Properties of an affine transport equation and its holonomy

Vines, Justin; Nichols, David A.

doi:10.1007/s10714-016-2118-2

Cited by 6 publications

(14 citation statements)

References 21 publications

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“…Note that Eq. (3.26) does not contain any acceleration terms at this order; moreover, it reduces to the results of [58] for the metric-compatible connection on the tangent bundle. Now, we consider the holonomy around a square, such as that given in Fig.…”

Section: = γC ∇Cmentioning

confidence: 77%

“…We then provided the machinery with which one can calculate these observables in an arbitrary spacetime (which included reviewing the very powerful technique of covariant bitensors for understanding how tensor fields evolve along curves). Extending the results of [42,58], we used these techniques to compute the holonomy with respect to an arbitrary connection around a variety of curves, as well as the evolution of the separation vector between two arbitrary worldlines. We then used these holonomies and the separation vector to compute our final results, which are in Eqs.…”

Section: Discussionmentioning

confidence: 99%

“…IV. In the following three subsections, we follow the formalisms of [33] for introducing the idea of using a connection on a vector bundle to understand linear and angular momentum transport, [37] for a brief review of bitensors, and [58] for computations of holonomies.…”

Section: Review Of Techniques Of Covariant Bitensorsmentioning

confidence: 99%

“…First, following [58], we show that the holonomy around a (nongeodesic) triangle is given by expressions involving the Riemann tensor associated with the connection on the vector bundle. Explicitly, consider the triangle depicted in Fig.…”

Section: Nongeodesic Polygonsmentioning

confidence: 99%

“…FIG.4. A nongeodesic triangle (generalizing Fig.3of[58]), where the two sides γ and γ are arbitrary curves (with affine parameter lengths and ¯ ), and where the third side is formed by joining the two end points by the unique geodesic extending between them.…”

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Persistent gravitational wave observables: General framework

et al. 2019

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In the first paper in this series, we introduced "persistent gravitational wave observables" as a framework for generalizing the gravitational wave memory effect. These observables are nonlocal in time and nonzero in the presence of gravitational radiation. We defined three specific examples of persistent observables: a generalization of geodesic deviation that allowed for arbitrary acceleration, a holonomy observable involving a closed curve, and an observable that can be measured using a spinning test particle. For linearized plane waves, we showed that our observables could be determined just from one, two, and three time integrals of the Riemann tensor along a central worldline, when the observers follow geodesics. In this paper, we compute these three persistent observables in nonlinear plane wave spacetimes, and we find that the fully nonlinear observables contain effects that differ qualitatively from the effects present in the observables at linear order. Many parts of these observables can be determined from two functions, the transverse Jacobi propagators, and their derivatives (for geodesic observers). These functions, at linear order in the spacetime curvature, reduce to the one, two, and three time integrals of the Riemann tensor mentioned above. CONTENTS

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Section: = γC ∇Cmentioning

confidence: 77%

Section: Discussionmentioning

confidence: 99%

Section: Review Of Techniques Of Covariant Bitensorsmentioning

confidence: 99%

Section: Nongeodesic Polygonsmentioning

confidence: 99%

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confidence: 99%

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Persistent gravitational wave observables: General framework

et al. 2019

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Persistent gravitational wave observables: nonlinearities in (non-)geodesic deviation

Grant

2024

Class. Quantum Grav.

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The usual gravitational wave memory effect can be understood as a change in the separation of two initially comoving observers due to a burst of gravitational waves. Over the past few decades, a wide variety of other, ‘persistent’ observables which measure permanent effects on idealized detectors have been introduced, each probing distinct physical effects. These observables can be defined in (regions of) any spacetime where there exists a notion of radiation, such as perturbation theory off of a fixed background, nonlinear plane wave spacetimes, or asymptotically flat spacetimes. Many of the persistent observables defined in the literature have only been considered in asymptotically flat spacetimes, and the perturbative nature of such calculations has occasionally obscured deeper relationships between these observables that hold more generally. The goal of this paper is to show how these more general results arise, and to do so we focus on two observables related to the separation between two, potentially accelerated observers. The first is the curve deviation, which is a natural generalization of the displacement memory, and also contains what this paper proposes to call drift memory (previously called ‘subleading displacement memory’) and ballistic memory. The second is a relative proper time shift that arises between the two observers, either at second order in their initial separation and relative velocity, or in the presence of relative acceleration. The results of this paper are, where appropriate, entirely non-perturbative in the curvature of spacetime, and so could be used beyond leading order in asymptotically flat spacetimes.

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Persistent gravitational wave observables: Curve deviation in asymptotically flat spacetimes

Grant

Nichols

2022

Phys. Rev. D

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The usual gravitational wave memory effect can be understood as a change in the separation of two initially comoving observers due to a burst of gravitational waves. Over the past few decades, a wide variety of other, "persistent" observables which measure permanent effects on idealized detectors have been introduced, each probing distinct physical effects. These observables can be defined in (regions of) any spacetime where there exists a notion of radiation, such as perturbation theory off of a fixed background, nonlinear plane wave spacetimes, or asymptotically flat spacetimes. Many of the persistent observables defined in the literature have been considered only in asymptotically flat spacetimes, and the perturbative nature of such calculations has occasionally obscured deeper relationships between these observables that hold more generally. The goal of this paper is to show how these more general results arise, and to do so we focus on two observables related to the separation between two, potentially accelerated observers. The first is the curve deviation, which is a natural generalization of the displacement memory, and also contains what this paper proposes to call drift memory (previously called "subleading displacement memory") and ballistic memory. The second is a relative proper time shift that arises between the two observers, either at second order in their initial separation and relative velocity, or in the presence of relative acceleration. The results of this paper are, where appropriate, entirely non-perturbative in the curvature of spacetime, and so could be used beyond leading order in asymptotically flat spacetimes.

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Properties of an affine transport equation and its holonomy

Cited by 6 publications

References 21 publications

Persistent gravitational wave observables: General framework

Persistent gravitational wave observables: General framework

Persistent gravitational wave observables: nonlinearities in (non-)geodesic deviation

Persistent gravitational wave observables: Curve deviation in asymptotically flat spacetimes

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