MU puzzle

2013-12-24T16:38:47Z

68.62.96.177: /* The puzzle */ Less vague, how it was made it solvable

{{electromagnetism|cTopic=Covariant formulation}}
{{main|Mathematical descriptions of the electromagnetic field}}
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.

This article uses [[SI units]] for the purely spatial components of tensors (including vectors), the [[classical treatment of tensors]] and the [[Einstein summation convention]] throughout, and the [[Minkowski metric]] has the form diag (+1, −1, −1, −1). Where the equations are specified as holding in a vacuum, one could instead regard them as the formulation of Maxwell's equations in terms of ''total'' charge and current.

For a more general overview of the relationships between classical electromagnetism and special relativity, including various conceptual implications of this picture, see [[Classical electromagnetism and special relativity]].

==Covariant objects==

===Preliminary 4-vectors===
{{Main|Lorentz covariance}}
For background purposes, we present here three other relevant four-vectors, which are not directly connected to electromagnetism, but which will be useful in this article:
*In [[meter]], the "position" or "coordinate" four-vector is
::<math>x^\alpha = (ct, x, y, z) \,.</math>
*In [[meter]]·[[second]]−1, the [[velocity four-vector]] (or [[four-velocity]]) is
::<math>u^\alpha = \gamma(c,\bold{u}) \,</math>
:where γ('''u''') is the [[Lorentz factor]] at the 3-velocity '''u'''.
*In [[kilogram]]·[[meter]]·[[second]]−1, the [[four-momentum]] (or [[momentum four-vector]]) of a particle is
::<math>p_\alpha = ( E/c, - \bold{p}) = mu_{\alpha} \,</math>
:where '''p''' is the 3-momentum, ''E'' is the [[kinetic energy|energy]], and ''m'' is the particle's [[rest mass]].
*In [[meter]]−1 the [[four-gradient]] is
:<math>\partial^{\nu} = \frac{\partial}{\partial x_{\nu}} = \left( \frac{1}{c} \frac{\partial}{\partial t}, - \bold{\nabla} \right) \,,</math>
*In [[meter]]−2 the [[d'Alembertian]] operator is denoted: <math> \Box </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 <tt>+---</tt>, corresponding to the [[Minkowski space#Standard basis|Minkowski metric tensor]]:
:<math>\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]]. In [[volt]]·[[second]]s·[[meter]]−2, the field strength tensor is written in terms of fields as:<ref name=Vanderlinde>{{Citation
| last = Vanderlinde
| first = Jack
| title = classical electromagnetic theory
| publisher = Springer
| year = 2004
| pages = 313–328
| url = http://books.google.com/books?id=HWrMET9_VpUC&pg=PA316&dq=electromagnetic+field+tensor+vanderlinde
| isbn = 9781402026997}}</ref>
:<math>F_{\alpha \beta} = \left( \begin{matrix}
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{matrix} \right)\,</math>

and the result of raising its indices is
:<math>F^{\mu \nu} \, \stackrel{\mathrm{def}}{=} \, \eta^{\mu \alpha} \, F_{\alpha \beta} \, \eta^{\beta \nu} = \left( \begin{matrix}
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{matrix} \right)\,.</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 current density]] '''J''' and [[electric charge density]] ρ. In [[ampere]]s·[[meter]]−2, it is given by

:<math>J^{\alpha} = \, (c \rho, \bold{J} ) \,</math>

===Four-potential===
{{Main|Four-potential}}
In [[volt]]·[[second]]s·[[meter]]−1, 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>A_{\alpha} = \left(\phi/c, - \bold{A} \right)\,</math>

The relation between the electromagnetic potentials and the electromagnetic fields is given by the following equation:
:<math>F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha} \,</math>

===Electromagnetic stress-energy tensor===
{{Main|Electromagnetic stress-energy tensor}}
The electromagnetic stress-energy tensor can be interpreted as the flux (density) of the momentum 4-vector, and is a contravariant symmetric tensor which is the contribution of the electromagnetic fields to the overall [[stress-energy tensor]]. In [[joule]]·[[meter]]−3, it is given by
:<math>T^{\alpha\beta} = \begin{pmatrix} \epsilon_{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 ε0 is the [[electric constant|electric permittivity of vacuum]], μ0 is the [[magnetic constant|magnetic permeability of vacuum]], the [[Poynting vector]] in [[watt]]·[[meter]]−2 is

