Energy and electromagnetism of a differential k-form

• Published in: Journal of mathematical physics 2012-10, Vol.53 (10)
• Main Authors: Navarro, J., Sancho, J. B.
• Format: Article
• Language: English
• Subjects: Charged particles; Divergence; Electromagnetic fields; Electromagnetism; Exact sciences and technology; Grounds; Manifolds; Mathematical analysis; Mathematical methods in physics; Mathematics; Physics; Sciences and techniques of general use; Tensors
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.4754817

Let X be a smooth manifold of dimension 1 + n endowed with a Lorentzian metric g. The energy tensor of a 2-form F is locally defined as \documentclass[12pt]{minimal}\begin{document}$T_{ab}\break := \, - \left( {F_a}^{i} F_{bi} - \frac{1}{4} \, F^{ij} F_{ij} g_{ab} \right)$\end{document} T a b : = − F a i F b i − 1 4 F i j F i j g a b . In this paper we characterize this tensor as the only 2-covariant natural tensor associated to a Lorentzian metric and a 2-form that is independent of the unit of scale and satisfies certain condition on its divergence. This characterization is motivated on physical grounds, and can be used to justify the Einstein–Maxwell field equations. More generally, we characterize in a similar manner the energy tensor associated to a differential form of arbitrary order k. Finally, we develop a generalized theory of electromagnetism where charged particles are not punctual, but of an arbitrary fixed dimension p. In this theory, the electromagnetic field F is a differential form of order 2 + p and its electromagnetic energy tensor is precisely the energy tensor associated to F.