论文标题
通过仿射载体束的重力理论中曲率和扭转的统一观点
A unified view of curvature and torsion in metric-affine gauge theory of gravity through affine-vector bundles
论文作者
论文摘要
公制植物量规理论的最吸引人的结果之一是黎曼曲率两种形式和纸笛扭转两种形式之间的相似之处：虽然前者是Lorentz-Group连接一型的场强，但后者可以理解为Coframe Onform的场强度。不幸的是，直到人们采用Trautman引入仿射矢量价值零从零from引入的想法之前，这种相似之处才被完全确定，其含义并没有令人满意地阐明。本文旨在从没有任何临时处方的第一原理中得出这一相似之处。我们提出了一个新的数学框架，将相关的仿射矢量捆绑包作为比传统矢量束更合适的舞台，并严格地在正式的ehresmann-connection方法中，对仿射矢量束的本地部分的协变量。 Riemann曲率与曲折扭转之间的相似之处自然出现在仿载体束上，它们的几何和物理含义变得透明。清晰的图片还导致了关于cartan扭转的运动学效应的猜想，从原则上讲，可以测量aharonov-bohm效应。
One of the most appealing results of metric-affine gauge theory of gravity is a close parallel between the Riemann curvature two-form and the Cartan torsion two-form: While the former is the field strength of the Lorentz-group connection one-form, the latter can be understood as the field strength of the coframe one-form. This parallel, unfortunately, is not fully established until one adopts Trautman's idea of introducing an affine-vector-valued zero-from, the meaning of which has not been satisfactorily clarified. This paper aims to derive this parallel from first principles without any ad hoc prescriptions. We propose a new mathematical framework of an associated affine-vector bundle as a more suitable arena for the affine group than a conventional vector bundle, and rigorously derive the covariant derivative of a local section on the affine-vector bundle in the formal Ehresmann-connection approach. The parallel between the Riemann curvature and the Cartan torsion arises naturally on the affine-vector bundle, and their geometric and physical meanings become transparent. The clear picture also leads to a conjecture about a kinematical effect of the Cartan torsion that in principle can be measured à la the Aharonov-Bohm effect.