:<math>\bold{S} = \frac{1}{\mu_{0}} \bold{E} \times \bold{B} </math>

and the [[Maxwell stress tensor]] in [[joule]]·[[meter]]−3 is given by

:<math>\sigma_{ij} = \epsilon_{0}E_{i}E_{j} + \frac{1}{\mu_{0}}B_{i}B_{j} -
\left(\frac12\epsilon_{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:
:<math>T^{\alpha\beta} = \frac{1}{\mu_{0}} \left( \eta_{\gamma \nu}F^{\alpha \gamma}F^{\nu \beta} - \frac{1}{4}\eta^{\alpha\beta}F_{\gamma \nu}F^{\gamma \nu}\right)</math>
where η is the [[Minkowski metric]] tensor. Notice that we use the fact that
:<math>\epsilon_{0} \mu_{0} c^2 = 1\,</math>
which is predicted by Maxwell's equations.

== Maxwell's equations in vacuum ==
{{Main|Maxwell's equations}}
In a 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 +--- metric):<ref>Classical Electrodynamics by Jackson, 3rd Edition, Chapter 11 Special Theory of Relativity</ref>

{{Equation box 1
|indent=:
|title='''[[Gauss's Law|Gauss]]-[[Ampère's circuital law|Ampère]] law''' ''(vacuum)''
|equation=<math>\partial_{\alpha}F^{\alpha\beta} = \mu_{0} J^{\beta} </math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

while the homogeneous equations - [[Faraday's law of induction]] and [[Gauss's law for magnetism]] combine to form:

{{Equation box 1
|indent=:
|title='''[[Gauss's law for magnetism|Gauss]]-[[Faraday's law of induction|Faraday]] law''' ''(vacuum)''
|equation=<math>\partial_{\alpha}(\tfrac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}) = 0 </math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

where ''F''αβ is the [[electromagnetic tensor]], ''J''α is the [[4-current]], εαβγδ is the [[Levi-Civita symbol]], and the indices behave according to the [[Einstein summation convention]].

The first tensor equation corresponds to four scalar equations, one for each value of β. The second tensor equation actually corresponds to 43 = 64 different scalar equations, but only four of these are independent. Using the antisymmetry of the electromagnetic field one can either reduce to an identity (0 = 0) or render redundant all the equations except for those with λ, μ, ν = either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.

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> F_{[\alpha \beta , \gamma]} =0 </math>

In the absence of sources, Maxwell's equations reduce to:

:<math>\partial^{\nu} \partial_{\nu} F^{\alpha\beta} \,\ \stackrel{\mathrm{def}}{=}\ \, \Box F^{\alpha\beta} \,\ \stackrel{\mathrm{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===

{{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]]; if it holds in one [[inertial frame]] it will generally not hold in any other.) It is expressed in terms of the four-potential as follows:

:<math>\partial_{\alpha} A^{\alpha} = \partial^{\alpha} A_{\alpha}=0 \,.</math>

In the Lorenz gauge, the microscopic Maxwell's equations can be written as:

:<math>\Box 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 (in [[Newton (unit)|newton]]s) 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]] (not proper time) ''t'' in [[second]]s, it is:
:<math> { d p_{\alpha} \over { d t } } = q \, F_{\alpha \beta} \, \frac{d x^\beta}{d t} \,</math>

where ''p''α is the [[four-momentum]] (see above), ''q'' is the [[Electric charge|charge]] (in [[coloumb]]s), and ''x''β is the position in [[meter]]s.

In the co-moving reference frame, this yields the so-called 4-force

:<math> \frac{d p_{\alpha}}{d \tau} \, = q \, F_{\alpha \beta} \, u^\beta </math>

where ''u''β is the [[four-velocity]] (see above), and τ is the particle's [[proper time]] which is related to coordinate time by ''dt'' = γ''d''τ.

===Charge continuum===
[[File:Lorentz force continuum.svg|200px|thumb|Lorentz force (per unit 3-volume) '''f''' on a continuous [[charge distribution]] ([[charge density]] ρ) in motion. The 3-[[current density]] '''J''' corresponds to the motion of the charge element ''dq'' in [[volume element]] ''dV'' and varies throughout the continuum.]]

