May, Taillte2025-09-222025-09-222025-09-222025-09-20https://hdl.handle.net/10012/22508This thesis employs numerical methods to study black hole perturbations beyond the leading order. This includes vacuum gravitational perturbations as well as superradiant scalar, vector, and tensor boson perturbations. Understanding and quantifying beyond leading order effects enables more accurate tests of General Relativity and of physics beyond the Standard Model. First, we explore the nonlinear behavior of gravitational perturbations on a Kerr black hole. The ringdown gravitational wave signal, during the final stage of binary black hole mergers, contains important information about the properties of the remnant black hole, and can be used to perform clean tests of general relativity. However, interpreting the loudest portion of the ringdown signal requires understanding the role of nonlinearities and their potential impact on modeling this phase using quasinormal modes. Here, we focus on a particular nonlinear effect arising from the change in the black hole's mass and spin due to the partial absorption of a quasinormal perturbation. We estimate the size and characteristics of this third-order effect using numerical techniques. Quantifying these effects, we find that they may be relevant in analyzing the ringdown in black hole mergers. Next, we discuss self-gravity corrections to beyond the standard model particle dynamics around black holes. Specifically, for scalar and vector bosons forming superradiant clouds. Oscillating clouds of ultralight bosons can grow around spinning black holes through superradiance, extracting energy and angular momentum, and eventually dissipating through gravitational radiation. This makes gravitational wave detectors powerful probes of ultralight bosonic fields. Here, we use fully general-relativistic solutions of the black hole-boson cloud systems to study the self-gravity effects of scalar and vector boson clouds. We calculate the self-gravity shift in the cloud oscillation frequency, which determines the frequency evolution of the gravitational wave signal, improving the accuracy of gravitational wave searches for physics beyond the Standard Model. We also perform an analysis of the spacetime geometry of these systems, we compute how the presence of the cloud changes the innermost stable circular orbit and light ring. Lastly, we investigate the nonlinear phenomenology of the spin-2 boson's superradiant instability. Models that result in a massive spin-2 boson at low energies have been proposed as solutions to the dark matter problem or as modifications to general relativity. The existence of ultralight scalar and vector bosons is constrained using measurements of black hole spins, due to the mechanism of black hole superradiance, and attempts have been made to place constraints on the existence of spin-2 bosons using the same approach. However, those constraints so far have relied on the assumption that the spin-2 superradiant behavior matches that of lower-spin fields. Here we consider a particular nonlinear theory, quadratic gravity, to study the behavior of spin-2 particles that undergo superradiance. We find that the phenomenology is different from that of spin zero or one bosons, increasing the spin of the central black hole and resulting in an extremal black hole horizon.enDoctoral Thesis