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In [[differential geometry]] and [[theoretical physics]], the '''classification of electromagnetic fields''' is a pointwise classification of [[bivector]]s at each point of a [[Lorentzian manifold]]. It is used in the study of solutions of [[Maxwell's equations]] and has applications in Einstein's theory of [[general relativity]].

==The classification theorem==

A (real) bivector field may be viewed, at any given event in a spacetime, as a ''skew-symmetric'' [[linear operator]] on a four-dimensional (real) [[vector space]], ''r''''a'' → ''F''''a''''b''''r''''b''. Here, the vector space is the tangent space at the given event, and thus isomorphic as a (real) inner product space to E1,3. That is, it has the same notion of vector [[magnitude (mathematics)|magnitude]] and [[angle]] (or inner product) as [[Minkowski spacetime]].

In the remainder of this section (and in the next section), we'll assume our spacetime ''is'' Minkowski spacetime. This simplifies the mathematics (but tends to blur the distinction between the tangent space at an event and the underlying manifold). Fortunately, nothing will be lost by this apparently drastic specialization, for reasons we discuss as the end of the article.

In studying any linear operator, we confront the ''eigenvalue problem'', that is, the problem of finding [[eigenvalues]] ''λ'' and [[eigenvectors]] ''r'' which satisfy the ''eigenvalue equation''
: <math>F^a{}_br^b \, =\lambda r^a</math>
The skew-symmetry of the operator we are interested in now implies that one of the following must hold:
*''r'' is a [[null vector (Minkowski space)|null vector]] belonging to a nonzero eigenvalue
*''r'' is a nonnull eigenvector belonging to the eigenvalue zero
*''r'' is a null eigenvector belonging to the eigenvalue zero
The linearly independent null [[eigenspace]]s are called the ''principal null directions'' 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:
* one ''repeated'' principal null direction, in this case, the bivector is said to be ''null'',
* two ''distinct'' principal null directions, in this case, the bivector is said to be ''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, ''λ'' = ±''ν'', so we have three subclasses of non-null bivectors:
:*''spacelike'': ''ν'' = 0
:*''timelike'' : ''ν'' ≠ 0 and rank ''F'' = 2
:*''non-simple'': ''ν'' ≠ 0 and rank ''F'' = 4
where the rank refers to the [[rank (linear algebra)|rank]] of the linear operator ''F''. Every nonsimple bivector can be written as a ''sum'' of at most two simple ones.

==Physical interpretation==

The algebraic classification of bivectors given above has an important application in [[relativistic physics]]: the [[electromagnetic field]] is represented by a skew-symmetric second rank tensor (the electromagnetic field tensor) so we immediately obtain an algebraic classification of electromagnetic fields.

Recall that for in a cartesian chart on Minkowski spacetime, the [[electromagnetic field tensor]] has components
:<math>F_{ab} = \left(
\begin{matrix}
0 & B_z & -B_y & E_x/c \\
-B_z & 0 & B_x & E_y/c \\
B_y & -B_x & 0 & E_z/c \\
-E_x/c & -E_y/c & -E_z/c & 0
\end{matrix}
\right) </math>
where <math>E_x, E_y, E_z</math> and <math>B_x, B_y, B_z</math> 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 [[geometrized unit system|geometrised units]] in which <math>c=1</math>. In the "tensor gymanastics" formalism of special relativity, the [[Minkowski metric]] <math>\eta</math> is used to raise and lower indices.

===Invariants===

The fundamental invariants of the electromagnetic field are:
:<math> P \equiv \frac{1}{2} F_{ab} \, F^{ab} = \| \vec{B} \|^2 - \| \vec{E} \|^2 = -\frac{1}{2}{}^* F_{ab} \, {}^* F^{ab}</math>
:<math>Q \equiv \frac{1}{4} F_{ab} \, {}^*F^{ab} =\frac{1}{4}\epsilon^{abcd}F_{ab}F_{cd}= \vec{E} \cdot \vec{B}</math>.
(Fundamental means that every other invariant can be expressed in terms of these two.)

A '''null electromagnetic field''' is characterised by <math>P = Q =0</math>. 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 wave|plane electromagnetic wave]] in [[Minkowski space]].

A '''non-null field''' is characterised by <math>P^2+Q^2 \neq \, 0</math>. If <math>P \neq 0 = Q</math>, there exists an [[inertial frame]] for which either the electric or magnetic field vanishes. (These correspond respectively to ''magnetostatic'' and ''electrostatic'' fields.) If <math>Q \neq 0</math>, there exists an inertial frame in which electric and magnetic fields are proportional.

==Curved Lorentzian manifolds==

So far we have discussed only flat spacetime, i.e. the [[Minkowski spacetime|Minkowski vacuum]]. Fortunately, according to the (strong) equivalence principle, if we simply replace "inertial frame" above with a [[frame fields in general relativity|frame field]], everything works out exactly the same way on curved manifolds.

==See also==

*[[Electromagnetic peeling theorem]]
*[[Electrovacuum solution]]
*[[Lorentz group]]
*[[Petrov classification]]

==References==

*{{cite book | author=Landau, Lev D.; and Lifshitz, E. M. | title=The Classical Theory of Fields | location=New York | publisher=Pergamon | year=1973 | isbn=0-08-025072-6}} See ''section 25''.

[[Category:Mathematical physics]]
[[Category:Electromagnetism]]
[[Category:Lorentzian manifolds]]

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