Leveraging numerical relativity formulations for perturbative analysis: applications to quasinormal modes in general relativity

Nkele, Sipho

Leveraging numerical relativity formulations for perturbative analysis: applications to quasinormal modes in general relativity

Thesis / Dissertation

2025

Publisher

University of Cape Town

Department

Department of Mathematics and Applied Mathematics

Faculty

Faculty of Science

Abstract

Numerical relativity has become an essential tool for studying highly dynamic and strong-field regimes of general relativity, enabling the simulation of compact object mergers, gravitational col-lapse, and other nonlinear phenomena. At the same time, perturbation theory provides a powerful analytical framework for understanding small deviations from equilibrium configurations, offering insights into gravitational wave emission, stability, and fundamental mode structures of relativistic systems. Despite their complementary strengths, numerical relativity and perturbation theory are often treated as distinct approaches, with limited interaction between them. Bridging this gap is crucial for improving our ability to extract physical information from numerical simulations and for validating approximations used in perturbative studies. In this work, we study two key problems on quasinormal modes of compact objects, as a case study in unifying numerical relativity with perturbation theory. We discuss a new approach for analyzing linearized perturbations of a Schwarzschild black hole using the characteristic formulation of numerical relativity, focusing on the computation of quasinormal modes (QNMs). Unlike traditional methods based on the Regge-Wheeler and Zerilli equations, this approach focuses on deriving the master equation governing gravitational perturbations within the characteristic formulation of numerical relativity. We analyze the singular points of this equation, and we derive series solutions with coefficients determined by three-term recurrence relations. These allow for the application of Leaver's continued fraction method, leading to the standard Schwarzschild quasinormal modes (QNMs). In addition, we investigate linearized ADM perturbations on a Tolman-Oppenheimer-Volkoff (TOV) background solution to study radial perturbations. Within this framework, the perturbation equations take the form of three coupled partial differential equations, in contrast to the usual Sturm-Lioville problem that arises in the traditional approach. Using the Weighted Essentially Non-Oscillatory (WENO) finite difference method, we analyze three models derived from a polytropic equation of state: one stable, one marginally stable near the onset of instability, and one unstable. Our results consistent with those derived from standard methods, confirming the expected stability characteristics of these models.

Keywords

Quasinormal modes

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