Covariant formulation of classical electromagnetism

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

• four-displacement: $${\displaystyle x^{\alpha }=(ct,\mathbf {x} )=(ct,x,y,z)\,.}$$
• Four-velocity: $${\displaystyle u^{\alpha }=\gamma (c,\mathbf {u} ),}$$ where γ(u) is the Lorentz factor at the 3-velocity u.
• Four-momentum: $${\displaystyle p^{\alpha }=(E/c,\mathbf {p} )=m_{0}u^{\alpha }}$$ where ${\displaystyle \mathbf {p} }$ is 3-momentum, ${\displaystyle E}$ is the total energy, and ${\displaystyle m_{0}}$ is rest mass.
• Four-gradient: $${\displaystyle \partial ^{\nu }={\frac {\partial }{\partial x_{\nu }}}=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},-\mathbf {\nabla } \right)\,,}$$
• The d'Alembertian operator is denoted ${\displaystyle {\partial }^{2}}$, $${\displaystyle \partial ^{2}=\partial ^{\nu }\partial _{\nu }={\frac {1}{c^{2}}}{\partial ^{2} \over \partial t^{2}}-\nabla ^{2}.}$$

The signs in the following tensor analysis depend on the convention used for the metric tensor. The convention used here is (+ − − −), corresponding to the Minkowski metric tensor: $${\displaystyle \eta ^{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}}$$

The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities.^[1]$${\displaystyle F_{\alpha \beta }={\begin{pmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}}$$ and the result of raising its indices is $${\displaystyle F^{\mu \nu }\mathrel {\stackrel {\mathrm {def} }{=}} \eta ^{\mu \alpha }\,F_{\alpha \beta }\,\eta ^{\beta \nu }={\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}\,,}$$ where E is the electric field, B the magnetic field, and c the speed of light.

The four-current is the contravariant four-vector which combines electric charge densityρ and electric current densityj: $${\displaystyle J^{\alpha }=(c\rho ,\mathbf {j} )\,.}$$

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

The differential of the electromagnetic potential is $${\displaystyle F_{\alpha \beta }=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }\,.}$$

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

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: $${\displaystyle T^{\alpha \beta }={\begin{pmatrix}\varepsilon _{0}E^{2}/2+B^{2}/2\mu _{0}&S_{x}/c&S_{y}/c&S_{z}/c\\S_{x}/c&-\sigma _{xx}&-\sigma _{xy}&-\sigma _{xz}\\S_{y}/c&-\sigma _{yx}&-\sigma _{yy}&-\sigma _{yz}\\S_{z}/c&-\sigma _{zx}&-\sigma _{zy}&-\sigma _{zz}\end{pmatrix}}\,,}$$ where ${\displaystyle \varepsilon _{0}}$ is the electric permittivity of vacuum, μ₀ is the magnetic permeability of vacuum, the Poynting vector is $${\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} }$$ and the Maxwell stress tensor is given by $${\displaystyle \sigma _{ij}=\varepsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-\left({\frac {1}{2}}\varepsilon _{0}E^{2}+{\frac {1}{2\mu _{0}}}B^{2}\right)\delta _{ij}\,.}$$

The electromagnetic field tensor F constructs the electromagnetic stress–energy tensor T by the equation:^[2]$${\displaystyle T^{\alpha \beta }={\frac {1}{\mu _{0}}}\left(\eta ^{\alpha \nu }F_{\nu \gamma }F^{\beta \gamma }-{\frac {1}{4}}\eta ^{\alpha \beta }F_{\gamma \nu }F^{\gamma \nu }\right)}$$ where η is the Minkowski metric tensor (with signature (+ − − −)). Notice that we use the fact that $${\displaystyle \varepsilon _{0}\mu _{0}c^{2}=1\,,}$$ which is predicted by Maxwell's equations.

Another way to covariant expression for the electromagnetic stress-energy tensor which may be simpler since it does not involve covariant and contravariant indices is this one: $${\displaystyle T=-{\frac {1}{\mu _{0}}}(F*\eta *F'-{\frac {1}{4}}\operatorname {tr} (F*\eta *F'*\eta ))}$$ Where F' is the transposed electromagnetic tensor or equivalently -F and the asterisk denotes matrix multiplication.

