Covariant formulation of classical electromagnetism#Maxwell's equations in vacuum

{{Main|Lorentz covariance}}

Lorentz tensors of the following kinds may be used in this article to describe bodies or particles:

• four-displacement: $$x^\backslash alpha\; =\; (ct,\; \backslash mathbf\{x\})\; =\; (ct,\; x,\; y,\; z)\; \backslash ,.$$
• Four-velocity: $$u^\backslash alpha\; =\; \backslash gamma(c,\backslash mathbf\{u\})\; ,$$ where γ(u) is the Lorentz factor at the 3-velocity u.
• Four-momentum: $$p^\backslash alpha\; =\; (\; E/c,\; \backslash mathbf\{p\})\; =\; m\_0\; u^\{\backslash alpha\}$$ where $\backslash mathbf\{p\}$ is 3-momentum, $E$ is the total energy, and $m\_0$ is rest mass.
• Four-gradient: $$\backslash partial^\{\backslash nu\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_\{\backslash nu\}\}\; =\; \backslash left(\; \backslash frac\{1\}\{c\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\},\; -\; \backslash mathbf\{\backslash nabla\}\; \backslash right)\; \backslash ,,$$
• The d'Alembertian operator is denoted $\{\backslash partial\}^2$, $$\backslash partial^2\; =\; \backslash frac\{1\}\{c^2\}\; \{\backslash partial^2\; \backslash over\; \backslash partial\; t^2\}\; -\; \backslash nabla^2.$$

The signs in the following tensor analysis depend on the convention used for the metric tensor. The convention used here is {{nowrap|(+ − − −)}}, corresponding to the Minkowski metric 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.

{{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

}}

$$F\_\{\backslash alpha\; \backslash beta\}\; =\; \backslash begin\{pmatrix\}$$

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}

and the result of raising its indices is

$$F^\{\backslash mu\; \backslash nu\}\; \backslash mathrel\{\backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\}\; \backslash eta^\{\backslash mu\; \backslash alpha\}\; \backslash ,\; F\_\{\backslash alpha\; \backslash beta\}\; \backslash ,\; \backslash eta^\{\backslash beta\; \backslash nu\}\; =\; \backslash begin\{pmatrix\}$$

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} \,,

where E is the electric field, B the magnetic field, and c the speed of light.

{{Main|Four-current}}

The four-current is the contravariant four-vector which combines electric charge density ρ and electric current density j:

$$J^\{\backslash alpha\}\; =\; (\; c\; \backslash rho,\; \backslash mathbf\{j\}\; )\; \backslash ,.$$

{{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:

$$A^\{\backslash alpha\}\; =\; \backslash left(\backslash phi/c,\; \backslash mathbf\{A\}\; \backslash right)\backslash ,.$$

The differential of the electromagnetic potential is

$$F\_\{\backslash alpha\; \backslash beta\}\; =\; \backslash partial\_\{\backslash alpha\}\; A\_\{\backslash beta\}\; -\; \backslash partial\_\{\backslash beta\}\; A\_\{\backslash alpha\}\; \backslash ,.$$

In the language of differential forms, which provides the generalisation to curved spacetimes, these are the components of a 1-form $A\; =\; A\_\{\backslash alpha\}\; dx^\{\backslash alpha\}$ and a 2-form $F\; =\; dA\; =\; \backslash frac\{1\}\{2\}\; F\_\{\backslash alpha\; \backslash beta\}\; dx^\{\backslash alpha\}\; \backslash wedge\; dx^\{\backslash beta\}$ respectively. Here, $d$ is the exterior derivative and $\backslash wedge$ the wedge product.

