标题： 仿射传输方程及其整体性的性质 显示英文标题

标题： Properties of an affine transport equation and its holonomy

摘要： 最近，一个仿射传输方程被用来研究广义相对论中角动量和引力波记忆效应的性质。 在本文中，我们更详细地研究了这个传输方程的局部性质。 与这个传输方程相关的是曲线上两点切空间之间的映射。 该映射由沿曲线的平行传输映射给出的齐次（线性）部分以及一个非齐次部分组成，该部分与流形中曲线发展到仿射切空间有关。 对于闭合曲线，仿射传输方程定义了一个“广义全纯性”，它在切空间上表现为一个仿射映射。 我们通过使用协变双张量方法来计算测地多边形环路周围的广义全纯性，以探索这种广义全纯性的局部性质。 我们关注由测地线段组成的三角形和“平行六面体”结构。 对于小环路，我们恢复了关于一阶线性全纯性的已知结果（$\sim$ Riemann $\times$ 面积），并推导了广义全纯性的首阶非齐次部分（$\sim$ Riemann $\times$ 面积$^{3/2}$）。 我们的双张量方法使我们能够自然地计算这些首阶结果的高阶修正。 这些修正揭示了进入较大环路全纯性的有限尺寸效应的形式；它们也可能为有限环路的首阶结果提供定量误差。 显示英文摘要

摘要： An affine transport equation was used recently to study properties of angular momentum and gravitational-wave memory effects in general relativity. In this paper, we investigate local properties of this transport equation in greater detail. Associated with this transport equation is a map between the tangent spaces at two points on a curve. This map consists of a homogeneous (linear) part given by the parallel transport map along the curve plus an inhomogeneous part, which is related to the development of a curve in a manifold into an affine tangent space. For closed curves, the affine transport equation defines a "generalized holonomy" that takes the form of an affine map on the tangent space. We explore the local properties of this generalized holonomy by using covariant bitensor methods to compute the generalized holonomy around geodesic polygon loops. We focus on triangles and "parallelogramoids" with sides formed from geodesic segments. For small loops, we recover the well-known result for the leading-order linear holonomy ($\sim$ Riemann $\times$ area), and we derive the leading-order inhomogeneous part of the generalized holonomy ($\sim$ Riemann $\times$ area$^{3/2}$). Our bitensor methods let us naturally compute higher-order corrections to these leading results. These corrections reveal the form of the finite-size effects that enter into the holonomy for larger loops; they could also provide quantitative errors on the leading-order results for finite loops.