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: $${\displaystyle \partial _{\alpha }A^{\alpha }=\partial ^{\alpha }A_{\alpha }=0\,.}$$

In the Lorenz gauge, the microscopic Maxwell's equations can be written as: $${\displaystyle {\partial }^{2}A^{\sigma }=\mu _{0}\,J^{\sigma }\,.}$$

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.^[4]

Expressed in terms of coordinate timet, it is: $${\displaystyle {dp_{\alpha } \over {dt}}=q\,F_{\alpha \beta }\,{\frac {dx^{\beta }}{dt}},}$$ 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 $${\displaystyle {\frac {dp_{\alpha }}{d\tau }}\,=q\,F_{\alpha \beta }\,u^{\beta },}$$ where u^β is the four-velocity, and τ is the particle's proper time, which is related to coordinate time by dt = γdτ.

The density of force due to electromagnetism, whose spatial part is the Lorentz force, is given by $${\displaystyle f_{\alpha }=F_{\alpha \beta }J^{\beta }.}$$ and is related to the electromagnetic stress–energy tensor by $${\displaystyle f^{\alpha }=-{T^{\alpha \beta }}_{,\beta }\equiv -{\frac {\partial T^{\alpha \beta }}{\partial x^{\beta }}}.}$$

The continuity equation: $${\displaystyle {J^{\beta }}_{,\beta }\mathrel {\overset {\text{def}}{\mathop {=} }} \partial _{\beta }J^{\beta }=\partial _{\beta }\partial _{\alpha }F^{\alpha \beta }/\mu _{0}=0.}$$ 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 $${\displaystyle {T^{\alpha \beta }}_{,\beta }+F^{\alpha \beta }J_{\beta }=0}$$ or $${\displaystyle \eta _{\alpha \nu }{T^{\nu \beta }}_{,\beta }+F_{\alpha \beta }J^{\beta }=0,}$$ 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; $${\displaystyle J^{\nu }={J^{\nu }}_{\text{free}}+{J^{\nu }}_{\text{bound}}\,,}$$ where $${\displaystyle {\begin{aligned}{J^{\nu }}_{\text{free}}&={\begin{pmatrix}c\rho _{\text{free}},&\mathbf {J} _{\text{free}}\end{pmatrix}}\\[0.5ex]&={\begin{pmatrix}c\nabla \cdot \mathbf {D} ,&-{\frac {\partial \mathbf {D} }{\partial t}}+\nabla \times \mathbf {H} \end{pmatrix}}\,,\\[1.5ex]{J^{\nu }}_{\text{bound}}&={\begin{pmatrix}c\rho _{\text{bound}},&\mathbf {J} _{\text{bound}}\end{pmatrix}}\\[0.5ex]&={\begin{pmatrix}-c\nabla \cdot \mathbf {P} ,&{\frac {\partial \mathbf {P} }{\partial t}}+\nabla \times \mathbf {M} \end{pmatrix}}\,.\end{aligned}}}$$

Maxwell's macroscopic equations have been used, in addition the definitions of the electric displacementD and the magnetic intensityH: $${\displaystyle {\begin{aligned}\mathbf {D} &=\varepsilon _{0}\mathbf {E} +\mathbf {P} ,\\\mathbf {H} &={\frac {1}{\mu _{0}}}\mathbf {B} -\mathbf {M} \,.\end{aligned}}}$$ 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 ^[1][5][6][7]$${\displaystyle {\mathcal {M}}^{\mu \nu }={\begin{pmatrix}0&P_{x}c&P_{y}c&P_{z}c\\-P_{x}c&0&-M_{z}&M_{y}\\-P_{y}c&M_{z}&0&-M_{x}\\-P_{z}c&-M_{y}&M_{x}&0\end{pmatrix}},}$$ which determines the bound current $${\displaystyle {J^{\nu }}_{\text{bound}}=\partial _{\mu }{\mathcal {M}}^{\mu \nu }\,.}$$

If this is combined with F^μν we get the antisymmetric contravariant electromagnetic displacement tensor which combines the D and H fields as follows: $${\displaystyle {\mathcal {D}}^{\mu \nu }={\begin{pmatrix}0&-D_{x}c&-D_{y}c&-D_{z}c\\D_{x}c&0&-H_{z}&H_{y}\\D_{y}c&H_{z}&0&-H_{x}\\D_{z}c&-H_{y}&H_{x}&0\end{pmatrix}}.}$$

The three field tensors are related by: $${\displaystyle {\mathcal {D}}^{\mu \nu }={\frac {1}{\mu _{0}}}F^{\mu \nu }-{\mathcal {M}}^{\mu \nu }}$$ which is equivalent to the definitions of the D and H fields given above.

