Del Piano, Manuel (2025) The Effective Metric Description: a parametrization for black hole spacetimes beyond General Relativity. [Tesi di dottorato]

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Abstract

This thesis develops a comprehensive and model-independent framework for studying static and spherically symmetric black holes (BHs) subjected to deformations of the classical solutions. Central to this construction is the Effective Metric Description (EMD), which parametrizes deviations from classical geometries, such as Schwarzschild and Reissner–Nordström, through one or two functions of a physical quantity, typically the proper distance from the event horizon. By defining the deformation functions in terms of this physical quantity, the EMD ensures coordinate invariance of the deformations and consistency with the classical symmetries of the spacetime. We provide a systematic method to solve the self-consistency equations near the horizon via series expansions, imposing regularity conditions that guarantee finite curvature scalars and a well-defined Hawking temperature. To extend the applicability of the framework beyond the near-horizon region, we introduce Padé approximants, allowing accurate computation of astrophysical observables, including the photon-sphere radius and the black hole shadow. Our results show that effective expressions capture the leading corrections to these observables and can be directly compared with experimental measurements, enabling model-independent constraints on the deformation functions. The EMD framework is further generalized to incorporate charged black holes, yielding consistent expressions for the Hawking temperature and clarifying the role of extremal configurations. Additionally, we establish explicit mappings between EMDs based on other spacetime invariants and other widely used parametrization schemes, such as Johannsen–Psaltis and Rezzolla–Zhidenko, demonstrating the fundamental equivalence of these approaches and facilitating the translation of quantum-gravity–inspired deformations across different frameworks. Applications to quasi normal modes in the eikonal limit and various regular black hole models illustrate the versatility and predictive power of the framework. Overall, this thesis provides a unified and flexible basis to study black hole spacetimes in classical and quantum regimes. The formalism developed here not only advances theoretical understanding but also bridges the gap to phenomenology, offering practical tools to confront upcoming high-precision observations with predictions of quantum gravity and general relativity.

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