In this paper we first show that any coupled system consisting of a gravitational plus a free electromagnetic field can be described geometrically in the sense that both Maxwell equations and Einstein equation having as source term the energy-momentum of the electromagnetic field can be derived from a geometrical Lagrangian proportional to the scalar curvature R of a particular kind of Riemann-Cartan spacetime structure, where those fields are identified as geometrical objects of the structure. We show moreover that the contorsion tensor of the particular Riemann-Cartan spacetime structure of our theory encodes the same information as the one contained in Chern-Simons term that is proportional to the spin density of the electromagnetic field. Next we show that by adding to the geometrical Lagrangian a term describing the interaction of a electromagnetic current with a general electromagnetic plus the gravitational field and a term describing the matter carrier of the current we get Maxwell equations with source term and Einstein equation having as source term the sum of the energy-momentum tensors of the electromagnetic and matter terms. Finally modeling by dust charged matter the carrier of the electromagnetic current we get the Lorentz force equation. Moreover, we prove that our theory is gauge invariant. We also briefly discuss our reasons for the present enterprise.
The mathematical foundations of general relativity revisited
本文的目的是第一次提出一个初步的总结，通过应用偏微分方程和李伪群的形式理论，以重温广义相对论的数学基础所获得的一些最近的结果。为了说明我们将提供的三个基本结果，还将考虑其他工程实例（控制理论、弹性理论、电磁学）。因此，本文分为三个部分，对应于不同的形式化方法。1)Cartan VS VESSIOT：根据“Vessiot结构方程“出现在“Riemann张量“中的二次项不能等同于李群的著名“Cartan结构方程“中出现的二次项，对于“Weyl张量“也可以做类似的评论。特别是，“曲率+挠率”（Cartan）不能被认为是“仅曲率”（Vessiot）的推广。粗略地说，嘉当和追随者还没有能够“商下降到基流形“，结果只获得了斯宾塞在1970年通过“非线性斯宾塞序列”，但在一种方式完全不同的一个其次是维西奥在1903年为同一目的，仍然被忽视。2)珍妮特VS斯宾塞：“Ricci张量“只依赖于非线性变换（Cartan在1922年称之为“elations“），它描述了外尔群（庞加莱子群的10个参数+ 1个伸缩）和时空共形群（15个参数）之间存在的“差异“。它可以通过正则分裂来定义，也就是说，不使用导致黎曼张量的标准收缩或迹的指数。同时，我们将得到Riemann张量和Weyl张量的分支数，而不需要任何关于指数交换的组合论论证。因此，共形Killing系统的Spencer序列及其形式伴随完全描述了Cosse…
