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| {{electromagnetism|cTopic=Covariant formulation}}
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| {{main|Mathematical descriptions of the electromagnetic field}}
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| 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. | |
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| 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.
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| 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]].
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| ==Covariant objects==
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| ===Preliminary 4-vectors===
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| {{Main|Lorentz covariance}}
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| 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:
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| *In [[meter]], the "position" or "coordinate" four-vector is
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| ::<math>x^\alpha = (ct, x, y, z) \,.</math>
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| *In [[meter]]·[[second]]<sup>−1</sup>, the [[velocity four-vector]] (or [[four-velocity]]) is
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| ::<math>u^\alpha = \gamma(c,\bold{u}) \,</math>
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| :where γ('''u''') is the [[Lorentz factor]] at the 3-velocity '''u'''.
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| *In [[kilogram]]·[[meter]]·[[second]]<sup>−1</sup>, the [[four-momentum]] (or [[momentum four-vector]]) of a particle is
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| ::<math>p_\alpha = ( E/c, - \bold{p}) = mu_{\alpha} \,</math>
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| :where '''p''' is the 3-momentum, ''E'' is the [[kinetic energy|energy]], and ''m'' is the particle's [[rest mass]].
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| *In [[meter]]<sup>−1</sup> the [[four-gradient]] is
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| :<math>\partial^{\nu} = \frac{\partial}{\partial x_{\nu}} = \left( \frac{1}{c} \frac{\partial}{\partial t}, - \bold{\nabla} \right) \,,</math>
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| *In [[meter]]<sup>−2</sup> the [[d'Alembertian]] operator is denoted: <math> \Box </math>.
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| 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]]:
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| :<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>
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| ===Electromagnetic tensor===
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| {{Main|Electromagnetic tensor}}
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| The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant [[antisymmetric tensor]]. In [[volt]]·[[second]]s·[[meter]]<sup>−2</sup>, the field strength tensor is written in terms of fields as:<ref name=Vanderlinde>{{Citation
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| | last = Vanderlinde
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| | first = Jack
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| | title = classical electromagnetic theory
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| | publisher = Springer
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| | year = 2004
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| | pages = 313–328
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| | url = http://books.google.com/books?id=HWrMET9_VpUC&pg=PA316&dq=electromagnetic+field+tensor+vanderlinde
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| | isbn = 9781402026997}}</ref>
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| :<math>F_{\alpha \beta} = \left( \begin{matrix}
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| 0 & E_x/c & E_y/c & E_z/c \\
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| -E_x/c & 0 & -B_z & B_y \\
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| -E_y/c & B_z & 0 & -B_x \\
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| -E_z/c & -B_y & B_x & 0
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| \end{matrix} \right)\,</math>
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| and the result of raising its indices is
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| :<math>F^{\mu \nu} \, \stackrel{\mathrm{def}}{=} \, \eta^{\mu \alpha} \, F_{\alpha \beta} \, \eta^{\beta \nu} = \left( \begin{matrix}
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| 0 & -E_x/c & -E_y/c & -E_z/c \\
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| E_x/c & 0 & -B_z & B_y \\
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| E_y/c & B_z & 0 & -B_x \\
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| E_z/c & -B_y & B_x & 0
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| \end{matrix} \right)\,.</math>
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| where '''E''' is the [[electric field]], '''B''' the [[magnetic field]], and ''c'' the [[speed of light]].
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| ===Four-current===
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| {{Main|Four-current}}
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| The four-current is the contravariant four-vector which combines [[electric current density]] '''J''' and [[electric charge density]] ρ. In [[ampere]]s·[[meter]]<sup>−2</sup>, it is given by
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| :<math>J^{\alpha} = \, (c \rho, \bold{J} ) \,</math>
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| ===Four-potential===
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| {{Main|Four-potential}}
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| In [[volt]]·[[second]]s·[[meter]]<sup>−1</sup>, 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:
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| :<math>A_{\alpha} = \left(\phi/c, - \bold{A} \right)\,</math>
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| The relation between the electromagnetic potentials and the electromagnetic fields is given by the following equation:
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| :<math>F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha} \,</math>
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| ===Electromagnetic stress-energy tensor===
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| {{Main|Electromagnetic stress-energy tensor}}
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| 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]]<sup>−3</sup>, it is given by
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| :<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} \\
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| S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\
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| S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{pmatrix}\,</math>
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| where ε<sub>0</sub> is the [[electric constant|electric permittivity of vacuum]], μ<sub>0</sub> is the [[magnetic constant|magnetic permeability of vacuum]], the [[Poynting vector]] in [[watt]]·[[meter]]<sup>−2</sup> is | |
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| :<math>\bold{S} = \frac{1}{\mu_{0}} \bold{E} \times \bold{B} </math>
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| and the [[Maxwell stress tensor]] in [[joule]]·[[meter]]<sup>−3</sup> is given by | |
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| :<math>\sigma_{ij} = \epsilon_{0}E_{i}E_{j} + \frac{1}{\mu_{0}}B_{i}B_{j} -
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| \left(\frac12\epsilon_{0}E^2 + \frac{1}{2\mu_{0}}B^2\right)\delta_{ij} \,.</math>
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| The electromagnetic field tensor ''F'' constructs the electromagnetic stress-energy tensor ''T'' by the equation:
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| :<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>
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| where η is the [[Minkowski metric]] tensor. Notice that we use the fact that
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| :<math>\epsilon_{0} \mu_{0} c^2 = 1\,</math>
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| which is predicted by Maxwell's equations.
