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Singular perturbation techniques in the gravitational self-force problem

Singular perturbation techniques in the gravitational self-force problem
Singular perturbation techniques in the gravitational self-force problem
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.
1550-7998
124009 -[40pp]
Pound, Adam
5aac971a-0e07-4383-aff0-a21d43103a70
Pound, Adam
5aac971a-0e07-4383-aff0-a21d43103a70

Pound, Adam (2010) Singular perturbation techniques in the gravitational self-force problem. Physical Review D, 81 (12), 124009 -[40pp]. (doi:10.1103/PhysRevD.81.124009).

Record type: Article

Abstract

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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Published date: 1 June 2010
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 339867
URI: http://eprints.soton.ac.uk/id/eprint/339867
ISSN: 1550-7998
PURE UUID: 69aa833a-d820-460a-b1ed-a1d5a44cacd9
ORCID for Adam Pound: ORCID iD orcid.org/0000-0001-9446-0638

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Date deposited: 31 May 2012 15:53
Last modified: 15 Mar 2024 03:41

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