It is used in the study of solutions of Maxwell's equations and has applications in Einstein's theory of relativity.
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 η = ηab by defining and examining the so-called "principal null directions".
Any nonzero bivector over a 4-dimensional vector space either is simple, or can be written as F = v ∧ w + x ∧ y, 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.
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: 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: where the rank refers to the rank of the linear operator F.[clarification needed] 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 where
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 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.
, there exists an inertial reference frame for which either the electric or magnetic field vanishes.