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 fieldThe electromagnetic field is a physical field produced by electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. Light is the electromagnetic field in a certain frequency range...
of a physical system in Maxwell's theory of
electromagnetismElectromagnetism is the physics of the electromagnetic field, a field that exerts a force on particles with the property of electric charge and is reciprocally affected by the presence and motion of such particles....
. The field tensor was first used after the 4-dimensional
tensorTensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
formulation of
special relativitySpecial relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"...
introduced by
Hermann MinkowskiHermann Minkowski was a German mathematician of Polish Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born in...
. The tensor allows some physical laws to be written in a very concise form.
- Mathematical note: In this article, the abstract index notation
Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis, and in particular are non-numerical...
will be used.
The electromagnetic tensor is commonly written as a matrix:
or
-
- where
- E is the electric field
In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. This electric field exerts a force on other electrically charged objects...
,
- B the magnetic field
Magnetic fields surround magnetic materials and electric currents and are detected by the force they exert on other magnetic materials and moving electric charges...
, and
- c the speed of light
In physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
.
- Caution: The signs in the tensor above depend on the convention used for the metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
.
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 fieldThe electromagnetic field is a physical field produced by electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. Light is the electromagnetic field in a certain frequency range...
of a physical system in Maxwell's theory of
electromagnetismElectromagnetism is the physics of the electromagnetic field, a field that exerts a force on particles with the property of electric charge and is reciprocally affected by the presence and motion of such particles....
. The field tensor was first used after the 4-dimensional
tensorTensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
formulation of
special relativitySpecial relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"...
introduced by
Hermann MinkowskiHermann Minkowski was a German mathematician of Polish Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born in...
. The tensor allows some physical laws to be written in a very concise form.
Details
- Mathematical note: In this article, the abstract index notation
Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis, and in particular are non-numerical...
will be used.
The electromagnetic tensor is commonly written as a matrix:
or
-
- where
- E is the electric field
In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. This electric field exerts a force on other electrically charged objects...
,
- B the magnetic field
Magnetic fields surround magnetic materials and electric currents and are detected by the force they exert on other magnetic materials and moving electric charges...
, and
- c the speed of light
In physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
.
- Caution: The signs in the tensor above depend on the convention used for the metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
. The convention used here is +---, corresponding to the metric tensor:
Properties
From the matrix form of the field tensor, it becomes clear that the electromagnetic tensor satisfies the following properties:
- antisymmetry
In set theory, the adjective antisymmetric usually refers to an antisymmetric relation.The term "antisymmetric function" is sometimes used for odd function, although some meanings of antisymmetric are essentiality f = −f...
: (hence the name bivector).
- six independent components.
If one forms an inner product of the field strength tensor a Lorentz invariant is formed:
The product of the tensor with its dual tensor gives the
pseudoscalarIn physics, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as improper rotations while a true scalar does not.The prototypical example of a pseudoscalar is the scalar triple product...
invariant:
where is the completely antisymmetric unit pseudotensor of the fourth rank or
Levi-Civita symbolThe Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus. It is named after the Italian mathematician and physicist Tullio Levi-Civita.-Definition:In three dimensions, the Levi-Civita...
. Caution: the sign for the above invariant depends on the convention used for the Levi-Civita symbol. The convention used here is = +1.
Notice that:
More formally, the electromagnetic tensor may be written in terms of the
4-vector potentialThe electromagnetic four-potential is a covariant four-vector consisting of the electric potential and magnetic vector potential. All formulas in this article are given in SI units, and in parentheses in Gaussian-cgs units...
:
Where the 4-vector potential is:
-
- and its covariant form is found by multiplying by the Minkowski metric :
Derivation
To derive all the elements in the electromagnetic tensor we need to define the derivative operator:
and the
4-vector potentialThe electromagnetic four-potential is a covariant four-vector consisting of the electric potential and magnetic vector potential. All formulas in this article are given in SI units, and in parentheses in Gaussian-cgs units...
:
where
-
- is the vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
and are its components
- is the scalar potential
A scalar potential is a fundamental concept in vector analysis and physics...
and
- is the speed of light.
