Properties of an affine transport equation and its holonomy

TL;DR

This paper studies the properties of an affine transport equation in general relativity, focusing on its generalized holonomy around loops, and derives higher-order corrections to the well-known linear holonomy results.

Contribution

It provides a detailed analysis of the local properties of the affine transport equation and computes higher-order corrections to the holonomy using covariant bitensor methods.

Findings

Recovered the leading-order linear holonomy for small loops

Derived the leading-order inhomogeneous part of the generalized holonomy

Computed higher-order corrections revealing finite-size effects

Abstract

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.