Properties of an affine transport equation and its holonomy
Abstract
An affine transport equation was used recently to study properties of angular momentum and gravitationalwave 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 wellknown result for the leadingorder linear holonomy (̃ Riemann × area), and we derive the leadingorder inhomogeneous part of the generalized holonomy (̃ Riemann × area^{3/2}). Our bitensor methods let us naturally compute higherorder corrections to these leading results. These corrections reveal the form of the finitesize effects that enter into the holonomy for larger loops; they could also provide quantitative errors on the leadingorder results for finite loops.
 Publication:

General Relativity and Gravitation
 Pub Date:
 October 2016
 DOI:
 10.1007/s1071401621182
 arXiv:
 arXiv:1412.4077
 Bibcode:
 2016GReGr..48..127V
 Keywords:

 Bitensors;
 Holonomies;
 Transport equations;
 General Relativity and Quantum Cosmology
 EPrint:
 18 pages, 4 figures, new short section (Sec. 5) in v3