The
covariant formulation of
classical electromagnetism
Classical electromagnetism or classical electrodynamics is a branch of physics focused on the study of interactions between electric charges and electrical current, currents using an extension of the classical Newtonian model. It is, therefore, a ...
refers to ways of writing the laws of classical electromagnetism (in particular,
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
and the
Lorentz force
In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation ...
) in a form that is manifestly invariant under
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
s, in the formalism of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity,
"On the Ele ...
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
Maxwell's, last known as Maxwell's Tavern, was a bar/restaurant and Music venue, music club in Hoboken, New Jersey. Over several decades the venue attracted a wide variety of acts looking for a change from the New York City concert spaces across ...
or non-rectilinear coordinate systems.
Covariant objects
Preliminary four-vectors
Lorentz tensors of the following kinds may be used in this article to describe bodies or particles:
*
four-displacement:
*
Four-velocity:
where ''γ''(u) is the
Lorentz factor
The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
at the 3-velocity u.
*
Four-momentum
In special relativity, four-momentum (also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum i ...
:
where
is 3-momentum,
is the
total energy, and
is
rest mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
.
*
Four-gradient:
* The
d'Alembertian operator is denoted
,
The signs in the following tensor analysis depend on the
convention used for the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
. The convention used here is , corresponding to the
Minkowski metric tensor:
Electromagnetic tensor
The electromagnetic tensor is the combination of the electric and magnetic fields into a covariant
antisymmetric tensor In mathematics and theoretical physics, a tensor is antisymmetric or alternating on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally ...
whose entries are B-field quantities.
[
]
and the result of raising its indices is
where E is the
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
, B the
magnetic field
A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
, and ''c'' the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
.
Four-current
The four-current is the contravariant four-vector which combines
electric charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
''ρ'' and
electric current density j:
Four-potential
The electromagnetic four-potential is a covariant four-vector containing the
electric potential
Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
(also called the
scalar potential
In mathematical physics, scalar potential describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one p ...
) ''ϕ'' and
magnetic vector potential
In classical electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the ma ...
(or
vector potential) A, as follows:
The differential of the electromagnetic potential is
In the language of
differential forms
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
, which provides the generalisation to curved spacetimes, these are the components of a 1-form
and a 2-form
respectively. Here,
is the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
and
the
wedge product
A wedge is a triangular shaped tool, a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converting a fo ...
.
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
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
:
where
is the
electric permittivity of vacuum, ''μ''
0 is the
magnetic permeability of vacuum, the
Poynting vector
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area, per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the wat ...
is
and the
Maxwell stress tensor is given by
The electromagnetic field tensor ''F'' constructs the electromagnetic stress–energy tensor ''T'' by the equation:
where ''η'' is the
Minkowski metric
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model ...
tensor (with signature ). Notice that we use the fact that
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 not involve covariant and contravariant indices is this one:
Where F' is the trasposed electromagnetic tensor or equivalently -F and the asterisk denotes matrix multipliaction.
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):
The homogeneous equations –
Faraday's law of induction and
Gauss's law for magnetism
In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
combine to form
, which may be written using Levi-Civita duality as:
where ''F''
''αβ'' is the
electromagnetic tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
, ''J''
''α'' is the
four-current
In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the current density, with units of charge per unit time per unit area. Also known as vector current, it is used in the ...
, ''ε''
''αβγδ'' is the
Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers , for some ...
, and the indices behave according to the
Einstein summation convention
In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies s ...
.
Each of these tensor equations corresponds to four scalar equations, one for each value of ''β''.
Using the
antisymmetric tensor In mathematics and theoretical physics, a tensor is antisymmetric or alternating on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally ...
notation and comma notation for the partial derivative (see
Ricci calculus Ricci () is an Italian surname. Notable Riccis Arts and entertainment
* Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin
* Christina Ricci (born 1980), American actress
* Clara Ross Ricci (1858-1954), British ...
), the second equation can also be written more compactly as:
In the absence of sources, Maxwell's equations reduce to:
which is an
electromagnetic wave equation
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous for ...
in the field strength tensor.
