零场电场线和磁场线的协变描述：对Hopf-Ranada解的应用
The covariant description of electric and magnetic field lines of null fields: application to Hopf-Ranada solutions
摘要：
电场线和磁场线的概念本质上是非相对论的。然而，对于满足某些几何性质的某些类型的场，场线可以协变地定义。更精确地说，可以定义时空中的两个洛伦兹不变的2D表面，使得对于任何观察者来说，磁场线和电场线都可以通过这些表面与类空超平面的相交来确定。这种类型的场的一个例子是由无源麦克斯韦方程组的所谓的霍普夫-拉纳达解构成的，由于它们有趣的拓扑性质，即它们的场线的链接，它们已经被研究。为了以简洁的方式描述几何和拓扑性质，我们采用几何代数（又名Clifford代数）的工具，并使用Clebsch表示的矢量势以及欧拉表示的磁场和电场。这种描述很容易协变，从而使我们能够定义电力和磁场线协变在一个紧凑的几何语言。场线的定义可以用空间中的2D表面来表述。我们在不同的参考系中显示这些表面，显示这些表面在洛伦兹变换下如何变化，同时保持其拓扑特性。作为副产品，我们还获得光学螺旋度，光学手征和推广，和他们的守恒定律之间的关系。
英文摘要：
The concept of electric and magnetic field lines is intrinsically non-relativistic. Nonetheless, for certain types of fields satisfying certain geometric properties, field lines can be defined covariantly. More precisely, two Lorentz-invariant 2D surfaces in spacetime can be defined such that magnetic and electric field lines are determined, for any observer, by the intersection of those surfaces with spacelike hyperplanes. An instance of this type of field is constituted by the so-called Hopf-Ranada solutions of the source-free Maxwell equations, which have been studied because of their interesting topological properties, namely, linkage of their field lines. In order to describe both geometric and topological properties in a succinct manner, we employ the tools of Geometric Algebra (aka Clifford Algebra) and use the Clebsch representation for the vector potential as well as the Euler representation for both magnetic and electric fields. This description is easily made covariant, thus allowing us to define electric and magnetic field lines covariantly in a compact geometric language. The definitions of field lines can be phrased in terms of 2D surfaces in space. We display those surfaces in different reference frames, showing how those surfaces change under Lorentz transformations while keeping their topological properties. As a byproduct we also obtain relations between optical helicity, optical chirality and generalizations thereof, and their conservation laws.
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