The Lagrangian density for classical electrodynamics is composed by two components: a field component and a source component: $${\displaystyle {\mathcal {L}}\,=\,{\mathcal {L}}_{\text{field}}+{\mathcal {L}}_{\text{int}}=-{\frac {1}{4\mu _{0}}}F^{\alpha \beta }F_{\alpha \beta }-A_{\alpha }J^{\alpha }\,.}$$

The source-free term ${\displaystyle {\mathcal {L}}_{\text{field}}=-{\frac {1}{4\mu _{0}}}F^{\alpha \beta }F_{\alpha \beta }}$ is the Maxwell Lagrangian. In the interaction term ${\displaystyle {\mathcal {L}}_{\text{int}}=-A_{\alpha }J^{\alpha }}$, the four-current ${\displaystyle J^{\alpha }}$ 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 ${\displaystyle {\mathcal {L}}{\mathord {\left(A_{\alpha },\partial _{\beta }A_{\alpha }\right)}}}$ can be stated as follows: $${\displaystyle \partial _{\beta }\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\beta }A_{\alpha })}}\right]-{\frac {\partial {\mathcal {L}}}{\partial A_{\alpha }}}=0\,.}$$

Noting $${\displaystyle {\begin{aligned}F^{\lambda \sigma }&=F_{\mu \nu }\eta ^{\mu \lambda }\eta ^{\nu \sigma },\\F_{\mu \nu }&=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }\,\\{\partial \left(\partial _{\mu }A_{\nu }\right) \over \partial \left(\partial _{\rho }A_{\sigma }\right)}&=\delta _{\mu }^{\rho }\delta _{\nu }^{\sigma }\end{aligned}}}$$ the expression inside the square bracket is $${\displaystyle {\begin{aligned}{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\beta }A_{\alpha })}}&=-\ {\frac {1}{4\mu _{0}}}\ {\frac {\partial \left(F_{\mu \nu }\eta ^{\mu \lambda }\eta ^{\nu \sigma }F_{\lambda \sigma }\right)}{\partial \left(\partial _{\beta }A_{\alpha }\right)}}\\&=-\ {\frac {1}{4\mu _{0}}}\ \eta ^{\mu \lambda }\eta ^{\nu \sigma }\left(F_{\lambda \sigma }\left(\delta _{\mu }^{\beta }\delta _{\nu }^{\alpha }-\delta _{\nu }^{\beta }\delta _{\mu }^{\alpha }\right)+F_{\mu \nu }\left(\delta _{\lambda }^{\beta }\delta _{\sigma }^{\alpha }-\delta _{\sigma }^{\beta }\delta _{\lambda }^{\alpha }\right)\right)\\&=-\ {\frac {F^{\beta \alpha }}{\mu _{0}}}\,.\end{aligned}}}$$

The second term is $${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial A_{\alpha }}}=-J^{\alpha }\,.}$$

Therefore, the electromagnetic field's equations of motion are $${\displaystyle {\frac {\partial F^{\beta \alpha }}{\partial x^{\beta }}}=\mu _{0}J^{\alpha }\,.}$$ 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: $${\displaystyle {\mathcal {L}}\,=\,-{\frac {1}{4\mu _{0}}}F^{\alpha \beta }F_{\alpha \beta }-A_{\alpha }J_{\text{free}}^{\alpha }+{\frac {1}{2}}F_{\alpha \beta }{\mathcal {M}}^{\alpha \beta }\,.}$$

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

The equivalent expression in vector notation is: $${\displaystyle {\mathcal {L}}\,=\,{\frac {1}{2}}\left(\varepsilon _{0}E^{2}-{\frac {1}{\mu _{0}}}B^{2}\right)-\phi \,\rho _{\text{free}}+\mathbf {A} \cdot \mathbf {J} _{\text{free}}+\mathbf {E} \cdot \mathbf {P} +\mathbf {B} \cdot \mathbf {M} \,.}$$