{{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:

$$T^\{\backslash alpha\backslash beta\}\; =\; \backslash begin\{pmatrix\}$$

\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}\,,

where $\backslash varepsilon\_0$ is the electric permittivity of vacuum, μ₀ is the magnetic permeability of vacuum, the Poynting vector is

$$\backslash mathbf\{S\}\; =\; \backslash frac\{1\}\{\backslash mu\_\{0\}\}\; \backslash mathbf\{E\}\; \backslash times\; \backslash mathbf\{B\}$$

and the Maxwell stress tensor is given by

$$\backslash sigma\_\{ij\}\; =\; \backslash varepsilon\_0\; E\_i\; E\_j\; +\; \backslash frac\{1\}\{\backslash mu\_0\}\; B\_i\; B\_j\; -\; \backslash left(\backslash frac\; 1\; 2\; \backslash varepsilon\_0\; E^2\; +\; \backslash frac\{1\}\{2\backslash mu\_0\}\; B^2\backslash right)\; \backslash delta\_\{ij\}\; \backslash ,.$$

The electromagnetic field tensor F constructs the electromagnetic stress–energy tensor T by the equation:Classical Electrodynamics, Jackson, 3rd edition, page 609

$$T^\{\backslash alpha\backslash beta\}\; =\; \backslash frac\{1\}\{\backslash mu\_\{0\}\}\; \backslash left(\; \backslash eta^\{\backslash alpha\; \backslash nu\}F\_\{\backslash nu\; \backslash gamma\}F^\{\backslash beta\; \backslash gamma\}\; -\; \backslash frac\{1\}\{4\}\; \backslash eta^\{\backslash alpha\backslash beta\}F\_\{\backslash gamma\; \backslash nu\}F^\{\backslash gamma\; \backslash nu\}\backslash right)$$

where η is the Minkowski metric tensor (with signature {{nowrap|(+ − − −)}}). Notice that we use the fact that

$$\backslash varepsilon\_\{0\}\; \backslash mu\_\{0\}\; c^2\; =\; 1\backslash ,,$$

which is predicted by Maxwell's equations.

{{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 law (with Maxwell's correction) combine into (with {{nowrap|(+ − − −)}} metric):Classical Electrodynamics by Jackson, 3rd Edition, Chapter 11 Special Theory of Relativity

{{Equation box 1

|indent=:

|title=Gauss–Ampère law

|equation=$\backslash partial\_\{\backslash alpha\}F^\{\backslash alpha\backslash beta\}\; =\; \backslash mu\_\{0\}\; J^\{\backslash beta\}$

|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 $\backslash partial^\{\backslash sigma\}\; F^\{\backslash mu\; \backslash nu\}\; +\; \backslash partial^\{\backslash mu\}\; F^\{\backslash nu\; \backslash sigma\}\; +\; \backslash partial^\{\backslash nu\}\; F^\{\backslash sigma\; \backslash mu\}\; =\; 0$, which may be written using Levi-Civita duality as:

{{Equation box 1

|indent=:

|title=Gauss–Faraday law

|equation=$\backslash partial\_\{\backslash alpha\}\backslash left(\backslash varepsilon^\{\backslash alpha\backslash beta\backslash gamma\backslash delta\}F\_\{\backslash gamma\backslash delta\}\backslash right)\; =\; 0$

|cellpadding

|border

|border colour = #50C878

|background colour = #ECFCF4

}}

where F^αβ is the electromagnetic tensor, J^α is the four-current, ε^αβγδ 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:

$$F\_\{[\backslash alpha\; \backslash beta,\; \backslash gamma]\}\; =\; 0\; .$$

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

\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 \,,

which is an electromagnetic wave equation in the field strength tensor.

{{Main|Lorenz gauge condition}}

The Lorenz gauge condition is a Lorentz-invariant gauge condition. (This can be contrasted with other 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:

$$\backslash partial\_\{\backslash alpha\}\; A^\{\backslash alpha\}\; =\; \backslash partial^\{\backslash alpha\}\; A\_\{\backslash alpha\}\; =\; 0\; \backslash ,.$$

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

$$\{\backslash partial\}^2\; A^\{\backslash sigma\}\; =\; \backslash mu\_\{0\}\; \backslash ,\; J^\{\backslash sigma\}\backslash ,.$$

{{Main|Lorentz force}}

File:Lorentz force particle.svg f on a charged particle (of charge q) in motion (instantaneous velocity v). The E field and B field vary in space and time.]]

Electromagnetic (EM) fields affect the motion of electrically charged matter: due to the Lorentz force. In this way, EM fields can be detected (with applications in particle physics, and natural occurrences such as in aurorae). In relativistic form, the Lorentz force uses the field strength tensor as follows.The assumption is made that no forces other than those originating in E and B are present, that is, no gravitational, weak or strong forces.