{{see also|continuum mechanics}}
In a continuous medium, the 3D ''density of force'' combines with the ''density of power'' to form a covariant 4-vector, ''f''μ. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is 1/''c'' times the power transferred to that cell divided by the volume of the cell. The density of [[Lorentz force]] is the part of the density of force due to electromagnetism. Its spatial part is
:<math> - \bold{f} = - (\rho \bold{E} + \bold{J} \times \bold{B})\,</math>.

In manifestly covariant notation it becomes:
:<math>f_{\alpha} = F_{\alpha\beta}J^{\beta} .\!</math>

The relationship between Lorentz force and electromagnetic stress-energy tensor is
:<math>f^{\alpha} = - {T^{\alpha\beta}}_{,\beta} \equiv - \frac{\partial T^{\alpha\beta}}{\partial x^\beta}.</math>

==Conservation laws==

===Electric charge===

The [[continuity equation]]:
:<math>{J^{\alpha}}_{,\alpha} \, \stackrel{\mathrm{def}}{=} \, \partial_{\alpha} J^{\alpha} \, = \, 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>{T^{\alpha\beta}}_{,\beta} + F^{\alpha\beta} J_{\beta} = 0</math>

or
:<math>\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 4-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''ν 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>J^{\nu} = {J^{\nu}}_{\text{free}} + {J^{\nu}}_{\text{bound}} \,,</math>

where
:<math>{J^{\nu}}_{\text{free}}=(c\rho_{\text{free}},\mathbf{J}_{\text{free}})=\left(c \nabla \cdot \mathbf{D}, - \ \frac{\partial \mathbf{D}}{\partial t}+\nabla\times\mathbf{H}\right) \,,</math>
:<math>{J^{\nu}}_{\text{bound}}=(c\rho_{\text{bound}},\mathbf{J}_{\text{bound}})=\left(- \ c \nabla \cdot \mathbf{P}, \frac{\partial \mathbf{P}}{\partial t}+\nabla\times\mathbf{M}\right) \,.</math>

[[Maxwell's equations#Maxwell's "macroscopic" equations|Maxwell's macroscopic equations]] have been used, in addition the definitions of the [[electric displacement]] '''D''' (in [[coloumb]]·[[meter]]−1) and the [[magnetic field|magnetic intensity]] '''H''' (in [[ampere]]·[[meter]]−1):

:<math>\bold{D} = \epsilon_0 \bold{E} + \bold{P} \,</math>
:<math>\bold{H} = \frac{1}{\mu_{0}} \bold{B} - \bold{M} \,.</math>

where '''M''' is the [[magnetization]] (in [[ampere]]·[[meter]]−2) and '''P''' the [[electric polarization]] (in [[coulomb]]·[[meter]]−2).

===Magnetization-polarization tensor===

The bound current is derived from the '''P''' and '''M''' fields which form an antisymmetric contravariant magnetization-polarization tensor (in [[ampere]]·[[meter]]2)<ref name=Vanderlinde />
:<math>
\mathcal{M}^{\mu \nu} =
\begin{pmatrix}
0 & P_xc & P_yc & P_zc \\
- P_xc & 0 & - M_z & M_y \\
- P_yc & M_z & 0 & - M_x \\
- P_zc & - M_y & M_x & 0
\end{pmatrix},
</math>

which determines the bound current
:<math>{J^{\nu}}_{\text{bound}}=\partial_{\mu} \mathcal{M}^{\mu \nu} \,.</math>

===Electric displacement tensor===

If this is combined with ''F''μν we get the antisymmetric contravariant electromagnetic displacement tensor (in [[ampere]]·[[meter]]−1) which combines the '''D''' and '''H''' fields as follows:
:<math>
\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>\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>\bold{\nabla} \times \bold{H} = \bold{J}_{\text{free}} + \frac{\partial \bold{D}} {\partial t}</math>,
and [[Gauss's law]],
:<math>\bold{\nabla} \cdot \bold{D} = \rho_{\text{free}}</math>,

combine into one equation:

{{Equation box 1
|indent=:
|title='''[[Gauss's Law|Gauss]]-[[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
|background colour=#F5FFFA}}

The bound current and free current as defined above are automatically and separately conserved
:<math>\partial_{\nu} {J^{\nu}}_{\text{bound}} = 0 \,</math>
:<math>\partial_{\nu} {J^{\nu}}_{\text{free}} = 0 \,.</math>

===Constitutive equations===

{{main|Constitutive equation}}

====Vacuum====

In a vacuum, the constitutive relations between the field tensor and displacement tensor are:

:<math>\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''μν by
:<math>F^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha \beta} \eta^{\beta \nu} \,</math>

the constitutive equations may, in a ''vacuum'', be combined with Gauss-Ampère law to get:
:<math>\partial_\beta F^{\alpha \beta} = \mu_0 J^{\alpha}. \!</math>

The electromagnetic stress-energy tensor in terms of the displacement is:

:<math>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 δαπ 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.

