Covariant formulation of classical electromagnetism explained

The 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 transformations, in the formalism of special relativity using rectilinear inertial 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 as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

Covariant objects

Preliminary four-vectors

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

• four-displacement: $$x^\backslash alpha\; =\; (ct,\; \backslash mathbf)\; =\; (ct,\; x,\; y,\; z)\; \backslash ,.$$
• Four-velocity: $$u^\backslash alpha\; =\; \backslash gamma(c,\backslash mathbf),$$ where γ(u) is the Lorentz factor at the 3-velocity u.
• Four-momentum: $$p^\backslash alpha\; =\; (E/c,\; \backslash mathbf)\; =\; m\_0\; u^$$ where

is 3-momentum, is the total energy, and is rest mass.

• Four-gradient: $$\backslash partial^\; =\; \backslash frac\; =\; \backslash left(\backslash frac\; \backslash frac,\; -\; \backslash mathbf\; \backslash right)\; \backslash ,,$$
• The d'Alembertian operator is denoted

, $$\backslash partial^2\; =\; \backslash frac\; -\; \backslash nabla^2.$$

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:

Electromagnetic tensor

See main article: 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.and the result of raising its indices iswhere E is the electric field, B the magnetic field, and c the speed of light.

Four-current

See main article: Four-current. The four-current is the contravariant four-vector which combines electric charge density ρ and electric current density j:$$J^\; =\; (c\; \backslash rho,\; \backslash mathbf)\; \backslash ,.$$

Four-potential

See main article: 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:$$A^\; =\; \backslash left(\backslash phi/c,\; \backslash mathbf\; \backslash right)\backslash ,.$$

The differential of the electromagnetic potential is$$F\_\; =\; \backslash partial\_\; A\_\; -\; \backslash partial\_\; A\_\; \backslash ,.$$

In the language of differential forms, which provides the generalisation to curved spacetimes, these are the components of a 1-form

and a 2-form $F\; =\; dA\; =\; \backslash frac\; F\_\; dx^\; \backslash wedge\; dx^$ respectively. Here, is the exterior derivative and the wedge product.

Electromagnetic stress–energy tensor

See main article: 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:where

is the electric permittivity of vacuum, μ₀ is the magnetic permeability of vacuum, the Poynting vector is$$\backslash mathbf\; =\; \backslash frac\; \backslash mathbf\; \backslash times\; \backslash mathbf$$and the Maxwell stress tensor is given by$$\backslash sigma\_\; =\; \backslash varepsilon\_0\; E\_i\; E\_j\; +\; \backslash frac\; B\_i\; B\_j\; -\; \backslash left(\backslash frac\; 1\; 2\; \backslash varepsilon\_0\; E^2\; +\; \backslash frac\; B^2\backslash right)\; \backslash delta\_\; \backslash ,.$$

The electromagnetic field tensor F constructs the electromagnetic stress–energy tensor T by the equation:^[1] $$T^\; =\; \backslash frac\; \backslash left(\backslash eta^F\_F^\; -\; \backslash frac\; \backslash eta^F\_F^\backslash right)$$where η is the Minkowski metric tensor (with signature). Notice that we use the fact that$$\backslash varepsilon\_\; \backslash mu\_\; c^2\; =\; 1\backslash ,,$$which is predicted by Maxwell's equations.

Maxwell's equations in vacuum

See main article: 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 metric):^[2]

The homogeneous equations – Faraday's law of induction and Gauss's law for magnetism combine to form

, which may be written using Levi-Civita duality as: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:$$F\_\; =\; 0\; .$$

In the absence of sources, Maxwell's equations reduce to:$$\backslash partial^\; \backslash partial\_\backslash nu\; F^\; \backslash mathrel\; \backslash partial^2\; F^\; \backslash mathrel\; -\; \backslash nabla^2\; F^\; =\; 0\; \backslash ,,$$which is an electromagnetic wave equation in the field strength tensor.

Maxwell's equations in the Lorenz gauge

See main article: 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:$$\backslash partial\_\; A^\; =\; \backslash partial^\; A\_\; =\; 0\; \backslash ,.$$

In the Lorenz gauge, the microscopic Maxwell's equations can be written as:$$^2\; A^\; =\; \backslash mu\_\; \backslash ,\; J^\backslash ,.$$

Lorentz force

See main article: 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.^[3]

Expressed in terms of coordinate time t, it is:$$=\; q\; \backslash ,\; F\_\; \backslash ,\; \backslash frac,$$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$$\backslash frac\; \backslash ,\; =\; q\; \backslash ,\; F\_\; \backslash ,\; u^\backslash beta,$$where u^β is the four-velocity, and τ is the particle's proper time, which is related to coordinate time by .

