NDLI: Properties of an affine transport equation and its holonomy

Content Provider	Springer Nature Link
Author	Vines, Justin Nichols, David A.
Copyright Year	2016
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.
Starting Page	1
Ending Page	18
Page Count	18
File Format	PDF
ISSN	00017701
Journal	General Relativity and Gravitation
Volume Number	48
Issue Number	10
e-ISSN	15729532
Language	English
Publisher	Springer US
Publisher Date	2016-08-31
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Bitensors Holonomies Transport equations Theoretical, Mathematical and Computational Physics Classical and Quantum Gravitation, Relativity Theory Differential Geometry Astronomy, Astrophysics and Cosmology Quantum Physics
Content Type	Text
Resource Type	Article
Subject	Physics and Astronomy

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