Part two by Lydia Bieri , David Garfinkle , and

Introduction In 2015 gravitational waves were detected for the first time by the LIGO team [1]. This triumph happened 100 years after Albert Einstein’s formulation of the theory of general relativity and 99 years after his prediction of gravitational waves [4]. This article focuses on the mathematics of Einstein’s gravitational waves, from the properties of the Einstein vacuum equations and the initial value problem (Cauchy problem), to the various approximations used to obtain quantitative predictions from these equations, and eventually an experimental detection. General relativity is studied as a branch of astronomy, physics, and mathematics. At its core are the Einstein equations, which link the physical content of our universe to geometry. By solving these equations, we construct the spacetime itself, a continuum that relates space, time, geometry, and matter (including energy). The dynamics of the gravitational field are studied in the Cauchy problem for the Einstein equations, relying on the theory of nonlinear partial differential equations (pde) and geometric analysis. The connections between astronomy, physics, and mathematics are richly illustrated by the story of gravitational radiation.