Exact Results in Quantum Theory & Gravity

I will review the motivations (from classical as well as quantum field theory) for studying electromagnetism and linearized gravity in harmonic gauge (locality and regularity, simplicity of residual gauge freedom, renormalization of interactions). Unfortunately, even after successful separation of variables, this choice of gauge leads to technical difficulties on non-flat backgrounds, like the Schwarzschild black hole, due to rather complicated radial mode equations. I will then describe a recent series of works, in which I have taken steps to overcome these difficulties by explicitly decoupling the radial mode equations into sparse triangular form. This simplification opens the door to immediate applications: solving the equations of motion with sources, proving the absence of exponentially growing modes, constructing Green functions, explicit formulas for Debye potentials, all directly in harmonic gauge.