In
differential geometry and
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, the classification of electromagnetic fields is a
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
classification of
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
s at each point of a
Lorentzian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
. It is used in the study of solutions of
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, and electric circuits.
...
and has applications in Einstein's
theory of relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
.
The classification theorem
The electromagnetic field at a point ''p'' (i.e. an event) of a Lorentzian spacetime is represented by a
real
Real may refer to:
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* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
bivector defined over the tangent space at ''p''.
The tangent space at ''p'' is isometric as a real inner product space to E
1,3. That is, it has the same notion of vector
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
and
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
as
Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
. To simplify the notation, we will assume the spacetime ''is'' Minkowski spacetime. This tends to blur the distinction between the tangent space at ''p'' and the underlying manifold; fortunately, nothing is lost by this specialization, for reasons we discuss as the end of the article.
The classification theorem for electromagnetic fields characterizes the bivector ''F'' in relation to the Lorentzian metric by defining and examining the so-called "principal null directions". Let us explain this.
The bivector ''F''
''ab'' yields a
skew-symmetric linear operator defined by lowering one index with the metric. It acts on the tangent space at ''p'' by . We will use the symbol ''F'' to denote either the bivector or the operator, according to context.
We mention a dichotomy drawn from exterior algebra. A bivector that can be written as , where ''v'', ''w'' are linearly independent, is called ''simple''. Any nonzero bivector over a 4-dimensional vector space either is simple, or can be written as , where ''v'', ''w'', ''x'', and ''y'' are linearly independent; the two cases are mutually exclusive. Stated like this, the dichotomy makes no reference to the metric ''η'', only to exterior algebra. But it is easily seen that the associated skew-symmetric linear operator ''F''
''a''''b'' has rank 2 in the former case and rank 4 in the latter case.
[The rank given here corresponds to that as a linear operator or tensor; the rank as defined for a ''k''-vector is half that given here.]
To state the classification theorem, we consider the ''eigenvalue problem'' for ''F'', that is, the problem of finding
eigenvalues ''λ'' and
eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
''r'' which satisfy the eigenvalue equation
:
The skew-symmetry of ''F'' implies that:
* ''either'' the eigenvector ''r'' is a
null vector
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which .
In the theory of real bilinear forms, definite quadratic forms an ...
(i.e. ), ''or'' the eigenvalue ''λ'' is zero, ''or both''.
A 1-dimensional subspace generated by a null eigenvector is called a ''principal null direction'' of the bivector.
The classification theorem characterizes the possible principal null directions of a bivector. It states that one of the following must hold for any nonzero bivector:
* the bivector has one "repeated" principal null direction; in this case, the bivector itself is said to be ''null'',
* the bivector has two distinct principal null directions; in this case, the bivector is called ''non-null''.
Furthermore, for any non-null bivector, the two eigenvalues associated with the two distinct principal null directions have the same magnitude but opposite sign, , so we have three subclasses of non-null bivectors:
:*''spacelike'': ''ν'' = 0
:*''timelike'' : ''ν'' ≠ 0 and
:*''non-simple'': ''ν'' ≠ 0 and ,
where the rank refers to the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
of the linear operator ''F''.
Physical interpretation
The algebraic classification of bivectors given above has an important application in
relativistic physics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...
: the
electromagnetic field is represented by a skew-symmetric second rank tensor field (the
electromagnetic field tensor) so we immediately obtain an algebraic classification of electromagnetic fields.
In a cartesian chart on
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 ...
time, the electromagnetic field tensor has components
:
where
and
denote respectively the components of the electric and magnetic fields, as measured by an inertial observer (at rest in our coordinates). As usual in relativistic physics, we will find it convenient to work with
geometrised units in which
. In the "
Index gymnastics" formalism of special relativity, the
Minkowski metric
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 ...
is used to raise and lower indices.
Invariants
The fundamental invariants of the electromagnetic field are:
:
:
.
(Fundamental means that every other invariant can be expressed in terms of these two.)
A null electromagnetic field is characterised by
. In this case, the invariants reveal that the electric and magnetic fields are perpendicular and that they are of the same magnitude (in geometrised units). An example of a null field is a
plane electromagnetic wave in
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 ...
.
A non-null field is characterised by
. If
, there exists an
inertial reference frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
for which either the electric or magnetic field vanishes. (These correspond respectively to ''magnetostatic'' and ''electrostatic'' fields.) If
, there exists an inertial frame in which electric and magnetic fields are proportional.
Curved Lorentzian manifolds
So far we have discussed only
Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
. According to the (strong) equivalence principle, if we simply replace "inertial frame" above with a
frame field, everything works out exactly the same way on curved manifolds.
See also
*
Electromagnetic peeling theorem
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 o ...
*
Electrovacuum solution In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass–energy present is the field energy of an electromagnetic field, which must satisfy the (c ...
*
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
*
Petrov classification
Notes
References
*{{cite book , author1=Landau, Lev D. , author2=Lifshitz, E. M. , title=The Classical Theory of Fields , location=New York , publisher=Pergamon , year=1973 , isbn=0-08-025072-6 See ''section 25''.
Mathematical physics
Electromagnetism
Lorentzian manifolds
Scientific classification