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| == Maxwell's equations in vacuum ==
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| {{Main|Maxwell's equations}}
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| In a vacuum (or for the microscopic equations, not including macroscopic material descriptions) Maxwell's equations can be written as two tensor equations.
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| 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>
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| {{Equation box 1
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| |indent=:
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| |title='''[[Gauss's Law|Gauss]]-[[Ampère's circuital law|Ampère]] law''' ''(vacuum)''
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| |equation=<math>\partial_{\alpha}F^{\alpha\beta} = \mu_{0} J^{\beta} </math>
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| |cellpadding
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| while the homogeneous equations - [[Faraday's law of induction]] and [[Gauss's law for magnetism]] combine to form:
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| {{Equation box 1
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| |indent=:
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| |title='''[[Gauss's law for magnetism|Gauss]]-[[Faraday's law of induction|Faraday]] law''' ''(vacuum)''
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| |equation=<math>\partial_{\alpha}(\tfrac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}) = 0 </math>
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| |cellpadding
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| where ''F''<sup>αβ</sup> is the [[electromagnetic tensor]], ''J''<sup>α</sup> is the [[4-current]], ε<sup>αβγδ</sup> is the [[Levi-Civita symbol]], and the indices behave according to the [[Einstein summation convention]].
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| The first tensor equation corresponds to four scalar equations, one for each value of β. The second tensor equation actually corresponds to 4<sup>3</sup> = 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.
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| 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:
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| :<math> F_{[\alpha \beta , \gamma]} =0 </math>
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| In the absence of sources, Maxwell's equations reduce to:
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| :<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>
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| which is an [[electromagnetic wave equation]] in the field strength tensor.
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| ===Maxwell's equations in the Lorenz gauge===
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| <!--Lorenz is the correct name (not Lorentz). Ludvig Lorenz ≠ Hendrik Lorentz -->
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| {{Main|Lorenz gauge condition}}
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| 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:
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| :<math>\partial_{\alpha} A^{\alpha} = \partial^{\alpha} A_{\alpha}=0 \,.</math>
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| In the Lorenz gauge, the microscopic Maxwell's equations can be written as:
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| :<math>\Box A^{\sigma} = \mu_{0} \, J^{\sigma}\,</math>
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| ==Lorentz force==
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| {{Main|Lorentz force}}
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| ===Charged particle===
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| [[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.]]
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| 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>
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| Expressed in terms of [[coordinate time]] (not proper time) ''t'' in [[second]]s, it is:
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| :<math> { d p_{\alpha} \over { d t } } = q \, F_{\alpha \beta} \, \frac{d x^\beta}{d t} \,</math>
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| where ''p''<sub>α</sub> is the [[four-momentum]] (see above), ''q'' is the [[Electric charge|charge]] (in [[coloumb]]s), and ''x''<sup>β</sup> is the position in [[meter]]s.
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| In the co-moving reference frame, this yields the so-called 4-force
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| :<math> \frac{d p_{\alpha}}{d \tau} \, = q \, F_{\alpha \beta} \, u^\beta </math>
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| where ''u''<sup>β</sup> 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.
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| ===Charge continuum===
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| [[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.]]
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| {{see also|continuum mechanics}}
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| In a continuous medium, the 3D ''density of force'' combines with the ''density of power'' to form a covariant 4-vector, ''f''<sub>μ</sub>. 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
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| :<math> - \bold{f} = - (\rho \bold{E} + \bold{J} \times \bold{B})\,</math>.
