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 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.
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The 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:
and the result of raising its indices is
The fourcurrent is the contravariant fourvector which combines electric current and electric charge density. In amperes/meter^{2}, it is given by
where ρ is the charge density, is the current density, and c is the speed of light.
In volt·seconds/meter, the electromagnetic fourpotential is a covariant fourvector containing the electric potential and magnetic vector potential, as follows:
where φ 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
The 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 ε_{0} is the electric permittivity of vacuum, μ_{0} 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
For 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 a vacuum, Maxwell's equations can be written as two tensor equations
where F ^{αβ} is the electromagnetic tensor, J ^{α} is the 4current, є ^{αβγδ} 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:
Here, is the d'Alembertian operator.
Without the summation convention or the LeviCivita symbol, the equations would be written
where all indices range from 0 to 3 (or, more descriptively, x^{α} ranges over the set {ct,x,y,z}), where c is the speed of light in free space. The first tensor equation corresponds to four scalar equations, one for each value of β. The second tensor equation actually corresponds to 4^{3} = 64 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
The continuity equation which expresses the fact that charge is conserved is:
Fields 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^{[1]}
where p is the fourmomentum (see above), q is the charge, u is the fourvelocity (see above), and τ 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:
The electromagnetic stressenergy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current fourvector
which expresses the conservation of linear momentum and energy by electromagnetic interactions.
The 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:
In the Lorenz gauge, Maxwell's equations for a vacuum can be written as:
where denotes the d'Alembertian.
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.
The bound current is derived from the magnetization and electric polarization which form an antisymmetric contravariant magnetizationpolarization tensor
which is determines the bound current so
If this is combined with we get the antisymmetric contravariant electromagnetic displacement tensor which combines the electric displacement and the magnetic field 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.
In a vacuum, the Lagrangian 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.
If we separate free currents from bound currents, the Lagrangian becomes
The equivalent expression in nonrelativistic vector notation is
In general relativity, the metric, g_{αβ}, is no longer a constant (η_{αβ}) but can vary from place to place and time to time. The metric tensor is the potential of the gravitational field. In general relativity, the equations of electromagnetism in a vacuum become:
where f_{μ} is the density of Lorentz force, g^{αβ} is the reciprocal of the metric tensor g_{αβ}, and g is the determinant of the metric tensor. Notice that A_{α} and F_{αβ} are (ordinary) tensors while , J^{μ}, and f_{μ} 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.

