Second-order gravitational self-force in Kerr spacetime

Gravitational-wave astronomy has been a burgeoning field of research since the first detection of a merging black hole binary in 2015. As gravitational-wave detector sensitivity improves, our models must keep pace. The planned space-based detector LISA will be sensitive to new gravitational wave sources, such as extreme-mass-ratio inspirals (EMRIs). Precise extraction of EMRI parameters from LISA data will require highly accurate waveform templates. These templates need models which include, among other things, the dissipative piece of the second-order self-force in Kerr. This thesis formulates methods to help calculate the second-order self-force in Kerr. In the first part of the thesis, I develop a general framework for second-order calculations by deriving a new form of the second-order Teukolsky equation. I show that the source of this equation is well defined (in a highly regular gauge) for second-order self-force calculations. Additionally, I present methods for calculating second-order gauge-invariants. I produce an algebraic method for calculating a gauge-invariant. I also provide a formalism for calculating a gauge-invariant associated with the Bondi--Sachs gauge (with a fixed BMS frame). The asymptotically flat property of the Bondi--Sachs gauge is shown to circumvent infrared divergences that arise in generic second-order calculations. Next, I calculate a general formula for the second-order source, decomposed into spherical harmonics, in Schwarzschild. Using this formula, I help to implement a framework for quasi-circular inspirals in Schwarzschild. I transform the source to a near-Bondi--Sachs gauge, increasing the asymptotic falloff by two orders in r. My collaborator Ben Leather integrates the resulting source. From the resulting quantity, we will extract fluxes and evolve inspirals to first post-adiabatic accuracy. In the final part, I take a step toward implementation in Kerr by developing a new method of constructing a more regular first-order perturbation. To help formulate this method, I implement Green--Hollands--Zimmerman metric construction for a stationary point-mass in flat spacetime.