Expressed in terms of coordinate time t, it is:

$$\{\; d\; p\_\{\backslash alpha\}\; \backslash over\; \{\; d\; t\; \}\; \}\; =\; q\; \backslash ,\; F\_\{\backslash alpha\; \backslash beta\}\; \backslash ,\; \backslash frac\{d\; x^\backslash beta\}\{d\; t\}\; ,$$

where p_α is the four-momentum, q is the charge, and x^β is the position.

Expressed in frame-independent form, we have the four-force

$$\backslash frac\{d\; p\_\{\backslash alpha\}\}\{d\; \backslash tau\}\; \backslash ,\; =\; q\; \backslash ,\; F\_\{\backslash alpha\; \backslash beta\}\; \backslash ,\; u^\backslash beta\; ,$$

where u^β is the four-velocity, and τ is the particle's proper time, which is related to coordinate time by {{nowrap|1=dt = γdτ}}.

File:Lorentz force continuum.svg (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

$$f\_\{\backslash alpha\}\; =\; F\_\{\backslash alpha\backslash beta\}J^\{\backslash beta\}\; .$$

and is related to the electromagnetic stress–energy tensor by

$$f^\{\backslash alpha\}\; =\; -\; \{T^\{\backslash alpha\backslash beta\}\}\_\{,\backslash beta\}\; \backslash equiv\; -\; \backslash frac\{\backslash partial\; T^\{\backslash alpha\backslash beta\}\}\{\backslash partial\; x^\backslash beta\}.$$

The continuity equation:

{J^\beta}_{,\beta} \mathrel\overset{\text{def}}{\mathop{=}}

\partial_\beta J^\beta =

\partial_\beta \partial_\alpha F^{\alpha \beta}/\mu_0 =

0.

expresses charge conservation.

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

$$\{T^\{\backslash alpha\backslash beta\}\}\_\{,\backslash beta\}\; +\; F^\{\backslash alpha\backslash beta\}\; J\_\backslash beta\; =\; 0$$

or

$$\backslash eta\_\{\backslash alpha\; \backslash nu\}\; \{\; T^\{\backslash nu\; \backslash beta\; \}\; \}\_\{,\backslash beta\}\; +\; F\_\{\backslash alpha\; \backslash beta\}\; J^\backslash beta\; =\; 0,$$

which expresses the conservation of linear momentum and energy by electromagnetic interactions.

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;

$$J^\{\backslash nu\}\; =\; \{J^\{\backslash nu\}\}\_\{\backslash text\{free\}\}\; +\; \{J^\{\backslash nu\}\}\_\{\backslash text\{bound\}\}\; \backslash ,,$$

where

$$\backslash begin\{align\}$$

{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}

Maxwell's macroscopic equations have been used, in addition the definitions of the electric displacement D and the magnetic intensity H:

$$\backslash begin\{align\}$$

\mathbf{D} &= \varepsilon_0 \mathbf{E} + \mathbf{P}, \\

\mathbf{H} &= \frac{1}{\mu_0} \mathbf{B} - \mathbf{M} \,.

\end{align}

where M is the magnetization and P the electric polarization.

The bound current is derived from the P and M fields which form an antisymmetric contravariant magnetization-polarization tensor

However, the assumption that $M^\{\backslash mu\backslash nu\}$, $D^\{\backslash mu\backslash nu\}$, and even $F^\{\backslash mu\backslash nu\}$, are relativistic tensors in a polarizable medium, is without foundation. The quantity$$A^\{\backslash alpha\}\; =\; \backslash left(\backslash phi/c,\; \backslash mathbf\{A\}\; \backslash right)\backslash ,$$is not a four vector in a polarizable medium, so $$F\_\{\backslash alpha\; \backslash beta\}\; =\; \backslash partial\_\{\backslash alpha\}\; A\_\{\backslash beta\}\; -\; \backslash partial\_\{\backslash beta\}\; A\_\{\backslash alpha\}\; \backslash ,$$ does not produce a tensor.

