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 The electromagnetic field at a point p (i.e. an event) of a Lorentzian spacetime is represented by a real bivector defined over the tangent space at p. The tangent space at p is isometric as a real inner product space to E1,3. That is, it has the same notion of vector magnitude and angle as Minkowski spacetime. 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 Fab 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 Fab has rank 2 in the former case and rank 4 in the latter case. To state the classification theorem, we consider the eigenvalue problem for F, that is, the problem of finding eigenvalues λ and eigenvectors r which satisfy the eigenvalue equation The skew-symmetry of F implies that: either the eigenvector r is a null vector (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.