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. It is used in the study of solutions of Maxwell's equations and has applications in Einstein's theory of 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_br^b. Here, the vector space is the tangent space at the given event, and thus isomorphic as a (real) inner product space to E^1,3. That is, it has the same notion of vector 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

F^a{}_br^b \, =\lambda r^a

The skew-symmetry of the operator we are interested in now implies that one of the following must hold:

r is a 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 eigenspaces 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 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 field (the electromagnetic field tensor) so we immediately obtain an algebraic classification of electromagnetic fields.

In a cartesian chart on Minkowski spacetime, the electromagnetic field tensor has components

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)

where $E_x, E_y, E_z$ and $B_x, B_y, B_z$ 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 $c=1$ . In the "tensor gymanastics" formalism of special relativity, the Minkowski metric $\eta$ is used to raise and lower indices.

Invariants

The fundamental invariants of the electromagnetic field are:

P \equiv \frac{1}{2} F_{ab} \, F^{ab} = \| \vec{B} \|^2 - \frac{\| \vec{E} \|^2}{c^2} = -\frac{1}{2}{}^* F_{ab} \, {}^* F^{ab}

Q \equiv \frac{1}{4} F_{ab} \, {}^*F^{ab} =\frac{1}{8}\epsilon^{abcd}F_{ab}F_{cd}= \frac{\vec{E} \cdot \vec{B}}{c}

(Fundamental means that every other invariant can be expressed in terms of these two.)

A null electromagnetic field is characterised by $P = Q =0$ . 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.

A non-null field is characterised by $P^2+Q^2 \neq \, 0$ . If $P \neq 0 = Q$ , there exists an inertial reference frame for which either the electric or magnetic field vanishes. (These correspond respectively to magnetostatic and electrostatic fields.) If $Q \neq 0$ , 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 vacuum. Fortunately, 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.

References