Thecovariant formulation ofclassical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular,Maxwell's equations and theLorentz force) in a form that is manifestly invariant underLorentz transformations, in the formalism ofspecial relativity using rectilinearinertial coordinate systems. 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 asMaxwell's equations in curved spacetime or non-rectilinear coordinate systems.^[a]

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:$${\displaystyle u^{\alpha }=\gamma (c,\mathbf{u}),}$$ whereγ(u) is theLorentz factor at the 3-velocityu.
• Four-momentum:$${\displaystyle p^{\alpha }=(E/c,\mathbf{p})=m_0{u}^{\alpha }}$$ where${\displaystyle \mathbf{p}}$ is 3-momentum,${\displaystyle E}$ is thetotal energy, and${\displaystyle m_0}$ isrest mass.
• Four-gradient:$${\displaystyle {\mathrm{\partial }}^{\nu }=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_{\nu }}=\left(\frac{1}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t},-\mathrm{\nabla }\right),}$$
• Thed'Alembertian operator is denoted${\displaystyle {\mathrm{\partial }}^2}$,$${\displaystyle {\mathrm{\partial }}^2=\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}-{\mathrm{\nabla }}^2.}$$

The signs in the following tensor analysis depend on theconvention used for themetric tensor. The convention used here is(+ − − −), corresponding to theMinkowski metric tensor:

The electromagnetic tensor is the combination of the electric and magnetic fields into a covariantantisymmetric tensor whose entries areB-field quantities.^[1]and the result of raising its indices iswhereE is theelectric field,B themagnetic field, andc thespeed of light.

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

The electromagnetic four-potential is a covariant four-vector containing theelectric potential (also called thescalar potential)ϕ andmagnetic vector potential (orvector potential)A, as follows:$${\displaystyle A^{\alpha }=\left(\varphi /c,\mathbf{A}\right).}$$

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

In the language ofdifferential forms, which provides the generalisation to curved spacetimes, these are the components of a 1-form${\displaystyle A=A_{\alpha }d{x}^{\alpha }}$ and a 2-form$F=dA=\frac{1}{2}{F}_{\alpha \beta }d{x}^{\alpha }\wedge d{x}^{\beta }$ respectively. Here,${\displaystyle d}$ is theexterior derivative and${\displaystyle \wedge }$ thewedge 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 overallstress–energy tensor:where${\displaystyle {\epsilon }_0}$ is theelectric permittivity of vacuum,μ₀ is themagnetic permeability of vacuum, thePoynting vector is$${\displaystyle \mathbf{S}=\frac{1}{{\mu }_0}\mathbf{E}\times \mathbf{B}}$$and theMaxwell stress tensor is given by$${\displaystyle {\sigma }_{ij}={\epsilon }_0{E}_i{E}_j+\frac{1}{{\mu }_0}{B}_i{B}_j-\left(\frac{1}{2}{\epsilon }_0{E}^2+\frac{1}{2{\mu }_0}{B}^2\right){\delta }_{ij}.}$$

The electromagnetic field tensorF constructs the electromagnetic stress–energy tensorT by the equation:^[2]$${\displaystyle T^{\alpha \beta }=\frac{1}{{\mu }_0}\left({\eta }^{\alpha \nu }{F}_{\nu \gamma }{F}^{\beta \gamma }-\frac{1}{4}{\eta }^{\alpha \beta }{F}_{\gamma \nu }{F}^{\gamma \nu }\right)}$$whereη is theMinkowski metric tensor (with signature(+ − − −)). Notice that we use the fact that$${\displaystyle {\epsilon }_0{\mu }_0{c}^2=1,}$$which is predicted by Maxwell's equations.

Another way to covariant expression for the eletromagnetic stress-energy tensor which may be simpler since it does not involve covariant and contravariant indices is this one:$${\displaystyle T=-\frac{1}{{\mu }_0}(F\ast \eta \ast F^{\prime }-\frac{1}{4}trace(F\ast \eta \ast F^{\prime }\ast \eta ))}$$Where F' is the transposed electromagnetic tensor or equivalently -F and the asterisk denotes matrix multiplication.

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 andAmpère's law (with Maxwell's correction) combine into (with(+ − − −) metric):^[3]

The homogeneous equations –Faraday's law of induction andGauss'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:

whereF^αβ is theelectromagnetic tensor,J^α is thefour-current,ε^αβγδ is theLevi-Civita symbol, and the indices behave according to theEinstein summation convention.

Each of these tensor equations corresponds to four scalar equations, one for each value ofβ.

Using theantisymmetric tensor notation and comma notation for the partial derivative (seeRicci 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 anelectromagnetic wave equation in the field strength tensor.