{{citation

| last = Franklin

| first = Jerrold

| title = Can electromagnetic fields form tensors in a polarizable medium?

| url =https://www.academia.edu/104227822}}{{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

}}

$$\backslash mathcal\{M\}^\{\backslash mu\; \backslash nu\}\; =\; \backslash begin\{pmatrix\}$$

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},

which determines the bound current

$$\{J^\backslash nu\}\_\backslash text\{bound\}\; =\; \backslash partial\_\backslash mu\; \backslash mathcal\{M\}^\{\backslash mu\; \backslash nu\}\; \backslash ,.$$

If this is combined with F^μν we get the antisymmetric contravariant electromagnetic displacement tensor which combines the D and H fields as follows:

\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}.

The three field tensors are related by:

$$\backslash mathcal\{D\}^\{\backslash mu\; \backslash nu\}\; =\; \backslash frac\{1\}\{\backslash mu\_\{0\}\}\; F^\{\backslash mu\; \backslash nu\}\; -\; \backslash mathcal\{M\}^\{\backslash mu\; \backslash nu\}$$

which is equivalent to the definitions of the D and H fields given above.

The result is that Ampère's law,

$$\backslash mathbf\{\backslash nabla\}\; \backslash times\; \backslash mathbf\{H\}\; -\; \backslash frac\{\backslash partial\; \backslash mathbf\{D\}\}\; \{\backslash partial\; t\}\; =\; \backslash mathbf\{J\}\_\{\backslash text\{free\}\},$$

and Gauss's law,

$$\backslash mathbf\{\backslash nabla\}\; \backslash cdot\; \backslash mathbf\{D\}\; =\; \backslash rho\_\{\backslash text\{free\}\},$$

combine into one equation:

{{Equation box 1

|indent=:

|title=Gauss–Ampère law (matter)

|equation=$\{J^\{\backslash nu\}\}\_\{\backslash text\{free\}\}\; =\; \backslash partial\_\{\backslash mu\}\; \backslash mathcal\{D\}^\{\backslash mu\; \backslash nu\}$

|cellpadding

|border

|border colour = #0073CF

}}

The bound current and free current as defined above are automatically and separately conserved

$$\backslash begin\{align\}$$

\partial_\nu {J^\nu}_\text{bound} &= 0 \, \\

\partial_\nu {J^\nu}_\text{free} &= 0 \,.

\end{align}

{{main|Constitutive equation}}

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

$$\backslash mu\_0\; \backslash mathcal\{D\}^\{\backslash mu\; \backslash nu\}\; =\; \backslash eta^\{\backslash mu\; \backslash alpha\}\; F\_\{\backslash alpha\; \backslash beta\}\; \backslash eta^\{\backslash beta\; \backslash nu\}\; \backslash ,.$$

Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define F^μν by

$$F^\{\backslash mu\; \backslash nu\}\; =\; \backslash eta^\{\backslash mu\; \backslash alpha\}\; F\_\{\backslash alpha\; \backslash beta\}\; \backslash eta^\{\backslash beta\; \backslash nu\}\; ,$$

the constitutive equations may, in vacuum, be combined with the Gauss–Ampère law to get:

$$\backslash partial\_\backslash beta\; F^\{\backslash alpha\; \backslash beta\}\; =\; \backslash mu\_0\; J^\{\backslash alpha\}.$$

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

$$T\_\backslash alpha\{\}^\backslash pi\; =\; F\_\{\backslash alpha\backslash beta\}\; \backslash mathcal\{D\}^\{\backslash pi\backslash beta\}\; -\; \backslash frac\{1\}\{4\}\; \backslash delta\_\backslash alpha^\backslash pi\; F\_\{\backslash mu\backslash nu\}\; \backslash mathcal\{D\}^\{\backslash mu\backslash nu\}\; ,$$

where δ_α^π 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.

Thus we have reduced the problem of modeling the current, J^ν to two (hopefully) easier problems — modeling the free current, J^ν_free and modeling the magnetization and polarization, $\backslash mathcal\{M\}^\{\backslash mu\backslash nu\}$. For example, in the simplest materials at low frequencies, one has

$$\backslash begin\{align\}$$

\mathbf{J}_\text{free} &= \sigma \mathbf{E} \, \\

\mathbf{P} &= \varepsilon_0 \chi_e \mathbf{E} \, \\

\mathbf{M} &= \chi_m \mathbf{H} \,

\end{align}

where one is in the instantaneously comoving inertial frame of the material, σ is its electrical conductivity, χ_e is its electric susceptibility, and χ_m is its magnetic susceptibility.

