Epicycles and Poincaré resonances in general relativity

A longstanding approach to the dynamics of extended objects in curved space-time is given by the Mathisson-Papapetrou-Dixon (MPD) equations of motion for the so-called "pole-dipole approximation." This paper describes an analytic perturbation approach to the MPD equations via a power series expansion with respect to the particle's spin magnitude, in which the particle's kinematic and dynamical degrees of freedom are expressible in a completely general way to formally infinite order in the expansion parameter, and without any reference to pre-existent spacetime symmetries in the background. An important consequence to emerge from the formalism is that the particle's squared mass and spin magnitudes can shift based on a classical analogue of "radiative corrections" due to spin-curvature coupling, whose implications are investigated. It is explicitly shown how to solve for the linear momentum and spin angular momentum for the spinning particle, up to second order in the expansion. As well, this paper outlines two distinct approaches to address the study of many-body dynamics of spinning particles in curved space-time. The first example is the spin modification of the Raychaudhuri equation for worldline congruences, while the second example is the computation of neighbouring worldlines with respect to an arbitrarily chosen reference worldline.

Section: Generalization Of the Cmp Approximationmentioning

confidence: 99%

Section: Generalized Jacobi Equation and Computation Of Neighbouring mentioning

confidence: 99%

The MPD Equations in Analytic Perturbative Form

Singh

2015

“…and depends on the eccentricity, whereas in the first case the result was independent of the perturbation parameters. It should be mentioned that in the Newtonian solution (12) no perigee precession is present since there is only one frequency, the Keplerian frequency Ω K , involved. In Newtonian gravity (at least for a spherically symmetric potential) the Kepler ellipses are closed.…”

Section: Physical Effectsmentioning

confidence: 99%

“…It is by no means obvious why this linearization should be necessary, but it was used in [3]. Of course, for the Earth m/R is a very small quantity when R corresponds to radii above the surface, but the geodesic deviation equation works well even close to black holes, were m/R might be large [12]. However, Shirokov concludes that, due to the different periods of ϑ and r oscillations, the distance R (ϑ − π/2) to the equatorial plane, in which the reference orbit lies, is different from 0 after (several) radial oscillations and this is a new effect of GR.…”

Section: Shirokov's Effect Revisitedmentioning

confidence: 99%

On the Applicability of the Geodesic Deviation Equation in General Relativity

Philipp

Puetzfeld

Lämmerzahl

2019

Within the theory of General Relativity, we study the solution and range of applicability of the standard geodesic deviation equation in highly symmetric spacetimes. In the Schwarzschild spacetime, the solution is used to model satellite orbit constellations and their deviations around a spherically symmetric Earth model. We investigate the spatial shape and orbital elements of perturbations of circular reference curves. In particular, we reconsider the deviation equation in Newtonian gravity and then determine relativistic effects within the theory of General Relativity by comparison. The deviation of nearby satellite orbits, as constructed from exact solutions of the underlying geodesic equation, is compared to the solution of the geodesic deviation equation to assess the accuracy of the latter. Furthermore, we comment on the so-called Shirokov effect in the Schwarzschild spacetime and limitations of the first order deviation approach. *

“…a given geodesic [10,11,12]. The procedure described here has been worked out for Schwarzschild space-time up to and including the second-order deviations [6,13,14]. The results for orbits in the equatorial plane θ = π/2 are summarized by the parametrized expansions…”

Section: Geodesic Deviationsmentioning

confidence: 99%

World-Line Perturbation Theory

Holten

2019

The motion of a compact body in space and time is commonly described by the world line of a point representing the instantaneous position of the body. In General Relativity such a world-line formalism is not quite straightforward because of the strict impossibility to accommodate point masses and rigid bodies. In many situations of practical interest it can still be made to work using an effective hamiltonian or energy-momentum tensor for a finite number of collective degrees of freedom of the compact object. Even so exact solutions of the equations of motion are often not available. In such cases families of world lines of compact bodies in curved space-times can be constructed by a perturbative procedure based on generalized geodesic deviation equations. Examples for simple test masses and for spinning test bodies are presented. *