Covariant formulation of classical electromagnetism

Maxwell's equations in vacuum

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 (+ − − −) metric):

Gauss–Ampère law

${\displaystyle {\mathrm{\partial }}_{\alpha }{F}^{\alpha \beta }={\mu }_0{J}^{\beta }}$

The homogeneous equations – Faraday's law of induction and Gauss's law for magnetism combine to form ${\displaystyle {\mathrm{\partial }}^{\sigma }{F}^{\mu \nu }+{\mathrm{\partial }}^{\mu }{F}^{\nu \sigma }+{\mathrm{\partial }}^{\nu }{F}^{\sigma \mu }=0}$, which may be written using Levi-Civita duality as:

Gauss–Faraday law

${\displaystyle {\mathrm{\partial }}_{\alpha }\left({\epsilon }^{\alpha \beta \gamma \delta }{F}_{\gamma \delta }\right)=0}$

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: $${\displaystyle F_{[\alpha \beta ,\gamma ]}=0.}$$

In the absence of sources, Maxwell's equations reduce to: $${\displaystyle {\mathrm{\partial }}^{\nu }{\mathrm{\partial }}_{\nu }{F}^{\alpha \beta }\stackrel{\text{def}}{=}{\mathrm{\partial }}^2{F}^{\alpha \beta }\stackrel{\text{def}}{=}\frac{1}{c^2}\frac{{\mathrm{\partial }}^2{F}^{\alpha \beta }}{{\mathrm{\partial }t}^2}-{\mathrm{\nabla }}^2{F}^{\alpha \beta }=0,}$$ which is an electromagnetic wave equation in the field strength tensor.

Maxwell's equations in the Lorenz gauge

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

In the Lorenz gauge, the microscopic Maxwell's equations can be written as: $${\displaystyle {\mathrm{\partial }}^2{A}^{\sigma }={\mu }_0{J}^{\sigma }.}$$

Lorentz force

Charged particle

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.

Expressed in terms of coordinate time t, it is: $${\displaystyle \frac{d{p}_{\alpha }}{dt}=q{F}_{\alpha \beta }\frac{d{x}^{\beta }}{dt},}$$ 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 $${\displaystyle \frac{d{p}_{\alpha }}{d\tau }=q{F}_{\alpha \beta }{u}^{\beta },}$$ where u^β is the four-velocity, and τ is the particle's proper time, which is related to coordinate time by dt = γdτ.

Charge continuum

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

Conservation laws

Electric charge

The continuity equation: $${\displaystyle {J^{\beta }}_{,\beta }\stackrel{\text{def}}{=}{\mathrm{\partial }}_{\beta }{J}^{\beta }={\mathrm{\partial }}_{\beta }{\mathrm{\partial }}_{\alpha }{F}^{\alpha \beta }/{\mu }_0=0.}$$ 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 $${\displaystyle {T^{\alpha \beta }}_{,\beta }+F^{\alpha \beta }{J}_{\beta }=0}$$ 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.

Covariant objects in matter

Free and bound four-currents

In order to solve the equations of electromagnetism given here, it is necessary to add information about how to calculate the electric current, J^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 $${\displaystyle \begin{array}{cc}{J^{\nu }}_{\text{free}}& =\left(\begin{array}{cc}c{\rho }_{\text{free}},& {\mathbf{J}}_{\text{free}}\end{array}\right)\\ & =\left(\begin{array}{cc}c\mathrm{\nabla }\cdot \mathbf{D},& -\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}+\mathrm{\nabla }\times \mathbf{H}\end{array}\right),\\ {J^{\nu }}_{\text{bound}}& =\left(\begin{array}{cc}c{\rho }_{\text{bound}},& {\mathbf{J}}_{\text{bound}}\end{array}\right)\\ & =\left(\begin{array}{cc}-c\mathrm{\nabla }\cdot \mathbf{P},& \frac{\mathrm{\partial }\mathbf{P}}{\mathrm{\partial }t}+\mathrm{\nabla }\times \mathbf{M}\end{array}\right).\end{array}}$$

