 Covariant formulation of classical electromagnetism

Electromagnetism Electricity · Magnetism Lorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard–Wiechert potential · Maxwell tensor · Eddy currentCovariant formulationThe 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 which is "manifestly covariant" (i.e. in terms of covariant fourvectors and tensors), in the formalism of special relativity. 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.
The Minkowski metric used in this article is assumed to have the form diag (+1, −1, −1, −1). The purely spatial components of the tensors (including vectors) are given in SI units. This article uses the classical treatment of tensors and the Einstein summation convention throughout. Where the equations are specified as holding in a vacuum, one could instead regard them as the formulation of Maxwell's equations in terms of total charge and current.
For a more general overview of the relationships between classical electromagnetism and special relativity, including various conceptual implications of this picture, see the article: Classical electromagnetism and special relativity.
Covariant objects
Electromagnetic tensor
Main article: Electromagnetic tensorThe electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor. In volt·seconds/meter^{2}, the field strength tensor is written in terms of fields as^{[1]}:
and the result of raising its indices is
 where
 is the electric field,
 the magnetic field, and
 the speed of light.
 Caution: The signs in the tensor above depend on the convention used for the metric tensor. The convention used here is +, corresponding to the metric tensor :
FourCurrent
Main article: FourcurrentThe fourcurrent is the contravariant fourvector which combines electric current and electric charge density. In amperes/meter^{2}, it is given by
where is the charge density, is the current density, and is the speed of light.
Fourpotential
Main article: FourpotentialIn volt·seconds/meter, the electromagnetic fourpotential is a covariant fourvector containing the electric potential and magnetic vector potential, as follows:
where is the scalar potential and is the vector potential.
The relation between the electromagnetic potentials and the electromagnetic fields is given by the following equation:
where
Electromagnetic stressenergy tensor
Main article: Electromagnetic stressenergy tensorThe electromagnetic stressenergy tensor is a contravariant symmetric tensor which is the contribution of the electromagnetic fields to the overall stressenergy tensor. In joules/meters^{3}, it is given by
where is the electric permittivity of vacuum, is the magnetic permeability of vacuum, the Poynting vector is
and the Maxwell stress tensor is given by
The electromagnetic stressenergy tensor is related to the electromagnetic field tensor by the equation:
where is the Minkowski metric tensor. Notice that we use the fact that
Other, nonelectromagnetic objects
Main article: Lorentz covarianceFor background purposes, we present here three other relevant fourvectors, which are not directly connected to electromagnetism, but which will be useful in this article:
 In meters, the "position" or "coordinate" fourvector is
 In meters/second, the velocity fourvector (or fourvelocity) is

 where is the (threevector) velocity and is the Lorentz factor associated with
 In kilogram·meters/second, the fourmomentum (or momentum fourvector) of a particle is

