The Wave Equation on the Schwarzschild Exterior

Abstract

We study the covariant wave equation on Lorentzian manifolds. We begin by introducing Lorentzian geometry, presenting the fundamental constructions in the n + 1 dimensional formalism. We then continue our investigation in two main parts. In the first part, we discuss the Lagrangian structure of the equation and introduce the energy-momentum tensor. We develop fundamental properties of this construction and use it to discuss local and global uniqueness statements. We then consider the special case of Minkowski space to get global estimates for all time and to prove energy decay. In the second part, we develop Schwarzschild geometry from the classical metric to the maximally extended setting; we then focus on proving global boundedness-type results in a general class of spacetimes for which Schwarzschild will be our model. We finish by briefly discussing decay-type results on the Schwarzschild background.