Maxwell's equations in the Lorenz gauge
The
Lorenz gauge condition
In electromagnetism, the Lorenz gauge condition or Lorenz gauge (after Ludvig Lorenz) is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ...
is a Lorentz-invariant gauge condition. (This can be contrasted with other
gauge conditions such as the
Coulomb gauge
In the physics of gauge theory, gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant Degrees of freedom (physics and chemistry), degrees of freedom in field (physics), field variab ...
, which if it holds in one
inertial frame
In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative ...
will generally not hold in any other.) It is expressed in terms of the four-potential as follows:
In the Lorenz gauge, the microscopic Maxwell's equations can be written as:
Lorentz force
Charged particle

Electromagnetic (EM) fields affect the motion of
electrically charged matter: due to the
Lorentz force
In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation ...
. In this way, EM fields can be
detected (with applications in
particle physics
Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, 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:
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
where ''u''
''β'' is the four-velocity, and ''τ'' is the particle's
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
, which is related to coordinate time by .
Charge continuum

The density of force due to electromagnetism, whose spatial part is the Lorentz force, is given by
and is related to the electromagnetic stress–energy tensor by
Conservation laws
Electric charge
The
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity ...
:
expresses
charge conservation
In physics, charge conservation is the principle, of experimental nature, that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charg ...
.
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
or
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;
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
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Accordingly, physicists and engineers usually define magnetization as the quanti ...
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
which determines the bound current
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:
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,
and
Gauss's law,
combine into one equation:
The bound current and free current as defined above are automatically and separately conserved
Constitutive equations
Vacuum
In vacuum, the constitutive relations between the field tensor and displacement tensor are:
Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define ''F''
''μν'' by
the constitutive equations may, in ''vacuum'', be combined with the Gauss–Ampère law to get:
The electromagnetic stress–energy tensor in terms of the displacement is:
where ''δ
απ'' is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
. 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 has
where one is in the instantaneously comoving inertial frame of the material, ''σ'' is its
electrical conductivity
Electrical resistivity (also called volume resistivity or specific electrical resistance) is a fundamental specific property of a material that measures its electrical resistance or how strongly it resists electric current. A low resistivity in ...
, ''χ''
e is its
electric susceptibility
In electricity (electromagnetism), the electric susceptibility (\chi_; Latin: ''susceptibilis'' "receptive") is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applie ...
, and ''χ''
m is its
magnetic susceptibility
In electromagnetism, the magnetic susceptibility (; denoted , chi) is a measure of how much a material will become magnetized in an applied magnetic field. It is the ratio of magnetization (magnetic moment per unit volume) to the applied magnet ...
.
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
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more ...
and
permeability of the material (i.e. in rest frame of material),
and denotes the
Hodge star operator
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a Dimension (vector space), finite-dimensional orientation (vector space), oriented vector space endowed with a Degenerate bilinear form, nonde ...
.
Lagrangian for classical electrodynamics
Vacuum
The
Lagrangian density for classical electrodynamics is composed by two components: a field component and a source component:
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:
Noting
the expression inside the square bracket is
The second term is
Therefore, the electromagnetic field's equations of motion are
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:
Using Lagrange equation, the equations of motion for
can be derived.
The equivalent expression in vector notation is:
See also
*
Covariant classical field theory In mathematical physics, covariant classical field theory represents classical field theory, classical fields by Section (fiber bundle), sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of field ( ...
*
Electromagnetic tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
*
Electromagnetic wave equation
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous for ...
*
Liénard–Wiechert potential for a charge in arbitrary motion
*
Moving magnet and conductor problem
*
Inhomogeneous electromagnetic wave equation
*
Proca action
*
Quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
*
Relativistic electromagnetism
*
Stueckelberg action
*
Wheeler–Feynman absorber theory
The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is a theory of electrodynamics based on a relativistic correct ...
Notes
References
Further reading
The Feynman Lectures on Physics Vol. II Ch. 25: Electrodynamics in Relativistic Notation*
*
*
*
{{tensors
Concepts in physics
Electromagnetism
Special relativity