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  Properties of an affine transport equation and its holonomy

Vines, J., & Nichols, D. A. (2016). Properties of an affine transport equation and its holonomy. General Relativity and Gravitation, 48(10): 127. doi:10.1007/s10714-016-2118-2.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-002C-3F40-C Version Permalink: http://hdl.handle.net/21.11116/0000-0003-0211-7
Genre: Journal Article

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 Creators:
Vines, Justin1, Author              
Nichols, David A., Author
Affiliations:
1Astrophysical and Cosmological Relativity, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_1933290              

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Free keywords: 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.

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 Dates: 2014-12-122016-09-192016
 Publication Status: Published in print
 Pages: 18 pages, 4 figures, new short section (Sec. 5) in v3; updated to match article published in GRG
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 Rev. Method: -
 Identifiers: arXiv: 1412.4077
DOI: 10.1007/s10714-016-2118-2
URI: http://arxiv.org/abs/1412.4077
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Title: General Relativity and Gravitation
Source Genre: Journal
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Publ. Info: Dordrecht, etc. : Kluwer Academic Publishers [etc.]
Pages: - Volume / Issue: 48 (10) Sequence Number: 127 Start / End Page: - Identifier: ISSN: 0001-7701
CoNE: /journals/resource/954925263179