Abstract

The Einstein field equations (EFE) can be written in harmonic coordinates as a system of quasilinear wave equations. This allows for the study of the (EFE) within the theory of hyperbolic PDE . In particular, this can be used to study well-posedness and stability of the (EFE) for a large class of initial data.

I’ll focus on the linear wave equation, which is the prototype hyperbolic PDE . It can also be viewed as a “poor man’s linearisation” of the (EFE). Therefore the study of boundedness and decay of solutions of the wave equation on a fixed black hole background are a first step towards stability of the background as a solution of the (EFE).

I’ll start off by discussing this “poor man’s linearisation” and recalling what well-posedness means. We’ll then move onto the highlights of the proof of well-posedness the Cauchy problem for the wave equation.

Along the way, we’ll run into some Sobolev spaces and energy estimates, and I will try to convince you of their power and naturality. Time permitting, I’ll discuss the heuristics of my work on the linear stability of subextremal Kerr-Newman spacetimes.