal{L}}{\partial A_{\alpha}} = - J^{\alpha} \,.</math>

Therefore, the electromagnetic field's equations of motion are
<math display="block">\frac{\partial F^{\beta\alpha}}{\partial x^{\beta}} = \mu_0 J^{\alpha} \,.</math>
which is the Gauss–Ampère equation above.

=== Matter ===

Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows:
<math display="block"> \mathcal{L} \, = \, - \frac{1}{4 \mu_0} F^{\alpha \beta} F_{\alpha \beta} - A_{\alpha} J^{\alpha}_{\text{free}} + \frac 1 2 F_{\alpha \beta} \mathcal{M}^{\alpha \beta} \,.</math>

Using Lagrange equation, the equations of motion for <math> \mathcal{D}^{\mu\nu}</math> can be derived.

The equivalent expression in vector notation is:
<math display="block"> \mathcal{L} \, = \, \frac 1 2 \left(\varepsilon_{0} E^2 - \frac{1}{\mu_{0}} B^2\right) - \phi \, \rho_{\text{free}} + \mathbf{A} \cdot \mathbf{J}_{\text{free}} + \mathbf{E} \cdot \mathbf{P} + \mathbf{B} \cdot \mathbf{M} \,.</math>

== See also ==

* [[Covariant classical field theory]]
* [[Electromagnetic tensor]]
* [[Electromagnetic wave equation]]
* [[Liénard–Wiechert potential]] for a charge in arbitrary motion
* [[Moving magnet and conductor problem]]
* [[Inhomogeneous electromagnetic wave equation]]
* [[Proca action]]
* [[Quantum electrodynamics]]
* [[Relativistic electromagnetism]]
* [[Stueckelberg action]]
* [[Wheeler–Feynman absorber theory]]

== Notes ==
{{notelist}}

== References ==
{{reflist|2}}

== Further reading ==
* [https://feynmanlectures.caltech.edu/II_25.html The Feynman Lectures on Physics Vol. II Ch. 25: Electrodynamics in Relativistic Notation]
* {{cite book | author=Einstein, A. | title=Relativity: The Special and General Theory | location=New York | publisher=Crown | year=1961 | isbn=0-517-02961-8 | url-access=registration | url=https://archive.org/details/relativityspecia00eins_0 }}
* {{cite book |author1=Misner, Charles |author2=Thorne, Kip S. |author3=Wheeler, John Archibald | title=Gravitation | location=San Francisco | publisher=W. H. Freeman | year=1973 | isbn=0-7167-0344-0}}
* {{cite book |author1=Landau, L. D.  |author2=Lifshitz, E. M.| title=Classical Theory of Fields | location=Oxford | publisher=Pergamon | year=1975 | isbn=0-08-018176-7|edition=Fourth Revised English}}
* {{cite book |author1=R. P. Feynman |author2=F. B. Moringo |author3=W. G. Wagner | title=Feynman Lectures on Gravitation |url=https://archive.org/details/feynmanlectureso0000feyn_g4q1 |url-access=registration | publisher=Addison-Wesley | year=1995 | isbn=0-201-62734-5}}

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