Singular perturbation techniques in the gravitational self-force problem

Details Singular perturbation techniques in the gravitational self-force problem Singular perturbation techniques in the gravitational self-force problem

2010-03-20

arxiv.org/abs/1003.3954v2

Phys.Rev.D81:124009,2010

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

39 pages, 5 figures, final version to be published in Phys. Rev. D, several typos corrected, added discussion of order-reducti

Scientific paper

10.1103/PhysRevD.81.124009

Much of the progress in the gravitational self-force problem has involved the use of singular perturbation techniques. Yet the formalism underlying these techniques is not widely known. I remedy this situation by explicating the foundations and geometrical structure of singular perturbation theory in general relativity. Within that context, I sketch precise formulations of the methods used in the self-force problem: dual expansions (including matched asymptotic expansions), for which I identify precise matching conditions, one of which is a weak condition arising only when multiple coordinate systems are used; multiscale expansions, for which I provide a covariant formulation; and a self-consistent expansion with a fixed worldline, for which I provide a precise statement of the exact problem and its approximation. I then present a detailed analysis of matched asymptotic expansions as they have been utilized in calculating the self-force. Typically, the method has relied on a weak matching condition, which I show cannot determine a unique equation of motion. I formulate a refined condition that is sufficient to determine such an equation. However, I conclude that the method yields significantly weaker results than do alternative methods.

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