Metric Reconstruction and Photon Dynamics in Rotating Black Hole Spacetime

Abstract

Black holes are among the most remarkable predictions of general relativity. They arise as exact solutions to Einstein's field equations and represent regions of spacetime in which gravity becomes so strong that even light cannot escape. This dissertation investigates two closely related aspects of rotating black hole physics: gravitational perturbations and the dynamics of light. Together, these topics illuminate both the mathematical structure of black hole spacetimes and their observable signatures in the strong-field regime. The first part of this work develops methods for reconstructing metric perturbations of rotating black holes. Leveraging the Teukolsky formalism, which describes perturbations in terms of curvature scalars, we recover the underlying metric perturbation. Using the Hertz potential framework, we derive explicit, mode-by-mode reconstruction formulas in Kerr and extend these results to Kerr–(anti-)de Sitter spacetimes. These results enable extension to second-order perturbation theory and the direct computation of local geometric quantities from $h_{\mu \nu}$. The second part of the dissertation studies the propagation of light in black hole spacetimes, with an emphasis on visualization and dynamical structure. In the integrable Kerr geometry, null geodesics give rise to rich lensing phenomena, including photon rings and black hole shadows. We present an implementation of black hole lensing in a real-time visualization application, using analytic solutions to the geodesic equation. We then investigate how these dynamics change under perturbations that break integrability. In such settings, the phase space of photon trajectories develops chaotic behavior and fractal structure. Using tools from Hamiltonian dynamical systems, including Poincaré sections and return maps, we show how the photon ring emerges as an organized phase-space structure in Kerr and how it transitions to self-similar, chaotic behavior in deformed spacetimes.