Across the Horizon: On Gravitational Wave Flux Laws and Tests of Gravity

Motivated by the first detection of gravitational waves, this dissertation develops analytical, numerical, and data analysis techniques to address persistent blind spots in our understanding of gravity. Beginning with asymptotically flat spacetimes and the geometry of null geodesic congruences, the derivation of the shear tensor--encoding gravitational radiation--is revisited. The covariant phase space formulation of General Relativity is then employed to derive a non-conservation law associated with the symmetry group at null infinity, generalizing prior constructions of radiative phase space and yielding consistent results. These flux laws are used to derive constraint equations that enable the evaluation and comparison of state-of-the-art numerical waveform models, leading to new insights and a robust algorithm for benchmarking future improvements. These flux laws are further applied to compute quantum corrections to gravitational waveforms arising from the gravitational wave echo effect. Two leading phenomenological scenarios involving echoes from binary black hole mergers are analyzed. The structure of both the primary echo signal and its induced corrections to nonlinear features of the waveform are studied for observability by the upcoming LISA mission. The results indicate that LISA could detect such signals, potentially probing black hole area quantization. Finally, motivated by recent hints of a stochastic gravitational wave background from Pulsar Timing Array data, this work reviews its theoretical basis and studies several astrophysical and cosmological source scenarios. A forecast for detection prospects with LISA is presented using a modern data analysis pipeline. The results suggest LISA will constrain the extra-galactic stochastic background to a spectral energy density below $ \Omega_\text{GW} \lesssim 10^{-8}$.