Historically, black holes have been extensively studied using various frameworks, particularly due to advances in numerical relativity and global methods. The richness of the non-linear nature of black holes is beautifully captured in the intricacies of various fields of mathematics, including topology. The causal nature of spacetime, embedded with fundamental concepts of topology, gives rise to extraordinarily rich results such as the Uniqueness theorems [1]. Additionally, the laws of black hole mechanics strongly resemble the laws of thermodynamics, leading to a deeper understanding of the quantum nature of black holes and incorporating the tools of statistical mechanics.
The developments in gravitational physics characterized the nature of black holes through Event Horizons (except for the singularity theorems), which are globally motivated through topology. The main issue with this characterization is the non-local aspect of their determination. The idea of knowing everything to infinity in spacetime to define the causal past raises concerns about the practicality of the horizons. A paradigm shift in the exploration of black holes is formulated through the ideas of isolated [2] and dynamical horizons [1], in an entirely quasi-local framework [2]. These concepts enable the study of horizons without requiring global stationarity and symmetries. It may seem overly restrictive to assume that the entire spacetime outside the black hole is stationary for the definitions of event horizons. However, these ideas have yielded some of the most significant initial results, and they can now be expanded with additional conventions.
By introducing the notion of a quasi-local isolated horizon, we can construct the spacetime in the vicinity of the horizon. Starting with the intrinsic geometry of a quasi-local isolated horizon, we will integrate the Einstein field equations outward to evolve the entire spacetime. This approach overcomes the limitations of event horizons and provides a natural framework for investigating near-horizon phenomena, which includes perturbative analyses of tidal deformability and post-merger dynamics. The richness of this framework allows us to mediate the teleological issues in event horizons, obtain rich results on near-horizon geometry, and extend the ideas of black hole perturbations. The overarching goal of the thesis is to utilize a unified, quasi-local framework that offers new insights into the dynamical evolution and perturbative behaviour of black holes.