非线性共形电磁与引力
Nonlinear Conformal Electromagnetism and Gravitation
摘要：
1909年，E.和F. Cosserat发现了一种新的非线性群理论方法来研究弹性（EL），只需要测量EL常数。在现代语言中，他们的想法是使用非线性斯宾塞序列代替非线性珍妮特序列来定义李群胚，李群胚定义空间的刚性运动群。继H. Weyl，我们的目的是第一次计算定义时空共形群的李群胚的非线性Spencer序列，以提供电磁学（EM）和引力的物理基础，唯一的实验需要是测量真空中的EM常数和引力常量。对于维数为$n$的流形，困难在于处理被E. 1922年的卡尔丹。使用的事实，尺寸$n=4$具有非常具体的性能的斯宾塞上同调的计算，我们证明，有没有概念之间的差异Cosserat EL场或感应方程和麦克斯韦电磁场或感应方程。作为副产品，公知的场/物质耦合（压电、光弹性等）可以抽象地描述，只有实验需要测量相应的耦合常数。在引力的研究中，维数$n=4$也允许在除中心吸引质量外的任何地方定义一个共形因子，严格二阶喷流形成的子群胚的反演律将吸引转化为排斥。
英文摘要：
In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the nonlinear Spencer sequence for the Lie groupoid defining the conformal group of space-time in order to provide the physical foundations of both electromagnetism (EM) and gravitation, with the only experimental need to measure the EM constant in vacuum and the gravitational constant. With a manifold of dimension $n$, the difficulty is to deal with the $n$ nonlinear transformations that have been called "elations" by E. Cartan in 1922. Using the fact that dimension $n=4$ has very specific properties for the computation of the Spencer cohomology, we prove that there is no conceptual difference between the Cosserat EL field or induction equations and the Maxwell EM field or induction equations. As a byproduct, the well known field/matter couplings (piezzoelectricity, photoelasticity, ...) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. In the sudy of gravitation, the dimension $n=4$ also allows to have a conformal factor defined everywhere but at the central attractive mass and the inversion law of the subgroupoid made by strict second order jets transforms attraction into repulsion.
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