物质介质中的时空形变与电磁学
Spacetime Deformations and Electromagnetism in Material Media
摘要：
本文研究了各向异性介质中的电动力学与具有非零黎曼曲率张量的时空中的电动力学之间的关系。通过手征微分形式讨论了电磁学，指出了与拓扑自旋密切相关的旋光性和与拓扑挠率相关的法拉第旋转。这两个量都用磁势、电场、磁场和激励来定义。我们重新审视材料介质和相关的格林并矢的一些性质。还研究了铁素体的一些相关特性。众所周知，本构张量本质上等价于黎曼曲率张量。为了研究电磁波在物质介质中的传播，我们证明了它类似于考虑电磁波在真空中的传播，但这次是在弯曲的时空中，这是通过变形闵可夫斯基时空的洛伦兹度规得到的。时空变形使麦克斯韦方程组的形式保持不变。另外，弯曲时空中的麦克斯韦方程组与各向异性介质中的麦克斯韦方程组之间存在着密切的联系，这表明电磁学与时空性质有着深刻的联系。此外，波传播的几何方面可以用一种有效几何来描述，它代表了闵可夫斯基时空的洛仑兹度规的变形。
英文摘要：
This paper is intended to investigate the relation between electrodynamics in anisotropic material media and its analogous formulation in an spacetime, with non-null Riemann curvature tensor. After discussing the electromagnetism via chiral differential forms, we point out the optical activity of a given material medium, closely related to topological spin, and the Faraday rotation, associated to topological torsion. Both quantities are defined in terms of the magnetic potential and the electric and magnetic fields and excitations. We revisit some properties of material media and the associated Green dyadics. Some related features of ferrite are also investigated. It is well-known that the constitutive tensor is essentially equivalent to the Riemann curvature tensor. In order to investigate the propagation of electromagnetic waves in material media, we prove that it is analogous to consider the electromagnetic wave propagation in the vacuum, but this time in a curved spacetime, which is obtained by a deformation of the lorenztian metric of Minkowski spacetime. Spacetime deformations leave invariant the form of Maxwell equations. Also, there exists a close relation between Maxwell equations in curved spacetime and in an anisotropic material medium, indicating that electromagnetism and spacetime properties are deeply related. Besides, the geometrical aspects of wave propagation can be described by an effective geometry which represents a deformation of the lorentzian metric of Minkowski spacetime.
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