In
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, 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. The field tensor was first used after the four-dimensional
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
formulation of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
was introduced by
Hermann Minkowski. The tensor allows related physical laws to be written very concisely, and allows for the
quantization of the electromagnetic field by Lagrangian formulation described
below.
Definition
The electromagnetic tensor, conventionally labelled ''F'', is defined as the
exterior derivative 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—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form,
:
where
is the
four-gradient and
is the
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 ...
.
SI units for Maxwell's equations and the
particle physicist's sign convention for the
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) 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. The writer of a ...
of
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, will be used throughout this article.
Relationship with the classical fields
The Faraday
differential 2-form is given by
:
This is the
exterior derivative 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
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
\nabla \cdot \mathbf ...
).
Hereis a video series b
Michael Pennexplaining the Faraday 2-form and its relations to
Maxwell's equations.
Note that
:
where
is the exterior derivative,
is the
Hodge star,
(where
is the
electric current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional are ...
, 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 that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic fields can be obtained from the components of the electromagnetic tensor. The relationship is simplest in
Cartesian coordinates:
:
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 form,
:
The covariant form is given by
index lowering,
:
The Faraday tensor's
Hodge dual
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the a ...
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 syntactic theory presented in Richard S. Kayne's 1994 monograph ''The Antisymmetry of Syntax''. It asserts that grammatical hierarchies in natural language follow a universal order, namely specifier-head-comple ...
:
#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
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
is formed
meaning this number does not change from one
frame of reference 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.
Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. T ...
invariant: The product of the tensor
with its
Hodge dual
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the a ...
gives a
Lorentz invariant
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
:
where
is the rank-4
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 parity of a permutation, sign of a permutation of the n ...
. The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here is
.
#
Determinant:
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 as four vector calculus equations into two tensor field equations. In
electrostatics and
electrodynamic
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
s,
Gauss's law and
Ampère's circuital law are respectively:
:
and reduce to the inhomogeneous Maxwell equation:
:
, where
is the
four-current.
In
magnetostatics 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
Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic induct ...
are respectively:
:
which reduce to
Bianchi identity:
:
or using the
index notation with square brackets for the antisymmetric part of the tensor:
:
Relativity
The field tensor derives its name from the fact that the electromagnetic field is found to obey the
tensor transformation law
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
, 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 regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
. 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 related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
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 derivatives with
covariant derivatives:
:
and
where the semi-colon
notation
In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefore, ...
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):
:
Lagrangian formulation of classical electromagnetism
Classical electromagnetism
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
and
Maxwell's equations can be derived from the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
:
where
is over space and time.
This means the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
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
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
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 (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,
:
.
Quantum electrodynamics and field theory
The
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
of
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
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 , represents the
Dirac field. In
quantum field theory 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
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 that is manifestly invariant under Lorentz transformatio ...
*
Electromagnetic stress–energy tensor
*
Gluon field strength tensor
*
Ricci calculus
*
Riemann–Silberstein vector
In mathematical physics, in particular electromagnetism, the Riemann–Silberstein vector or Weber vector named after Bernhard Riemann, Heinrich Martin Weber and Ludwik Silberstein, (or sometimes ambiguously called the "electromagnetic field") is ...
Notes
References
*
*
*
{{tensors
Electromagnetism
Minkowski spacetime
Theory of relativity
Tensor physical quantities
Tensors in general relativity