Green Function Methods in Black Hole Spacetimes

In this thesis I present the development of a characteristic initial value problem approach to calculating the Green function for applications to Extreme Mass Ratio Inspirals. I demonstrate the approach with calculations of the scalar self-force in Schwarzschild spacetime. This method is extended to include the gravitational Regge-Wheeler and Zerilli Green functions, from which I compute gravitational wave energy fluxes. I apply this method to three additional problems: (i) the computation of scattering orbit deflection angle corrections at first-order in the mass ratio, (ii) calculation of contributions at second-order in the mass ratio to the orbital evolution, and (iii) application of the numerical techniques to the Teukolsky equation in both Schwarzschild and Kerr spacetimes. I present results of each of these applications and discuss potential future improvements and extensions.