Motion of small bodies in general relativity: foundations and implementations of the self-force
arXiv:1006.3903
Abstract
Extreme mass-ratio inspirals, in which solar-mass compact bodies spiral into supermassive black holes, are an important potential source for gravitational wave detectors. Because of the extreme mass-ratio, one can model these systems using perturbation theory. However, in order to relate the motion of the small body to the emitted waveform, one requires a model that is accurate on extremely long timescales. Additionally, in order to avoid intractable divergences, one requires a model that treats the small body as asymptotically small rather than exactly pointlike. Both of these difficulties can be resolved by using techniques of singular perturbation theory. I begin this dissertation with an analysis of singular perturbation theory on manifolds, including the common techniques of matched asymptotic expansions and two-timescale expansions. I then formulate a systematic asymptotic expansion in which the metric perturbation due to the body is expanded while a representative worldline is held fixed, and I contrast it with a regular expansion in which both the metric and the worldline must be expanded. This results in an approximation that is potentially uniformly accurate on long timescales. The equation of motion for the body's fixed worldline is determined by performing a local-in-space expansion in the neighbourhood of the body. Using this local expansion as boundary data, I construct a global solution to the perturbative Einstein equation. To concretely characterize orbits, I next devise a relativistic generalization of the Newtonian method of osculating orbits. Making use of this method and two-timescale expansions, I examine the utility of adiabatic approximations that can forgo an explicit calculation of the force.
246 pages, 15 figures, PhD thesis, abstract truncated to meet arXiv requirements
References in corpus (24)
- Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors
- Gravitational wave snapshots of generic extreme mass ratio inspirals
- The use of Generalised Functions and Distributions in General Relativity
- A Rigorous Derivation of Electromagnetic Self-force
- Towards adiabatic waveforms for inspiral into Kerr black holes: I. A new model of the source for the time domain perturbation equation
- Osculating orbits in Schwarzschild spacetime, with an application to extreme mass-ratio inspirals
- Towards adiabatic waveforms for inspiral into Kerr black holes: II. Dynamical sources and generic orbits
- Nonrotating black hole in a post-Newtonian tidal environment
- Scalar self-force on eccentric geodesics in Schwarzschild spacetime: a time-domain computation
- Self-Force Calculations with Matched Expansions and Quasinormal Mode Sums
- Self-force on extreme mass ratio inspirals via curved spacetime effective field theory
- m-Mode Regularization Scheme for the Self Force in Kerr Spacetime
- Second-order gravitational self-force
- Mode-sum regularization of the scalar self-force: Formulation in terms of a tetrad decomposition of the singular field
- Electromagnetic and gravitational self-force on a relativistic particle from quantum fields in curved space
- Multi-scale analysis of the electromagnetic self-force in a weak gravitational field
- Construction of the second-order gravitational perturbations produced by a compact object
- Self-forces from generalized Killing fields
- A light-cone gauge for black-hole perturbation theory
- Two-timescale adiabatic expansion of a scalar field model
- Post-Newtonian expansions for perfect fluids
- Regular second order perturbations of binary black holes: The extreme mass ratio regime
- Equations of Motion in General Relativity of a Small Charged Black Hole
- Existence of families of spacetimes with a Newtonian limit