Electromagnetic tensor
Encyclopedia
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
of a physical system in Maxwell's theory of electromagnetism
. The field tensor was first used after the 4-dimensional tensor
formulation of special relativity
was introduced by Hermann Minkowski
. The tensor allows some physical laws to be written in a very concise form.
The electromagnetic tensor starts with the Electromagnetic four-potential
:
where
Electric
and magnetic field
s are derived from the vector potentials and the scalar potential with two formulas:
By definition, the electromagnetic tensor is the exterior derivative of the differential 1-form
:
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
of a physical system in Maxwell's theory of electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
. The field tensor was first used after the 4-dimensional tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
formulation of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
was introduced by Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi 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...
. The tensor allows some physical laws to be written in a very concise form.
Definition
- Mathematical note: In this article, the abstract index notationAbstract index notationAbstract 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 starts with the Electromagnetic four-potential
Electromagnetic four-potential
The electromagnetic four-potential is a potential from which the electromagnetic field can be derived. It combines both the electric scalar potential and the magnetic vector potential into a single space-time four-vector. In a given reference frame, the first component is the scalar potential and...
:
-
and its covariant form is found by multiplying by the Minkowski metric η of signature (+,−,−,−) :
where
-
is the vector potentialVector potentialIn 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 potentialScalar potentialA scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...
and
is the speed of lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
.
Electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
and magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
s are derived from the vector potentials and the scalar potential with two formulas:
By definition, the electromagnetic tensor is the exterior derivative of the differential 1-form
:
-
F is therefore a differential 2-form on spacetime. In an inertial frame, the matrices of F read:
-
or
-
Properties
From the matrix form of the field tensor, it becomes clear that the electromagnetic tensor satisfies the following properties:
- antisymmetryAntisymmetricThe word antisymmetric refers to a change to an opposite quantity when another quantity is symmetrically changed. This concept is related to that of Symmetry and Asymmetry. The difference between these three concepts can be simply illustrated with Latin letters. The character "A" is symmetric about...
:
(hence the name bivectorBivectorIn mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors...
). - 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 pseudoscalarPseudoscalarIn 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 symbolLevi-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...
. 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:
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 4π 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
-
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 relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
. 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 tensorTensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s. The tensor formalism also leads to a mathematically elegant presentation of physical laws. For example, Maxwell's equationsMaxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
of electromagnetism may be written using the field tensor as:
-
and 
The second equation implies conservation of chargeContinuity equationA continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
:
These laws can be generalised to curved spacetime by simply replacing partial with covariant derivatives:
-
and 
where the semi-colon represents a covariant derivativeCovariant 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 given...
, as opposed to a partial derivative. These equations are sometimes referred to as the curved space Maxwell equationsMaxwell's equations in curved spacetimeIn 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):
Lagrangian formulation of classical electromagnetism without charges and currents
When there are no electric charges (ρ=0) and no electric currents (j=0), Classical electromagnetismClassical 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 equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
can be derived from the actionAction (physics)In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
defined:
where-
is over space and time.
This means the LagrangianLagrangianThe 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 by Irish mathematician William Rowan Hamilton known as...
density is
|
|
|-
|
|
|-
|
|
|}
The far left and far right terms are the same because
and 
are just dummy indicesEinstein notationIn mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...
after all. The two middle terms are also the same, so the Lagrangian density is
|
|
|}
We can then plug this into the Euler-Lagrange equationEuler-Lagrange equationIn calculus of variations, the Euler–Lagrange equation, Euler's 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:
The quantity in parentheses above 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 inhomogeneous Maxwell's equations as long as you make the substitutions:
where
and 
take on the values of 1, 2, and 3.
When there are charges or currents, the Lagrangian needs an extra term to account for the coupling between them and the electromagnetic field. In that case
is equal to the 4-current instead of zero.
Role in quantum electrodynamics and field theory
The LagrangianLagrangianThe 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 by Irish mathematician William Rowan Hamilton known as...
of quantum electrodynamicsQuantum electrodynamicsQuantum electrodynamics 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 special relativity is achieved...
extends beyond the classical Lagrangian established in relativity, from
to incorporate the creation and annihilation of photons (and electrons).
In quantum field theoryQuantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
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
- Application of tensor theory in physics
- Classification of electromagnetic fieldsClassification of electromagnetic fieldsIn 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
-
-
-
-
- antisymmetry
-
-