The constitutive relations between the $\backslash mathcal\{D\}$ and F tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are:

$$\backslash begin\{align\}$$

\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}

where u is the four-velocity of material, ε and μ are respectively the proper permittivity and permeability of the material (i.e. in rest frame of material), $\backslash star$ and denotes the Hodge star operator.

The Lagrangian density for classical electrodynamics is composed by two components: a field component and a source component:

$$\backslash mathcal\{L\}\; \backslash ,\; =\; \backslash ,\; \backslash mathcal\{L\}\_\backslash text\{field\}\; +\; \backslash mathcal\{L\}\_\backslash text\{int\}\; =\; -\; \backslash frac\{1\}\{4\; \backslash mu\_0\}\; F^\{\backslash alpha\; \backslash beta\}\; F\_\{\backslash alpha\; \backslash beta\}\; -\; A\_\{\backslash alpha\}\; J^\{\backslash alpha\}\; \backslash ,.$$

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 Lagrange equations for the electromagnetic lagrangian density $\backslash mathcal\{L\}\backslash mathord\backslash left(A\_\{\backslash alpha\},\; \backslash partial\_\{\backslash beta\}\; A\_\{\backslash alpha\}\backslash right)$ can be stated as follows:

$$\backslash partial\_\{\backslash beta\}\backslash left[\backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; (\backslash partial\_\{\backslash beta\}A\_\{\backslash alpha\})\}\backslash right]\; -\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; A\_\{\backslash alpha\}\}=0\; \backslash ,.$$

Noting

$$\backslash begin\{align\}$$

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}

the expression inside the square bracket is

$$\backslash begin\{align\}$$

\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}

The second term is

$$\backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; A\_\{\backslash alpha\}\}\; =\; -\; J^\{\backslash alpha\}\; \backslash ,.$$

Therefore, the electromagnetic field's equations of motion are

$$\backslash frac\{\backslash partial\; F^\{\backslash beta\backslash alpha\}\}\{\backslash partial\; x^\{\backslash beta\}\}\; =\; \backslash mu\_0\; J^\{\backslash alpha\}\; \backslash ,.$$

which is the Gauss–Ampère equation above.

Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows:

$$\backslash mathcal\{L\}\; \backslash ,\; =\; \backslash ,\; -\; \backslash frac\{1\}\{4\; \backslash mu\_0\}\; F^\{\backslash alpha\; \backslash beta\}\; F\_\{\backslash alpha\; \backslash beta\}\; -\; A\_\{\backslash alpha\}\; J^\{\backslash alpha\}\_\{\backslash text\{free\}\}\; +\; \backslash frac\; 1\; 2\; F\_\{\backslash alpha\; \backslash beta\}\; \backslash mathcal\{M\}^\{\backslash alpha\; \backslash beta\}\; \backslash ,.$$

Using Lagrange equation, the equations of motion for $\backslash mathcal\{D\}^\{\backslash mu\backslash nu\}$ can be derived.

The equivalent expression in vector notation is:

$$\backslash mathcal\{L\}\; \backslash ,\; =\; \backslash ,\; \backslash frac\; 1\; 2\; \backslash left(\backslash varepsilon\_\{0\}\; E^2\; -\; \backslash frac\{1\}\{\backslash mu\_\{0\}\}\; B^2\backslash right)\; -\; \backslash phi\; \backslash ,\; \backslash rho\_\{\backslash text\{free\}\}\; +\; \backslash mathbf\{A\}\; \backslash cdot\; \backslash mathbf\{J\}\_\{\backslash text\{free\}\}\; +\; \backslash mathbf\{E\}\; \backslash cdot\; \backslash mathbf\{P\}\; +\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathbf\{M\}\; \backslash ,.$$

• 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

{{notelist}}

{{reflist|2}}

• [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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