Maxwell's macroscopic equations have been used, in addition the definitions of the electric displacement D and the magnetic intensity H: $${\displaystyle \begin{array}{cc}\mathbf{D}& ={\epsilon }_0\mathbf{E}+\mathbf{P},\\ \mathbf{H}& =\frac{1}{{\mu }_0}\mathbf{B}-\mathbf{M}.\end{array}}$$ where M is the magnetization and P the electric polarization.

Magnetization–polarization tensor

The bound current is derived from the P and M fields which form an antisymmetric contravariant magnetization-polarization tensor $${\displaystyle {\mathcal{M}}^{\mu \nu }=\left(\begin{array}{cccc}0& P_{x}c& P_{y}c& P_{z}c\\ -P_{x}c& 0& -M_z& M_y\\ -P_{y}c& M_z& 0& -M_x\\ -P_{z}c& -M_y& M_x& 0\end{array}\right),}$$ which determines the bound current $${\displaystyle {J^{\nu }}_{\text{bound}}={\mathrm{\partial }}_{\mu }{\mathcal{M}}^{\mu \nu }.}$$

Electric displacement tensor

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

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.

Maxwell's equations in matter

The result is that Ampère's law, $${\displaystyle \mathrm{\nabla }\times \mathbf{H}-\frac{\mathrm{\partial }\mathbf{D}}{\mathrm{\partial }t}={\mathbf{J}}_{\text{free}},}$$ and Gauss's law, $${\displaystyle \mathrm{\nabla }\cdot \mathbf{D}={\rho }_{\text{free}},}$$ combine into one equation:

Gauss–Ampère law(matter)

${\displaystyle {J^{\nu }}_{\text{free}}={\mathrm{\partial }}_{\mu }{\mathcal{D}}^{\mu \nu }}$

The bound current and free current as defined above are automatically and separately conserved $${\displaystyle \begin{array}{cc}{\mathrm{\partial }}_{\nu }{J^{\nu }}_{\text{bound}}& =0\\ {\mathrm{\partial }}_{\nu }{J^{\nu }}_{\text{free}}& =0.\end{array}}$$

Constitutive equations

Vacuum

In vacuum, the constitutive relations between the field tensor and displacement tensor are: $${\displaystyle {\mu }_0{\mathcal{D}}^{\mu \nu }={\eta }^{\mu \alpha }{F}_{\alpha \beta }{\eta }^{\beta \nu }.}$$

Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define F^μν by $${\displaystyle F^{\mu \nu }={\eta }^{\mu \alpha }{F}_{\alpha \beta }{\eta }^{\beta \nu },}$$ the constitutive equations may, in vacuum, be combined with the Gauss–Ampère law to get: $${\displaystyle {\mathrm{\partial }}_{\beta }{F}^{\alpha \beta }={\mu }_0{J}^{\alpha }.}$$

The electromagnetic stress–energy tensor in terms of the displacement is: $${\displaystyle T_{\alpha }{}^{\pi }=F_{\alpha \beta }{\mathcal{D}}^{\pi \beta }-\frac{1}{4}{\delta }_{\alpha }^{\pi }{F}_{\mu \nu }{\mathcal{D}}^{\mu \nu },}$$ 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.

