Study of the Relativistic Dynamics of Extreme-Mass-Ratio Inspirals

TL;DR

This thesis explores the relativistic dynamics of extreme-mass-ratio inspirals, focusing on gravitational self-force, entropy theorems, and numerical methods, with implications for gravitational wave detection by LISA.

Contribution

It introduces a novel approach to calculating gravitational self-force corrections and extends numerical methods for scalar fields to broader applications in applied mathematics.

Findings

Development of a conservation law-based method for self-force calculation

Numerical computation of scalar self-force using the Particle-without-Particle method

Application of extended numerical methods to partial differential equations

Abstract

The principal subject of this thesis is the gravitational two-body problem in the extreme-mass-ratio regime---that is, where one mass is significantly smaller than the other---in the full context of our contemporary theory of gravity, general relativity. We divide this work into two broad parts: the first provides an overview of the theory of general relativity along with the basic mathematical methods underlying it, focusing on its canonical formulation and perturbation techniques; the second presents our novel work in these areas, focusing on the problems of entropy, motion and the self-force in general relativity. We begin here with a study of entropy theorems in classical Hamiltonian systems, and in particular, the issue of the second law of thermodynamics in classical mechanics and general relativity. Then, we develop a general approach based on conservation laws for calculating the correction to the motion of a sufficiently small object due to gravitational perturbations in general relativity. When the perturbations are attributed to the small object itself, this effect is known as the gravitational self-force. It is what drives the orbital evolution of extreme-mass-ratio inspirals: compact binary systems where one mass is much smaller than---thus effectively orbiting and eventually spiralling into---the other, expected to be among the main sources for the future space-based gravitational wave detector LISA. Finally, we present some work on the numerical computation of the scalar self-force using an approach called the Particle-without-Particle method, as well as the generalization of this method to general partial differential equations and applications to other areas of applied mathematics.