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| In manifestly covariant notation it becomes:
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| :<math>f_{\alpha} = F_{\alpha\beta}J^{\beta} .\!</math>
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| The relationship between Lorentz force and electromagnetic stress-energy tensor is
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| :<math>f^{\alpha} = - {T^{\alpha\beta}}_{,\beta} \equiv - \frac{\partial T^{\alpha\beta}}{\partial x^\beta}.</math>
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| ==Conservation laws==
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| ===Electric charge===
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| The [[continuity equation]]:
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| :<math>{J^{\alpha}}_{,\alpha} \, \stackrel{\mathrm{def}}{=} \, \partial_{\alpha} J^{\alpha} \, = \, 0 \,.</math>
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| expresses [[charge conservation]].
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| ===Electromagnetic energy-momentum===
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| 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
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| :<math>{T^{\alpha\beta}}_{,\beta} + F^{\alpha\beta} J_{\beta} = 0</math>
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| or
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| :<math>\eta_{\alpha \nu} { T^{\nu \beta } }_{,\beta} + F_{\alpha \beta} J^{\beta} = 0,</math>
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| which expresses the conservation of linear momentum and energy by electromagnetic interactions.
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| ==Covariant objects in matter==
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| ===Free and bound 4-currents===
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| 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;
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| :<math>J^{\nu} = {J^{\nu}}_{\text{free}} + {J^{\nu}}_{\text{bound}} \,,</math>
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| where
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| :<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>
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| :<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>
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| [[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]]<sup>−1</sup>) and the [[magnetic field|magnetic intensity]] '''H''' (in [[ampere]]·[[meter]]<sup>−1</sup>):
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| :<math>\bold{D} = \epsilon_0 \bold{E} + \bold{P} \,</math>
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| :<math>\bold{H} = \frac{1}{\mu_{0}} \bold{B} - \bold{M} \,.</math>
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| where '''M''' is the [[magnetization]] (in [[ampere]]·[[meter]]<sup>−2</sup>) and '''P''' the [[electric polarization]] (in [[coulomb]]·[[meter]]<sup>−2</sup>).
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| ===Magnetization-polarization tensor===
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| The bound current is derived from the '''P''' and '''M''' fields which form an antisymmetric contravariant magnetization-polarization tensor (in [[ampere]]·[[meter]]<sup>2</sup>)<ref name=Vanderlinde />
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| :<math>
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| \mathcal{M}^{\mu \nu} =
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| \begin{pmatrix}
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| 0 & P_xc & P_yc & P_zc \\
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| - P_xc & 0 & - M_z & M_y \\
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| - P_yc & M_z & 0 & - M_x \\
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| - P_zc & - M_y & M_x & 0
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| \end{pmatrix},
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| </math>
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| which determines the bound current
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| :<math>{J^{\nu}}_{\text{bound}}=\partial_{\mu} \mathcal{M}^{\mu \nu} \,.</math>
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| ===Electric displacement tensor===
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| If this is combined with ''F''<sup>μν</sup> we get the antisymmetric contravariant electromagnetic displacement tensor (in [[ampere]]·[[meter]]<sup>−1</sup>) which combines the '''D''' and '''H''' fields as follows:
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| :<math>
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| \mathcal{D}^{\mu \nu} =
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| \begin{pmatrix}
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| 0 & - D_xc & - D_yc & - D_zc \\
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| D_xc & 0 & - H_z & H_y \\
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| D_yc & H_z & 0 & - H_x \\
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| D_zc & - H_y & H_x & 0
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| \end{pmatrix}.
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| </math>
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| The three field tensors are related by:
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| :<math>\mathcal{D}^{\mu \nu} = \frac{1}{\mu_{0}} F^{\mu \nu} - \mathcal{M}^{\mu \nu} \,</math>
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| which is equivalent to the definitions of the '''D''' and '''H''' fields given above.
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| ==Maxwell's equations in matter==
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| The result is that [[Ampère's circuital law|Ampère's law]],
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| :<math>\bold{\nabla} \times \bold{H} = \bold{J}_{\text{free}} + \frac{\partial \bold{D}} {\partial t}</math>,
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| and [[Gauss's law]],
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| :<math>\bold{\nabla} \cdot \bold{D} = \rho_{\text{free}}</math>,
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| combine into one equation:
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| {{Equation box 1
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| |indent=:
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| |title='''[[Gauss's Law|Gauss]]-[[Ampère's circuital law|Ampère]] law''' ''(matter)''
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| |equation=<math>{J^{\nu}}_{\text{free}} = \partial_{\mu} \mathcal{D}^{\mu \nu} </math>
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| |cellpadding
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| |border
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| |border colour = #0073CF
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| |background colour=#F5FFFA}}
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| The bound current and free current as defined above are automatically and separately conserved
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| :<math>\partial_{\nu} {J^{\nu}}_{\text{bound}} = 0 \,</math>
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| :<math>\partial_{\nu} {J^{\nu}}_{\text{free}} = 0 \,.</math>
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| ===Constitutive equations===
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| {{main|Constitutive equation}}
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| ====Vacuum====
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| In a vacuum, the constitutive relations between the field tensor and displacement tensor are:
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| :<math>\mu_0 \mathcal{D}^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha \beta} \eta^{\beta \nu} \,.</math>
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| Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define ''F''<sup>μν</sup> by
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| :<math>F^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha \beta} \eta^{\beta \nu} \,</math>
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| the constitutive equations may, in a ''vacuum'', be combined with Gauss-Ampère law to get:
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| :<math>\partial_\beta F^{\alpha \beta} = \mu_0 J^{\alpha}. \!</math>
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| The electromagnetic stress-energy tensor in terms of the displacement is:
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| :<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>
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| 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.