 where is the (threevector) momentum, is the energy, and is the particle's rest mass.
Maxwell's equations in vacuo
Main article: Maxwell's equationsIn a vacuum (or for the microscopic equations, not including macroscopic material descriptions) Maxwell's equations can be written as two tensor equations
where is the electromagnetic tensor, is the 4current, is the LeviCivita symbol (a mathematical construct), and the indices behave according to the Einstein summation convention.
The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss's Law and Ampere's Law (with Maxwell's correction). The second equation is an expression of the homogeneous equations, Faraday's law of induction and Gauss's law for magnetism.
In the absence of sources, Maxwell's equations reduce to a wave equation in the field strength:
where,
 is the d'Alembertian operator.
Other notation
Without the summation convention or the LeviCivita symbol, the equations would be written
where all indices range from 0 to 3 (or, more descriptively, ranges over the set {ct,x,y,z}), where is the speed of light in free space. The first tensor equation corresponds to four scalar equations, one for each value of . The second tensor equation actually corresponds to different scalar equations, but only four of these are independent.
For convenience, professionals often write the 4gradient (that is, the derivative with respect to x) using abbreviated notations; for instance,
Using the latter notation, Maxwell's equations can be written as and
Continuity equation
The continuity equation which expresses the fact that charge is conserved is:
Lorentz force
Main article: Lorentz forceFields are detected by their effect on the motion of matter. Electromagnetic fields affect the motion of particles through the Lorentz force. Using the Lorentz force, Newton's law of motion can be written in relativistic form using the field strength tensor as^{[2]}
where is the fourmomentum (see above), is the charge, is the fourvelocity (see above), and is the particle's proper time.
In terms of (normal) time instead of proper time, the equation is
In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4vector, The spatial part is the result of dividing the force on a small cell (in 3space) by the volume of that cell. The time component is 1/c times the power transferred to that cell divided by the volume of the cell. The density of Lorentz force is the part of the density of force due to electromagnetism. Its spatial part is . In manifestly covariant notation it becomes:
Differential equation for electromagnetic stressenergy tensor
The relationship between Lorentz force and electromagnetic stressenergy tensor should be:
Therefore, The electromagnetic stressenergy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current fourvector
or
which expresses the conservation of linear momentum and energy by electromagnetic interactions.
Lorenz gauge condition
Main article: Lorenz gauge conditionThe Lorenz gauge condition is a Lorentzinvariant 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 fourpotential as follows:
Maxwell's equations in the Lorenz gauge
In the Lorenz gauge, Maxwell's equations for a vacuum can be written as:
where denotes the d'Alembertian.
Bound current
^{[dubious – discuss]}
In order to solve the equations of electromagnetism given here, it is necessary to add information about how to calculate the electric current, Frequently, it is convenient to separate the current into two parts, the free current and the bound current, which are modeled by different equations.
where
The bound current is derived from the magnetization and electric polarization which form an antisymmetric contravariant magnetizationpolarization tensor^{[1]}
which determines the bound current
If this is combined with we get the antisymmetric contravariant electromagnetic displacement tensor which combines the electric displacement and the Hfield as follows
They are related by
which is equivalent to the constitutive equations and And the result is that Ampère's law, , and Gauss's law, , combine to form:
The bound current and free current as defined above are automatically and separately conserved
Thus we have reduced the problem of modeling the current, to two (hopefully) easier problems — modeling the free current, and modeling the magnetization and polarization, For example, in the simplest materials at low frequencies, one has
where one is in the instantaneouslycomoving inertial frame of the material, σ is its electrical conductivity, χ_{e} is its electric susceptibility, and χ_{m} is its magnetic susceptibility.
Lagrangian for classical electrodynamics
In a vacuum, the Lagrangian (Lagrangian density) for classical electrodynamics (in joules/meter^{3}) is
In the interaction term, the fourcurrent should be understood as an abbreviation of many terms expressing the electric currents of other charged fields in terms of their variables; the fourcurrent is not itself a fundamental field.
The EulerLagrange equation for the electromagnetic Lagrangian density can be stated as follows:
Noting , The expression inside the square bracket is
The second term is
Therefore, the electromagnetic field's equations of motion are
Separating the free currents from the bound currents, another way to write the Lagrangian density is as follows:
Using EulerLagrange equation, the equations of motion for can be derived.
The equivalent expression in nonrelativistic vector notation is
In general relativity
Main article: Maxwell's equations in curved spacetimeIn general relativity, the metric, , is no longer a constant () but can vary from place to place and time to time. In general relativity, the equations of electromagnetism in a vacuum become:
where is the density of Lorentz force, is the reciprocal of the metric tensor , and is the determinant of the metric tensor. Notice that and are (ordinary) tensors while , , and are tensor densities of weight +1. All derivatives are partial derivatives — if one replaced them with covariant derivatives, the extra terms thereby introduced would cancel out.
See also
 Relativistic electromagnetism
 Electromagnetic wave equation
 Liénard–Wiechert potential for a charge in arbitrary motion
 Nonhomogeneous electromagnetic wave equation
 Moving magnet and conductor problem
 Electromagnetic tensor
 Proca action
 Stueckelberg action
 Quantum electrodynamics
 WheelerFeynman absorber theory
Notes and references
 ^ ^{a} ^{b} Vanderlinde, Jack (2004), classical electromagnetic theory, Springer, pp. 313–328, ISBN 9781402026997, http://books.google.com/books?id=HWrMET9_VpUC&pg=PA316&dq=electromagnetic+field+tensor+vanderlinde
 ^ 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.
Further reading
 Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0517029618.
 Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0716703440.
 Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0080181767.
 R. P. Feynman, F. B. Moringo, and W. G. Wagner (1995). Feynman Lectures on Gravitation. AddisonWesley. ISBN 0201627345.
Categories: Fundamental physics concepts
 Electromagnetism
 Special relativity
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