Higher-order geodesic deviations applied to the Kerr metric

Abstract

Starting with an exact and simple geodesic, we generate approximate geodesics by summing up higher-order geodesic deviations within a General Relativistic setting, without using Newtonian and post-Newtonian approximations.We apply this method to the problem of closed orbital motion of test particles in the Kerr metric space-time. With a simple circular orbit in the equatorial plane taken as the initial geodesic we obtain finite eccentricity orbits in the form of Taylor series with the eccentricity playing the role of small parameter.The explicit expressions of these higher-order geodesic deviations are derived using successive systems of linear equations with constant coefficients, whose solutions are of harmonic oscillator type. This scheme gives best results when applied to the orbits with low eccentricities, but with arbitrary values of (GM/Rc 2 ), smaller than 1/6 in the Schwarzschild limit. PACS number(s): 04.25.-g, 04.25.Nx

World-line deviation and spinning particles

“…The equation for n µ is then obtained by inserting x µ + λn µ into the geodesic equation, comparing terms linear in λ and neglecting O(λ 2 ). This idea have been used in [1] to obtain generalized geodesic deviation equations by considering expansions containing higher orders of λ. These generalized equations have been applied, for example, in [1] to the problem of closed orbital motion of test particles in the Kerr space-time and in [2] to the orbital motion in Schwarzchild metric.…”

Epicycles and Poincaré resonances in general relativity

“…in Einstein-Maxwell theory [4,5] and non-Abelian backgrounds [6], or the effects of spin [7,8]. Moreover, the method can be extended to arbitrary precision by taking into account higher-order deviations [9,10]. In the literature cited [4][5][6][7][8][9][10], the method has been applied e.g.…”

“…Moreover, the method can be extended to arbitrary precision by taking into account higher-order deviations [9,10]. In the literature cited [4][5][6][7][8][9][10], the method has been applied e.g. to calculate particle orbits in pp-waves, and Schwarzschild, Reissner-Nordstrøm and Kerr space-times.…”

New approximation methods in General Relativity: geodesic deviations and deformations of embeddings

“…The second approach has been tested on the two-body problem in General Relativity quite recently, by one of the authors of this paper (RK), in collaboration with with A. Balakin, J.-W. van Holten, R. Colistete Jr. and C. Leygnac ( [6], [7], [8], [9]). .…”

“…It should be stressed now that although the first deviation δ x µ transforms as a vector, the second and higher-order deviations do not; for example, after a coordinate change x µ → y λ (x µ ) we shall have 9) including the terms quadratic in the first deviations. However, it is easy to introduce a covariant quantity of the same order; we shall denote it by b µ ; in order to make clear the infinitesimal character of this vectorial quantity, let us introduce the notation with small parameter ε in the following manner: let now…”