Linear, nondispersive matter

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, ${\displaystyle {\mathcal{M}}^{\mu \nu }}$. For example, in the simplest materials at low frequencies, one has $${\displaystyle \begin{array}{cc}{\mathbf{J}}_{\text{free}}& =\sigma \mathbf{E}\\ \mathbf{P}& ={\epsilon }_0{\chi }_e\mathbf{E}\\ \mathbf{M}& ={\chi }_m\mathbf{H}\end{array}}$$ 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 ${\displaystyle \mathcal{D}}$ and F tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are: $${\displaystyle \begin{array}{cc}{\mathcal{D}}^{\mu \nu }{u}_{\nu }& =c^2\epsilon F^{\mu \nu }{u}_{\nu }\\ \star {\mathcal{D}}^{\mu \nu }{u}_{\nu }& =\frac{1}{\mu }\star F^{\mu \nu }{u}_{\nu }\end{array}}$$ 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), ${\displaystyle \star }$ and denotes the Hodge star operator.

Lagrangian for classical electrodynamics

Vacuum

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 ${\displaystyle \mathcal{L}\left(A_{\alpha },{\mathrm{\partial }}_{\beta }{A}_{\alpha }\right)}$ can be stated as follows: $${\displaystyle {\mathrm{\partial }}_{\beta }\left[\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\beta }{A}_{\alpha })}\right]-\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{A}_{\alpha }}=0.}$$

Noting $${\displaystyle \begin{array}{cc}{F}^{\lambda \sigma }& =F_{\mu \nu }{\eta }^{\mu \lambda }{\eta }^{\nu \sigma },\\ F_{\mu \nu }& ={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }\\ \frac{\mathrm{\partial }\left({\mathrm{\partial }}_{\mu }{A}_{\nu }\right)}{\mathrm{\partial }\left({\mathrm{\partial }}_{\rho }{A}_{\sigma }\right)}& ={\delta }_{\mu }^{\rho }{\delta }_{\nu }^{\sigma }\end{array}}$$ the expression inside the square bracket is $${\displaystyle \begin{array}{cc}\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\beta }{A}_{\alpha })}& =-\frac{1}{4{\mu }_0}\frac{\mathrm{\partial }\left(F_{\mu \nu }{\eta }^{\mu \lambda }{\eta }^{\nu \sigma }{F}_{\lambda \sigma }\right)}{\mathrm{\partial }\left({\mathrm{\partial }}_{\beta }{A}_{\alpha }\right)}\\ & =-\frac{1}{4{\mu }_0}{\eta }^{\mu \lambda }{\eta }^{\nu \sigma }\left(F_{\lambda \sigma }\left({\delta }_{\mu }^{\beta }{\delta }_{\nu }^{\alpha }-{\delta }_{\nu }^{\beta }{\delta }_{\mu }^{\alpha }\right)+F_{\mu \nu }\left({\delta }_{\lambda }^{\beta }{\delta }_{\sigma }^{\alpha }-{\delta }_{\sigma }^{\beta }{\delta }_{\lambda }^{\alpha }\right)\right)\\ & =-\frac{F^{\beta \alpha }}{{\mu }_0}.\end{array}}$$

The second term is $${\displaystyle \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{A}_{\alpha }}=-J^{\alpha }.}$$

Therefore, the electromagnetic field's equations of motion are $${\displaystyle \frac{\mathrm{\partial }{F}^{\beta \alpha }}{\mathrm{\partial }{x}^{\beta }}={\mu }_0{J}^{\alpha }.}$$ which is the Gauss–Ampère equation above.

Matter

Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows: $${\displaystyle \mathcal{L}=-\frac{1}{4{\mu }_0}{F}^{\alpha \beta }{F}_{\alpha \beta }-A_{\alpha }{J}_{\text{free}}^{\alpha }+\frac{1}{2}{F}_{\alpha \beta }{\mathcal{M}}^{\alpha \beta }.}$$

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

The equivalent expression in vector notation is: $${\displaystyle \mathcal{L}=\frac{1}{2}\left({\epsilon }_0{E}^2-\frac{1}{{\mu }_0}{B}^2\right)-\varphi {\rho }_{\text{free}}+\mathbf{A}\cdot {\mathbf{J}}_{\text{free}}+\mathbf{E}\cdot \mathbf{P}+\mathbf{B}\cdot \mathbf{M}.}$$