World vectors, Jacobi vectors and Jacobi one-forms on a manifold with a linear symmetric connection

Schattner, Ruprecht; Trümper, Manfred

doi:10.1088/0305-4470/14/9/029

Cited by 9 publications

(6 citation statements)

References 6 publications

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“…are the "Jacobi propagators" [17,[72][73][74], with the second equalities giving their expansions in powers of the deviation vector [73]. Thus, Let us take the momentumpμ atz to be related to the new momentum p µ at z bỹ…”

Section: Covariant Shift Of the Worldlinementioning

confidence: 99%

Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin

et al. 2016

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We derive a Hamiltonian for an extended spinning test-body in a curved background spacetime, to quadratic order in the spin, in terms of three-dimensional position, momentum, and spin variables having canonical Poisson brackets. This requires a careful analysis of how changes of the spin supplementary condition are related to shifts of the body's representative worldline and transformations of the body's multipole moments, and we employ bitensor calculus for a precise framing of this analysis. We apply the result to the case of the Kerr spacetime and thereby compute an explicit canonical Hamiltonian for the test-body limit of the spinning two-body problem in general relativity, valid for generic orbits and spin orientations, to quadratic order in the test spin. This fully relativistic Hamiltonian is then expanded in post-Newtonian orders and in powers of the Kerr spin parameter, allowing comparisons with the test-mass limits of available post-Newtonian results. Both the fully relativistic Hamiltonian and the results of its expansion can inform the construction of waveform models, especially effective-one-body models, for the analysis of gravitational waves from compact binaries.

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Section: Covariant Shift Of the Worldlinementioning

confidence: 99%

Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin

et al. 2016

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“…In this paper, we define these bitensors in terms of derivatives of the world function σ(y, y ), and with these definitions they would seem to inherit the limited domain over which σ(y, y ) and its derivatives are well-defined, requiring the points y and y to be connected by a unique geodesic segment (to be in each other's normal convex neighborhood (NCN)). In a companion paper [30], we discuss how these and other important bitensors can be alternately defined in terms of the horizontal and vertical covariant derivatives of the exponential map [27,31], thereby extending the domain of validity of those bitensors and of the solution (1.6) beyond the NCN, as well as simplifying key expressions and derivations.…”

Section: Introductionmentioning

confidence: 99%

Geodesic deviation at higher orders via covariant bitensors

Vines

2015

Gen Relativ Gravit

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We review a simple but instructive application of the formalism of covariant bitensors, to use a deviation vector field along a fiducial geodesic to describe a neighboring worldline, in an exact and manifestly covariant manner, via the exponential map. Requiring the neighboring worldline to be a geodesic leads to the usual linear geodesic deviation equation for the deviation vector, plus corrections at higher order in the deviation and relative velocity. We show how these corrections can be efficiently computed to arbitrary orders via covariant bitensor expansions, deriving a form of the geodesic deviation equation valid to all orders, and producing its explicit expanded form through fourth order. We also discuss the generalized Jacobi equation, action principles for the higher-order geodesic deviation equations, results useful for describing accelerated neighboring worldlines, and the formal general solution to the geodesic deviation equation through second order.Comment: 15 pages, 4 figure

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“…In the next step, total force and torque acting on the body will be defined. The right hand side of (21) can be defined to be the force…”

Section: Point Particle Approximationmentioning

confidence: 99%

“…This can be found by differentiating them with respect to s and taking the coincidence limit, x → z afterwards. The resulting initial conditions, obtained by using (A16) are [3,21]…”

Section: Existence Proofmentioning

confidence: 99%

“…Useful definitions of vertical and horizontal covariant derivatives, defined for a two-point tensor, is discussed in Appendix B. Jacobi equation and its consequences are frequently used, and is discussed in Appendix C. Moreover, tensor extensions, which are used in the definitions of relativistic force and torque, is important, and its definition, in addition to normal coordinates, can be found in Appendix D . The appendices are mainly based on [3,[20][21][22], respectively.…”

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confidence: 99%

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Dynamics of Extended Objects in General Relativity

Ilhan

2009

Preprint

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In this thesis, multipole expansions of mass, momentum and stress density will be made for a body in Newtonian mechanics. Using these definitions; momentum, angular momentum, center of mass, force and torque are defined for N gravitationally interacting isolated bodies. Equations of motions of such a system are derived. Definitions of momentum, angular momentum, center of mass, force and torque are made in a relativistic theory. Dynamical (gravitational) skeleton is defined and the multipole moments of the dynamical skeleton are found. Equations of motion for a test body moving in a gravitational field are derived in terms of the multipole moments. Save the details of the derivations, no originality in this thesis is claimed: it is intended as a review of the subject.

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World vectors, Jacobi vectors and Jacobi one-forms on a manifold with a linear symmetric connection

Cited by 9 publications

References 6 publications

Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin

Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin

Geodesic deviation at higher orders via covariant bitensors

Dynamics of Extended Objects in General Relativity

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