Isolated Black Hole Horizons

Abstract

Black holes have been ubiquitous in current times, offering an unparalleled understanding of gravity and the mathematical elegance of General Relativity. Traditionally, the characterization of black holes has relied on global event horizons, a topological approach that encounters teleological issues and non-locality. To circumvent these conceptual hurdles, this thesis embraces a paradigm shift towards quasi-local definitions, focusing primarily on isolated horizons. By utilizing a quasi-local isolated horizon and treating its time-independent intrinsic geometry as initial data, we establish a mathematically rigorous framework to compute the spacetime beyond the horizon. To unravel the spinorial symmetries and the intricate causal structure of these spacetimes, this work extensively leverages the Newman-Penrose (NP) null tetrad formalism. We apply the Characteristic Initial Value formulation to specify free data on a two-dimensional spacelike cross-section. By integrating the Einstein equations radially outward along null generators, we systematically construct the near-horizon geometry. This conformal and spinorial approach captures the full curvature dynamics through spin coefficients and Weyl scalars, accommodating both vacuum and electromagnetic fields. A central pillar of this thesis is the exploration of black hole tidal deformabilities under external gravitational and electromagnetic perturbations. When subjected to an external tidal environment, the response of the black hole is quantified through relativistic Love numbers. By modeling tidally perturbed black holes as isolated horizons at leading order, we map the horizon’s geometric response to static linear perturbations. This method utilizes the unique algebraic properties of the NP formalism, providing a powerful complement to conventional metric perturbation techniques. Finally, we apply this robust theoretical machinery to specific analytical backgrounds, namely the Schwarzschild and Reissner-Nordström spacetimes. Through these applications, we explicitly compute the radial evolution of perturbations and determine the intrinsic geometry of the near-horizon structure to extract tidal Love numbers. Ultimately, this thesis demonstrates that uniting the isolated horizon framework with the algebraic elegance of the Newman-Penrose formalism yields a profound and highly effective probe into the extreme gravity and enigmatic nature of black holes.