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General Relativity and Quantum Cosmology, gr-qc
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.