TheLorenz gauge condition is a Lorentz-invariant gauge condition. (This can be contrasted with othergauge conditions such as theCoulomb gauge, which if it holds in oneinertial 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:$${\displaystyle {\mathrm{\partial }}^2{A}^{\sigma }={\mu }_0{J}^{\sigma }.}$$

Electromagnetic (EM) fields affect the motion ofelectrically charged matter: due to theLorentz force. In this way, EM fields can bedetected (with applications inparticle physics, and natural occurrences such as inaurorae). In relativistic form, the Lorentz force uses the field strength tensor as follows.^[4]

Expressed in terms ofcoordinate timet, it is:$${\displaystyle \frac{d{p}_{\alpha }}{dt}=q{F}_{\alpha \beta }\frac{d{x}^{\beta }}{dt},}$$wherep_α is the four-momentum,q is thecharge, andx^β is the position.

Expressed in frame-independent form, we have the four-force$${\displaystyle \frac{d{p}_{\alpha }}{d\tau }=q{F}_{\alpha \beta }{u}^{\beta },}$$whereu^β is the four-velocity, andτ is the particle'sproper time, which is related to coordinate time bydt =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$${\displaystyle f^{\alpha }=-{T^{\alpha \beta }}_{,\beta }\equiv -\frac{\mathrm{\partial }{T}^{\alpha \beta }}{\mathrm{\partial }{x}^{\beta }}.}$$

Thecontinuity equation:$${\displaystyle {J^{\beta }}_{,\beta }\stackrel{\text{def}}{=}{\mathrm{\partial }}_{\beta }{J}^{\beta }={\mathrm{\partial }}_{\beta }{\mathrm{\partial }}_{\alpha }{F}^{\alpha \beta }/{\mu }_0=0.}$$expressescharge conservation.

Using the Maxwell equations, one can see that theelectromagnetic 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

Maxwell's macroscopic equations have been used, in addition the definitions of theelectric displacementD and themagnetic intensityH:whereM is themagnetization andP theelectric polarization.

The bound current is derived from theP andM fields which form an antisymmetric contravariant magnetization-polarization tensor^[1][5][6][7]which determines the bound current$${\displaystyle {J^{\nu }}_{\text{bound}}={\mathrm{\partial }}_{\mu }{\mathcal{M}}^{\mu \nu }.}$$

If this is combined withF^μν we get the antisymmetric contravariant electromagnetic displacement tensor which combines theD andH fields as follows:

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 theD andH fields given above.

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

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

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 defineF^μν by$${\displaystyle F^{\mu \nu }={\eta }^{\mu \alpha }{F}_{\alpha \beta }{\eta }^{\beta \nu },}$$the constitutive equations may, invacuum, 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:$${\displaystyle T_{\alpha }{}^{\pi }=F_{\alpha \beta }{\mathcal{D}}^{\pi \beta }-\frac{1}{4}{\delta }_{\alpha }^{\pi }{F}_{\mu \nu }{\mathcal{D}}^{\mu \nu },}$$whereδ_α^π is theKronecker delta. When the upper index is lowered withη, 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,${\displaystyle {\mathcal{M}}^{\mu \nu }}$. For example, in the simplest materials at low frequencies, one haswhere one is in the instantaneously comoving inertial frame of the material,σ is itselectrical conductivity,χ_e is itselectric susceptibility, andχ_m is itsmagnetic susceptibility.

The constitutive relations between the${\displaystyle \mathcal{D}}$ andF tensors, proposed byMinkowski for a linear materials (that is,E isproportional toD andB proportional toH), are:whereu is the four-velocity of material,ε andμ are respectively the properpermittivity andpermeability of the material (i.e. in rest frame of material),${\displaystyle \star }$ and denotes theHodge star operator.

TheLagrangian 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.

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

Notingthe expression inside the square bracket is

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.

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

• 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

• The Feynman Lectures on Physics Vol. II Ch. 25: Electrodynamics in Relativistic Notation
• Einstein, A. (1961).Relativity: The Special and General Theory. New York: Crown.ISBN 0-517-02961-8.{{cite book}}:ISBN / Date incompatibility (help)
• Misner, Charles; Thorne, Kip S.; Wheeler, John Archibald (1973).Gravitation. San Francisco: W. H. Freeman.ISBN 0-7167-0344-0.
• Landau, L. D.; Lifshitz, E. M. (1975).Classical Theory of Fields (Fourth Revised English ed.). Oxford: Pergamon.ISBN 0-08-018176-7.
• R. P. Feynman; F. B. Moringo; W. G. Wagner (1995).Feynman Lectures on Gravitation. Addison-Wesley.ISBN 0-201-62734-5.

• Concepts in physics
• Electromagnetism
• Special relativity

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