Harmonic gauge perturbations of the Schwarzschild metric

TL;DR

This paper develops a method to solve the harmonic gauge perturbation equations of the Schwarzschild metric, enabling calculation of gravitational self-force effects for EMRIs, which are crucial for gravitational wave modeling.

Contribution

It presents a new approach to solving the coupled perturbation equations in harmonic gauge using separation of variables and Fourier transforms, leading to explicit solutions for self-force calculations.

Findings

Derived decoupled ordinary differential equations for radial functions

Expressed solutions in terms of Zerilli and Regge-Wheeler functions

Calculated first-order gravitational self-force for circular orbits

Abstract

The satellite observatory LISA will be capable of detecting gravitational waves from extreme mass ratio inspirals (EMRIs), such as a small black hole orbiting a supermassive black hole. The gravitational effects of the much smaller mass can be treated as the perturbation of a known background metric, here the Schwarzschild metric. The perturbed Einstein field equations form a system of ten coupled partial differential equations. We solve the equations in the harmonic gauge, also called the Lorentz gauge or Lorenz gauge. Using separation of variables and Fourier transforms, we write the frequency domain solutions in terms of six radial functions which satisfy decoupled ordinary differential equations. The six functions are the Zerilli and five generalized Regge-Wheeler functions of spin 2,1,0. We use the solutions to calculate the gravitational self-force for circular orbits. The self-force gives the first order perturbative corrections to the equations of motion. Section 1.2 of the thesis has a more detailed summary.