Energy and electromagnetism of a differential k-form

Abstract

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 asIn 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 .

Electromagnetism in the Electrical Engineering Classroom: Dominant trends in teaching classical electromagnetic field theory and innovation vectors

“…Casting the classical EM field theory into a differential forms formalism originates in the pioneering works [51]- [54] and was elaborated upon in [55]. Differential forms were applied in EM over a very broad range, starting with basic formulations [56] up to high-level studies [57]. The main difference with respect to the prevalent, vector mathematic representation of EM quantities is dropping the continuous r-dependence, which was unwarranted in this case, and replacing it with a small integrals interpretation.…”

Continuum mechanics, stresses, currents and electrodynamics

“…[20][21][22]), p-form electrodynamics (e.g. [23,24]), as a natural particular case of the theory. It is observed that current formulations of p-form electrodynamics use additional geometric structure and consider smooth fields.…”

“…[20][21][22]), p-form electrodynamics (e.g. [23,24]), as a natural particular case of the theory. It is observed that .…”

Noether's Second Theorem on natural bundles

“…In [5] it is proved, under some additional assumptions, that such a T has to be (E, ∂F ), where E ab := −(F a i F bi − 1 4 F ij F ij g ab ) stands for the energy tensor of the 2-form F = dA. As it is well-known, this source equation (E, ∂F ) is locally variational.…”

“…On the other hand, let E ab := − F a i F bi − 1 4 F ij F ij g ab be the usual energy tensor of the 2-form F = dA. This tensor E can be characterized as the only 2-tensor that is natural, satisfies div g E = −i ∂F F and fulfils certain homogeneity and normalization conditions ( [5]).…”