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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: x^\alpha = (ct, \mathbf) = (ct, x, y, z) \,. * Four-velocity: u^\alpha = \gamma(c,\mathbf) , 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 ...
: p^\alpha = ( E/c, \mathbf) = m_0 u^ where \mathbf is 3-momentum, E is the total energy, and m_0 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: \partial^ = \frac = \left( \frac \frac, - \mathbf \right) \,, * The d'Alembertian operator is denoted ^2 , \partial^2 = \frac - \nabla^2. 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: \eta^=\begin 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end


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. F_ = \begin 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end and the result of raising its indices is F^ \mathrel \eta^ \, F_ \, \eta^ = \begin 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end \,, 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: J^ = ( c \rho, \mathbf ) \,.


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: A^ = \left(\phi/c, \mathbf \right)\,. The differential of the electromagnetic potential is F_ = \partial_ A_ - \partial_ A_ \,. 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 A = A_ dx^ and a 2-form F = dA = \frac F_ dx^ \wedge dx^ respectively. Here, d 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 \wedge 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 ...
: T^ = \begin \varepsilon_ E^2/2 + B^2/2\mu_0 & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_ & -\sigma_ & -\sigma_ \\ S_y/c & -\sigma_ & -\sigma_ & -\sigma_ \\ S_z/c & -\sigma_ & -\sigma_ & -\sigma_ \end\,, where \varepsilon_0 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 \mathbf = \frac \mathbf \times \mathbf and the Maxwell stress tensor is given by \sigma_ = \varepsilon_0 E_i E_j + \frac B_i B_j - \left(\frac 1 2 \varepsilon_0 E^2 + \frac B^2\right) \delta_ \,. The electromagnetic field tensor ''F'' constructs the electromagnetic stress–energy tensor ''T'' by the equation: T^ = \frac \left( \eta^F_F^ - \frac \eta^F_F^\right) 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 \varepsilon_ \mu_ 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 not involve covariant and contravariant indices is this one: T = - \frac ( F*\eta*F' - \frac trace(F*\eta*F'*\eta)) 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 \partial^ F^ + \partial^ F^ + \partial^ F^ = 0, 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: F_ = 0 . In the absence of sources, Maxwell's equations reduce to: \partial^ \partial_\nu F^ \mathrel \partial^2 F^ \mathrel - \nabla^2 F^ = 0 \,, 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: \partial_ A^ = \partial^ A_ = 0 \,. In the Lorenz gauge, the microscopic Maxwell's equations can be written as: ^2 A^ = \mu_ \, J^\,.


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: = q \, F_ \, \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 \frac \, = q \, F_ \, u^\beta , 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 f_ = F_J^ . and is related to the electromagnetic stress–energy tensor by f^ = - _ \equiv - \frac.


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 ...
: _ \mathrel\overset \partial_\beta J^\beta = \partial_\beta \partial_\alpha F^/\mu_0 = 0. 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 _ + F^ J_\beta = 0 or \eta_ _ + F_ J^\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^ = _ + _ \,, where \begin _ = \begin c\rho_, & \mathbf_ \end &= \begin c \nabla \cdot \mathbf, & - \frac + \nabla\times\mathbf\end \,, \\ _ = \begin c\rho_, & \mathbf_ \end &= \begin - c \nabla \cdot \mathbf, & \frac + \nabla\times\mathbf \end \,. \end Maxwell's macroscopic equations have been used, in addition the definitions of the electric displacement D and the magnetic intensity H: \begin \mathbf &= \varepsilon_0 \mathbf + \mathbf, \\ \mathbf &= \frac \mathbf - \mathbf \,. \end 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 \mathcal^ = \begin 0 & P_x c & P_y c & P_z c \\ - P_x c & 0 & -M_z & M_y \\ - P_y c & M_z & 0 & -M_x \\ - P_z c & -M_y & M_x & 0 \end, which determines the bound current _\text = \partial_\mu \mathcal^ \,.


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: \mathcal^ = \begin 0 & -D_xc & -D_yc & -D_zc \\ D_xc & 0 & - H_z & H_y \\ D_yc & H_z & 0 & -H_x \\ D_zc & -H_y & H_x & 0 \end. The three field tensors are related by: \mathcal^ = \frac F^ - \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, \mathbf \times \mathbf - \frac = \mathbf_, and Gauss's law, \mathbf \cdot \mathbf = \rho_, combine into one equation: The bound current and free current as defined above are automatically and separately conserved \begin \partial_\nu _\text &= 0 \, \\ \partial_\nu _\text &= 0 \,. \end


Constitutive equations


Vacuum

In vacuum, the constitutive relations between the field tensor and displacement tensor are: \mu_0 \mathcal^ = \eta^ F_ \eta^ \,. Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual to define ''F''''μν'' by F^ = \eta^ F_ \eta^ , the constitutive equations may, in ''vacuum'', be combined with the Gauss–Ampère law to get: \partial_\beta F^ = \mu_0 J^. The electromagnetic stress–energy tensor in terms of the displacement is: T_\alpha^\pi = F_ \mathcal^ - \frac \delta_\alpha^\pi F_ \mathcal^ , where ''δαπ'' 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, \mathcal^. For example, in the simplest materials at low frequencies, one has \begin \mathbf_\text &= \sigma \mathbf \, \\ \mathbf &= \varepsilon_0 \chi_e \mathbf \, \\ \mathbf &= \chi_m \mathbf \, \end 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 \mathcal and ''F'' tensors, proposed by Minkowski for a linear materials (that is, E is proportional to D and B proportional to H), are: \begin \mathcal^u_\nu &= c^2\varepsilon F^ u_\nu \\ u_\nu &= \frac u_\nu \end 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), \star 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: \mathcal \, = \, \mathcal_\text + \mathcal_\text = - \frac F^ F_ - A_ J^ \,. 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 \mathcal\mathord\left(A_, \partial_ A_\right) can be stated as follows: \partial_\left frac\right - \frac=0 \,. Noting \begin F^ &= F_\eta^\eta^, \\ F_ &= \partial_ A_ - \partial_ A_\, \\ &= \delta_^ \delta_^ \end the expression inside the square bracket is \begin \frac & = - \ \frac\ \frac \\ & = - \ \frac\ \eta^\eta^ \left(F_\left(\delta^\beta_\mu \delta^\alpha_\nu - \delta^\beta_\nu \delta^\alpha_\mu\right) + F_\left(\delta^\beta_\lambda \delta^\alpha_\sigma - \delta^\beta_\sigma \delta^\alpha_\lambda\right) \right) \\ & = - \ \frac\,. \end The second term is \frac = - J^ \,. Therefore, the electromagnetic field's equations of motion are \frac = \mu_0 J^ \,. 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: \mathcal \, = \, - \frac F^ F_ - A_ J^_ + \frac 1 2 F_ \mathcal^ \,. Using Lagrange equation, the equations of motion for \mathcal^ can be derived. The equivalent expression in vector notation is: \mathcal \, = \, \frac 1 2 \left(\varepsilon_ E^2 - \frac B^2\right) - \phi \, \rho_ + \mathbf \cdot \mathbf_ + \mathbf \cdot \mathbf + \mathbf \cdot \mathbf \,.


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