Singular perturbation techniques in the gravitational self-force problem

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.

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

References in corpus (15)

• Relativistic tidal properties of neutron stars
• Relativistic theory of tidal Love numbers
• Intermediate and Extreme Mass-Ratio Inspirals -- Astrophysics, Science Applications and Detection using LISA
• Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital Motion
• The self-consistent gravitational self-force
• Nonrotating black hole in a post-Newtonian tidal environment
• Geometry and dynamics of a tidally deformed black hole
• Multi-scale analysis of the electromagnetic self-force in a weak gravitational field
• The Newtonian limit for perfect fluids
• A light-cone gauge for black-hole perturbation theory
• Post-Newtonian expansions for perfect fluids
• Two-timescale adiabatic expansion of a scalar field model
• Equations of Motion in General Relativity of a Small Charged Black Hole
• Existence of families of spacetimes with a Newtonian limit
• On the Equation of Motion for a Fast Moving Small Object in the Strong Field Point Particle Limit

Cited by in corpus (31)

• Black holes, gravitational waves and fundamental physics: a roadmap
• The motion of point particles in curved spacetime
• Second-order gravitational self-force
• Two-timescale evolution of extreme-mass-ratio inspirals: waveform generation scheme for quasicircular orbits in Schwarzschild spacetime
• Black hole perturbation theory and gravitational self-force
• Nonlinear gravitational self-force: second-order equation of motion
• Gravitational self-force from radiation-gauge metric perturbations
• Effective Gravitational Wave Stress-energy Tensor in Alternative Theories of Gravity
• A practical, covariant puncture for second-order self-force calculations
• Nonlinear gravitational self-force. I. Field outside a small body
• Tidal invariants for compact binaries on quasi-circular orbits
• Self-force via $m$-mode regularization and 2+1D evolution: III. Gravitational field on Schwarzschild spacetime
• Motion of small objects in curved spacetimes: An introduction to gravitational self-force
• Second-order perturbation theory: problems on large scales
• A Rosetta Stone for eccentric gravitational waveform models
• Pseudospectral Collocation Methods for the Computation of the Self-Force on a Charged Particle: Generic Orbits around a Schwarzschild Black Hole
• The Eccentric Behavior of Inspiraling Compact Binaries
• Self-force via Green functions and worldline integration
• New Kludge Scheme for the Construction of Approximate Waveforms for Extreme-Mass-Ratio Inspirals
• Gauge and motion in perturbation theory
• Metric perturbations from eccentric orbits on a Schwarzschild black hole: I. Odd-parity Regge-Wheeler to Lorenz gauge transformation and two new methods to circumvent the Gibbs phenomenon
• Octupolar invariants for compact binaries on quasi-circular orbits
• Gravitational self-force in nonvacuum spacetimes
• Gravitational backreaction on a cosmic string: Formalism
• Radiation-Reaction Force on a Small Charged Body to Second Order
• Gravitational spin-orbit dynamics at the fifth-and-a-half post-Newtonian order
• Resonant excitation of eccentricity in spherical extreme-mass-ratio inspirals
• Nature Abhors a Circle
• Motion of small bodies in general relativity: foundations and implementations of the self-force
• Self-force and fluid resonances
• Deformed Schwarzschild horizons in second-order perturbation theory: mass, geometry, and teleology