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| ====Matter====
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| Thus we have reduced the problem of modeling the current, ''J''<sup>ν</sup> to two (hopefully) easier problems — 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
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| :<math>\bold{J}_{\text{free}} = \sigma \bold{E} \,</math>
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| :<math>\bold{P} = \epsilon_0 \chi_e \bold{E} \,</math>
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| :<math>\bold{M} = \chi_m \bold{H} \,</math>
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| 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]].
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| 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>
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| :<math>\mathcal{D}^{\mu\nu}u_\nu= c^2\epsilon F^{\mu\nu} u_\nu</math>
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| :<math>{\star\mathcal{D}^{\mu\nu}}u_\nu= \frac{1}{\mu}{\star F^{\mu\nu}} u_\nu</math>
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| 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]].
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| ==Lagrangian for classical electrodynamics==
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| ===Vacuum===
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| The [[Lagrangian]] (Lagrangian density) for classical electrodynamics (in [[joule]]·[[meter]]<sup>−3</sup>) is
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| :<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>
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| 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.
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| The [[Euler-Lagrange equation]] for the electromagnetic Lagrangian density <math> \mathcal{L}(A_{\alpha},\partial_{\beta}A_{\alpha})\,</math> can be stated as follows:
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| :<math> \partial_{\beta}\left[\frac{\partial \mathcal{L}}{\partial (\partial_{\beta}A_{\alpha})}\right] - \frac{\partial \mathcal{L}}{\partial A_{\alpha}}=0 \,.</math>
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| Noting
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| :<math>F_{\mu \nu}=\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}\,</math>,
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| the expression inside the square bracket is
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| :<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})} \\
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| & = - \ \frac{1}{4 \mu_0}\ \eta^{\mu\lambda}\eta^{\nu\sigma}
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| \left(F_{\lambda\sigma}(\delta^{\beta}_{\mu}\delta^{\alpha}_{\nu} - \delta^{\beta}_{\nu}\delta^{\alpha}_{\mu})
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| +F_{\mu\nu}(\delta^{\beta}_{\lambda}\delta^{\alpha}_{\sigma} - \delta^{\beta}_{\sigma}\delta^{\alpha}_{\lambda})
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| \right) \\
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| & = - \ \frac{F^{\beta\alpha}}{\mu_0}\,.
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| \end{align}</math>
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| The second term is
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| :<math>\frac{\partial \mathcal{L}}{\partial A_{\alpha}}= - J^{\alpha} \,.</math>
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| Therefore, the electromagnetic field's equations of motion are
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| :<math> \frac{\partial F^{\beta\alpha}}{\partial x^{\beta}}=\mu_0 J^{\alpha} \,.</math>
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| which is one of the Maxwell equations above.
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| ===Matter===
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| Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows:
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| :<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>
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| Using Euler-Lagrange equation, the equations of motion for <math> \mathcal{D}^{\mu\nu}</math> can be derived.
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| The equivalent expression in non-relativistic vector notation is
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| :<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>
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| ==See also==
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| * [[Relativistic electromagnetism]]
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| * [[Electromagnetic wave equation]]
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| * [[Liénard–Wiechert potential]] for a charge in arbitrary motion
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| * [[Nonhomogeneous electromagnetic wave equation]]
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| * [[Moving magnet and conductor problem]]
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| * [[Electromagnetic tensor]]
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| * [[Proca action]]
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| * [[Stueckelberg action]]
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| * [[Quantum electrodynamics]]
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| * [[Wheeler-Feynman absorber theory]]
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| ==Notes and references==
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| <references/>
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| ==Further reading==
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| *{{cite book | author=Einstein, A. | title=Relativity: The Special and General Theory | location= New York | publisher=Crown| year=1961 | isbn=0-517-02961-8}}
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| *{{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}}
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| *{{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}}
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| *{{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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| {{Physics-footer}}
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| [[Category:Concepts in physics]]
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| [[Category:Electromagnetism]]
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| [[Category:Special relativity]]
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