经典电磁学透视
Inside Perspectives on Classical Electromagnetism
摘要：
本文标题中的“经典电磁学”概念指的是建立在三个基础上的理论：相对性原理、原始麦克斯韦方程组和外部微积分数学。在这个电磁学理论中，一般定律被表示为四维时空流形上的量之间的几何（无坐标）关系。基于这些定律和度量空间的外部微积分，我们推导出了在闵可夫斯基时空（普通的平坦时空）内，从洛伦兹观测者的内部视角描述电磁学的场和关系。它表明如何场和关系定义的时空被切成三维场和关系的时间片，洛伦兹观察者认为这些作为一个单一的三维欧几里德空间的时间依赖性的领域和关系。在外部微积分（度量空间）中，每个域都与四个域变量相关联，从中可以选择域的出现方式。利用这一特点，我们推导出替代方程系统描述电磁从内部的角度来看。在一个系统A中，所有的场量都是形式场。在另一个系统C中，场被表示为矢量场和标量场。但与传统的麦克斯韦方程组的div和curl算子相比，这里不需要伪矢量和伪标量。本文导出了适用于C系统的伪自由向量分析的若干定理。本文最后一节，我们分析的电磁场的性质和法律方面的镜像变换。
英文摘要：
The concept "Classical Electromagnetism" in the title of the paper here refers to a theory built on three foundations: relativity principles, the original Maxwell's equations, and the mathematics of exterior calculus. In this theory of electromagnetism the general laws are expressed as geometric (coordinate-free) relations between quantities on a four-dimensional spacetime manifold. Based on these laws, and exterior calculus on metric spaces, we derive fields and relations that describe electromagnetism from the inside perspective of Lorentz observers, inside the Minkowski spacetime (the ordinary flat spacetime). It is shown how fields and relations defined on the spacetime are sliced into three-dimensional fields and relations on time slices, and that Lorentz observers perceive these as time-dependent fields and relations on a single three-dimensional Euclidean space. In exterior calculus (on metric spaces) each field is associated to four field variants from which one can choose how a field is to appear. Using this feature, we derive alternative equation systems describing electromagnetism from the inside perspective. In one system, A, all field quantities are form fields. In another system, C, the fields are represented as vector and scalar fields. But in contrast to the traditional Maxwell's equations with div and curl operators, there is here no need for pseudo-vectors and pseudo-scalars. Some theorems of a pseudo-free vector analysis, adapted to the C-system, are derived. The paper concludes with a section where we analyse properties of the electromagnetic field and its laws with respect to mirror transformations.
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