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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 ''F'' = ''F''ab 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 ''η'' = ''η''ab by defining and examining the so-called "principal null directions". Let us explain this. The bivector ''F''ab yields a skew-symmetric linear operator ''F''''a''''b'' = ''F''ac''η''cb defined by lowering one index with the metric. It acts on the tangent space at ''p'' by ''r''''a'' → ''F''''a''''b''''r''''b''. 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 ''F'' = ''v'' ∧ ''w'', 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 ''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. 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. 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. ''η''(''r'',''r'') = 0), 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, ''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''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Classification of electromagnetic fields」の詳細全文を読む スポンサード リンク
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