Charge continuum

See also: continuum mechanics. The density of force due to electromagnetism, whose spatial part is the Lorentz force, is given by$$f\_\; =\; F\_J^\; .$$and is related to the electromagnetic stress–energy tensor by$$f^\; =\; -\; \_\; \backslash equiv\; -\; \backslash frac.$$

Conservation laws

Electric charge

The continuity equation:$$\_\; \backslash mathrel\backslash overset\; \backslash partial\_\backslash beta\; J^\backslash beta\; =\; \backslash partial\_\backslash beta\; \backslash partial\_\backslash alpha\; F^/\backslash mu\_0\; =\; 0.$$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$$\_\; +\; F^\; J\_\backslash beta\; =\; 0$$or$$\backslash eta\_\; \_\; +\; F\_\; J^\backslash beta\; =\; 0,$$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;$$J^\; =\; \_\; +\; \_\; \backslash ,,$$where

Maxwell's macroscopic equations have been used, in addition the definitions of the electric displacement D and the magnetic intensity H: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 ^[4] which determines the bound current$$\_\backslash text\; =\; \backslash partial\_\backslash mu\; \backslash mathcal^\; \backslash ,.$$

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:

The three field tensors are related by:$$\backslash mathcal^\; =\; \backslash frac\; F^\; -\; \backslash mathcal^$$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,$$\backslash mathbf\; \backslash times\; \backslash mathbf\; -\; \backslash frac\; =\; \backslash mathbf\_,$$and Gauss's law,$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; \backslash rho\_,$$combine into one equation:

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

Constitutive equations

See main article: Constitutive equation.

Vacuum

In vacuum, the constitutive relations between the field tensor and displacement tensor are:$$\backslash mu\_0\; \backslash mathcal^\; =\; \backslash eta^\; F\_\; \backslash eta^\; \backslash ,.$$

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

The electromagnetic stress–energy tensor in terms of the displacement is:$$T\_\backslash alpha^\backslash pi\; =\; F\_\; \backslash mathcal^\; -\; \backslash frac\; \backslash delta\_\backslash alpha^\backslash pi\; F\_\; \backslash mathcal^,$$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,

. For example, in the simplest materials at low frequencies, one haswhere 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

and F tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are: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), 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:$$\backslash mathcal\; \backslash ,\; =\; \backslash ,\; \backslash mathcal\_\backslash text\; +\; \backslash mathcal\_\backslash text\; =\; -\; \backslash frac\; F^\; F\_\; -\; A\_\; J^\; \backslash ,.$$

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

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

Notingthe expression inside the square bracket is

The second term is$$\backslash frac\; =\; -\; J^\; \backslash ,.$$

Therefore, the electromagnetic field's equations of motion are$$\backslash frac\; =\; \backslash mu\_0\; J^\; \backslash ,.$$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:$$\backslash mathcal\; \backslash ,\; =\; \backslash ,\; -\; \backslash frac\; F^\; F\_\; -\; A\_\; J^\_\; +\; \backslash frac\; 1\; 2\; F\_\; \backslash mathcal^\; \backslash ,.$$

Using Lagrange equation, the equations of motion for

can be derived.

The equivalent expression in vector notation is:$$\backslash mathcal\; \backslash ,\; =\; \backslash ,\; \backslash frac\; 1\; 2\; \backslash left(\backslash varepsilon\_\; E^2\; -\; \backslash frac\; B^2\backslash right)\; -\; \backslash phi\; \backslash ,\; \backslash rho\_\; +\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\_\; +\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; +\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash ,.$$

See also

• 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

Further reading

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

Notes and References

1. Classical Electrodynamics, Jackson, 3rd edition, page 609
2. Classical Electrodynamics by Jackson, 3rd Edition, Chapter 11 Special Theory of Relativity
3. 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.
4. However, the assumption that

   M^\mu\nu

   ,

   D^\mu\nu

   , and even

   F^\mu\nu

   , are relativistic tensors in a polarizable medium, is without foundation. The quantity$$A^\; =\; \backslash left(\backslash phi/c,\; \backslash mathbf\; \backslash right)\backslash ,$$is not a four vector in a polarizable medium, so $$F\_\; =\; \backslash partial\_\; A\_\; -\; \backslash partial\_\; A\_\; \backslash ,$$ does not produce a tensor.