In
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, 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
An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
in spacetime. The field tensor was developed by
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld (; 5 December 1868 – 26 April 1951) was a German Theoretical physics, theoretical physicist who pioneered developments in Atomic physics, atomic and Quantum mechanics, quantum physics, and also educated and ...
after the four-dimensional
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
formulation 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 ...
was introduced by
Hermann Minkowski
Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...
. The tensor allows related physical laws to be written concisely, and allows for the
quantization of the electromagnetic field by the Lagrangian formulation described
below.
Definition
The electromagnetic tensor, conventionally labelled ''F'', is defined as 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 ...
of the
electromagnetic four-potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
, ''A'', a differential 1-form:
:
Therefore, ''F'' is a
differential 2-form— an antisymmetric rank-2 tensor field—on Minkowski space. In component form,
:
where
is the
four-gradient and
is the
four-potential.
SI units for Maxwell's equations and the
particle physicist's sign convention for the
signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
of
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
, will be used throughout this article.
Relationship with the classical fields
The Faraday
differential 2-form is given by
:
where
is the time element times the speed of light
.
This 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 ...
of its 1-form antiderivative
:
,
where
has
(
is a scalar potential for the
irrotational/conservative vector field ) and
has
(
is a vector potential for the
solenoidal vector field ).
Note that
:
where
is the exterior derivative,
is the
Hodge star,
(where
is the
electric current density, and
is the
electric charge density) is the 4-current density 1-form, is the differential forms version of Maxwell's equations.
The
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
and
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 ...
s can be obtained from the components of the electromagnetic tensor. The relationship is simplest in
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
:
:
where ''c'' is the speed of light, and
:
where
is the
Levi-Civita tensor. This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor will
transform covariantly, and the fields in the new frame will be given by the new components.
In contravariant
matrix
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the m ...
form with metric signature (+,-,-,-),
:
The covariant form is given by
index lowering,
:
The Faraday tensor's
Hodge dual is
:
From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is assumed, and the electric and magnetic fields are with respect to the coordinate system's reference frame, as in the equations above.
Properties
The matrix form of the field tensor yields the following properties:
#
Antisymmetry
In linguistics, antisymmetry, is a theory of syntax described in Richard S. Kayne's 1994 book ''The Antisymmetry of Syntax''. Building upon X-bar theory, it proposes a universal, fundamental word order for phrases (Branching (linguistics), branchin ...
:
#Six independent components: In Cartesian coordinates, these are simply the three spatial components of the electric field (''E
x, E
y, E
z'') and magnetic field (''B
x, B
y, B
z'').
#Inner product: If one forms an inner product of the field strength tensor a
Lorentz invariant is formed
meaning this number does not change from one
frame of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system, whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale have been specified in physical space. It ...
to another.
#
Pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
A pseudoscalar, when multiplied by an ordinary vector, becomes a '' pseudovector'' ...
invariant: The product of the tensor
with its
Hodge dual gives a
Lorentz invariant:
where
is the rank-4
Levi-Civita symbol. The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here is
.
#
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
:
which is proportional to the square of the above invariant.
#
Trace:
which is equal to zero.
Significance
This tensor simplifies and reduces
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 ...
as four vector calculus equations into two tensor field equations. In
electrostatic
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word (), mean ...
s and
electrodynamics,
Gauss's law and
Ampère's circuital law are respectively:
:
and reduce to the inhomogeneous Maxwell equation:
:
, where
is the
four-current.
In
magnetostatic
Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equat ...
s and magnetodynamics,
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 ...
and
Maxwell–Faraday equation are respectively:
:
which reduce to the
Bianchi identity:
:
or using the
index notation with square brackets for the antisymmetric part of the tensor:
:
Using the expression relating the Faraday tensor to the four-potential, one can prove that the above antisymmetric quantity turns to zero identically (
). This tensor equation reproduces the homogeneous Maxwell's equations.
Relativity
The field tensor derives its name from the fact that the electromagnetic field is found to obey the
tensor transformation law, this general property of physical laws being recognised after the advent 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 ...
. This theory stipulated that all the laws of physics should take the same form in all coordinate systems – this led to the introduction of
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
s. The tensor formalism also leads to a mathematically simpler presentation of physical laws.
The inhomogeneous Maxwell equation leads to the
continuity equation:
:
implying
conservation of charge.
Maxwell's laws above can be generalised to
curved spacetime by simply replacing
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
s with
covariant derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
Statistics
* Covariance matrix, a matrix of covariances between a number of variables
* Covariance or cross-covariance between ...
s:
:
and
where the
semicolon notation represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the
curved space Maxwell equations. Again, the second equation implies charge conservation (in curved spacetime):
:
The stress-energy tensor of electromagnetism
:
satisfies
:
Lagrangian formulation of classical electromagnetism
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 ...
and
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 ...
can be derived from the
action:
where
is over space and time.
This means the
Lagrangian density is
:
The two middle terms in the parentheses are the same, as are the two outer terms, so the Lagrangian density is
:
Substituting this into the
Euler–Lagrange equation of motion for a field:
:
So the Euler–Lagrange equation becomes:
:
The quantity in parentheses above is just the field tensor, so this finally simplifies to
:
That equation is another way of writing the two inhomogeneous
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 ...
(namely,
Gauss's law and
Ampère's circuital law) using the substitutions:
:
where ''i, j, k'' take the values 1, 2, and 3.
Hamiltonian form
The
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
density can be obtained with the usual relation,
:
Here
are the fields and the momentum density of the EM field is
:
such that the conserved quantity associated with translation from
Noether's theorem
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ...
is the total momentum
:
The Hamiltonian density for the electromagnetic field is related to the
electromagnetic stress-energy tensor
:
as
:
where we have neglected the
energy density of matter, assuming only the EM field, and the last equality assumes the CGS system. The momentum of nonrelativistic charges interarcting with the EM field in the
Coulomb gauge (
) is
:
The total Hamiltonian of the matter + EM field system is
:
where for nonrelativistic point particles in the Coulomb gauge
:
where the last term is identically
where
and
:
where and
.
Quantum electrodynamics and field theory
The
Lagrangian of
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 ...
extends beyond the classical Lagrangian established in relativity to incorporate the creation and annihilation of photons (and electrons):
:
where the first part in the right hand side, containing the
Dirac spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain comb ...
, represents the
Dirac field
In quantum field theory, a fermionic field is a quantum field whose Quantum, quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relation ...
. In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
it is used as the template for the gauge field strength tensor. By being employed in addition to the local interaction Lagrangian it reprises its usual role in QED.
See also
*
Classification of electromagnetic fields
*
Covariant formulation of classical electromagnetism
*
Electromagnetic stress–energy tensor
*
Gluon field strength tensor
In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks.
The strong interaction is one of the fundamental interactions of nature, and the quantum fiel ...
*
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 ...
*
Riemann–Silberstein vector
Notes
References
*
*
*
{{tensors
Electromagnetism
Minkowski spacetime
Theory of relativity
Tensor physical quantities
Tensors in general relativity