Electric and magnetic fields are derived from the vector potentials and the scalar potential with two formulas:
-
As an example, the x components are just
-
Using the definitions we began with, we can rewrite these two equations to look like:
-
Evaluating all the components results in a second-rank, antisymmetric and covariant tensor:
Thus, for example,
and
Compare with the matrix above.
Significance
Hidden beneath the surface of this complex mathematical equation is an ingenious unification of Maxwell's equations for electromagnetism. Consider the electrostatic equation
which tells us that the divergence of the electric field vector is equal to the charge density, and the electrodynamic equation
that is the change of the electric field with respect to time, minus the curl of the magnetic field vector, is equal to negative 4Ï€ times the current density.
These two equations for electricity reduce to
where
-
- is the 4-current.
The same holds for magnetism. If we take the magnetostatic equation
which tells us that there are no "true" magnetic charges, and the magnetodynamics equation
which tells us the change of the magnetic field with respect to time plus the curl of the Electric field is equal to zero (or, alternatively, the curl of the electric field is equal to the negative change of the magnetic field with respect to time). With the electromagnetic tensor, the equations for magnetism reduce to
where the comma indicates a
partial derivativeIn 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...
Relation to Lagrangian formulation of classical electromagnetism
Classical electromagnetismClassical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...
and
Maxwell's equationsMaxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
can be derived from the
actionIn physics, action is an attribute of the development of a physical system. It is a functional which takes the trajectory of the system as its argument and returns a real number as the result....
defined:
where
-
- is over space and time.
This means the
LagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. In classical mechanics, the...
is
|
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|-
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|-
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The far left and far right term are the same, because and are just
variables of integrationIn mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place...
after all. The two middle terms are also the same, so the Lagrangian is
|
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We can then plug this into the
Euler-Lagrange equationIn calculus of variations, the Euler–Lagrange equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...
of motion for a field:
The second term is zero, because the Lagrangian in this case only contains derivatives. So the Euler-Lagrange equation becomes:
That term in the parenthesis is just the field tensor, so this finally simplifies to
-
- {|cellpadding="2" style="border:2px solid #ccccff"
|
|}
That equation is just another way of writing the two homogeneous Maxwell's equations as long as you make the substitutions:
-
where and take on the values of 1, 2, and 3.
The field tensor and 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 (non-gravitational) physical laws being recognised after the advent of
special relativitySpecial relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"...
. This theory stipulated that all the (non-gravitational) laws of physics should take the same form in all coordinate systems - this led to the introduction of
tensorTensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
s. The tensor formalism also leads to a mathematically elegant presentation of physical laws. For example,
Maxwell's equationsMaxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
of electromagnetism may be written using the field tensor as:
-
- and
The second equation implies
conservation of chargeA continuity equation in physics is a differential equation that describes the transport of some kind of conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved, a vast variety of physics may be described with continuity equations.Continuity...
:
In general relativity, these laws can be generalised in (what many physicists consider to be) an appealing way:
-
- and
where the semi-colon represents a
covariant derivativeIn mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach...
, as opposed to a partial derivative. The elegance of these equations stems from the simple replacing of partial with covariant derivatives. These equations are sometimes referred to as the
curved space Maxwell equationsIn physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system...
. Again, the second equation implies charge conservation (in curved spacetime):
Role in quantum electrodynamics and field theory
The
LagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. In classical mechanics, the...
of
quantum electrodynamicsQuantum electrodynamics is a relativistic quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s. It basically describes how light and matter interact. More specifically it deals with the interactions between electrons, positrons and photons...
extends beyond the classical Lagrangian established in relativity from to incorporate the creation and annihilation of photons (and electrons).
In
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
, it is used for the template of the gauge field strength tensor. That is used in addition to the local interaction Lagrangian, nearly identical to its role in QED.
See also
- Application of tensor theory in physics
Tensors are used in various parts of physics, both as abstract constructs in mathematical physics and for describing relations between quantities represented by matrices.- Common applications :* Electromagnetic tensor in electromagnetism...
- Classification of electromagnetic fields
In differential geometry and theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold....
- Covariant formulation of classical electromagnetism