====Matter====

Thus we have reduced the problem of modeling the current, ''J''ν to two (hopefully) easier problems — modeling the free current, ''J''νfree and modeling the magnetization and polarization, <math> \mathcal{M}^{\mu\nu}</math>. For example, in the simplest materials at low frequencies, one has
:<math>\bold{J}_{\text{free}} = \sigma \bold{E} \,</math>
:<math>\bold{P} = \epsilon_0 \chi_e \bold{E} \,</math>
:<math>\bold{M} = \chi_m \bold{H} \,</math>

where one is in the instantaneously comoving inertial frame of the material, σ is its [[electrical conductivity]], χe is its [[electric susceptibility]], and χm 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:<ref>{{cite book|title=Introduction to Electrodynamics|edition=3rd|author=D.J. Griffiths|publisher = Dorling Kindersley|year=2007|page=563|isbn=81-7758-293-3}}</ref>

:<math>\mathcal{D}^{\mu\nu}u_\nu= c^2\epsilon F^{\mu\nu} u_\nu</math>
:<math>{\star\mathcal{D}^{\mu\nu}}u_\nu= \frac{1}{\mu}{\star F^{\mu\nu}} u_\nu</math>

where ''u'' is the 4-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 dual]].

==Lagrangian for classical electrodynamics==

===Vacuum===
The [[Lagrangian]] (Lagrangian density) for classical electrodynamics (in [[joule]]·[[meter]]−3) is
:<math> \mathcal{L} \, = \, \mathcal{L}_{\mathrm{field}} + \mathcal{L}_{\mathrm{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]] for the electromagnetic Lagrangian density <math> \mathcal{L}(A_{\alpha},\partial_{\beta}A_{\alpha})\,</math> can be stated as follows:
:<math> \partial_{\beta}\left[\frac{\partial \mathcal{L}}{\partial (\partial_{\beta}A_{\alpha})}\right] - \frac{\partial \mathcal{L}}{\partial A_{\alpha}}=0 \,.</math>

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

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

Therefore, the electromagnetic field's equations of motion are
:<math> \frac{\partial F^{\beta\alpha}}{\partial x^{\beta}}=\mu_0 J^{\alpha} \,.</math>
which is one of the Maxwell equations above.

===Matter===

Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows:
:<math> \mathcal{L} \, = \, - \frac{1}{4 \mu_0} F^{\alpha \beta} F_{\alpha \beta} - A_{\alpha} J^{\alpha}_{\text{free}} + \frac12 F_{\alpha \beta} \mathcal{M}^{\alpha \beta} \,.</math>

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

The equivalent expression in non-relativistic vector notation is
:<math> \mathcal{L} \, = \, \frac12 \left(\epsilon_{0} E^2 - \frac{1}{\mu_{0}} B^2\right) - \phi \, \rho_{\text{free}} + \bold{A} \cdot \bold{J}_{\text{free}} + \bold{E} \cdot \bold{P} + \bold{B} \cdot \bold{M} \,.</math>

==See also==

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

==Notes and references==
<references/>

==Further reading==

*{{cite book | author=Einstein, A. | title=Relativity: The Special and General Theory | location= New York | publisher=Crown| year=1961 | isbn=0-517-02961-8}}

*{{cite book | author=Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald | title=Gravitation | location=San Francisco | publisher=W. H. Freeman | year=1973 | isbn=0-7167-0344-0}}

*{{cite book | author=Landau, L. D. and Lifshitz, E. M.| title=Classical Theory of Fields (Fourth Revised English Edition) | location=Oxford | publisher=Pergamon | year=1975 | isbn=0-08-018176-7}}

*{{cite book | author=R. P. Feynman, F. B. Moringo, and W. G. Wagner | title=Feynman Lectures on Gravitation | publisher=Addison-Wesley | year=1995 | isbn=0-201-62734-5}}

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