ABSTRACT
This chapter explores the local existence theory of the vacuum field equations. It focuses on the gap between an introductory course on Relativity and papers on the Cauchy problem, and on the Newtonian limit. The chapter describes how to demonstrate the existence of analytic solutions of the field equations. The field equations form a quasilinear second order system for the unknown components of the metric tensor. Systems for which the Cauchy-Kowalevskaya theorem shows existence of analytic solutions have the property that one can solve for the highest derivative with respect to one coordinate, and no higher derivative with respect to the other coordinates appears. The coefficients in the equations and the initial data are found approximately to control convergence. The basic tools are ‘energy estimates’ which can be derived for hyperbolic systems. The harmonic coordinate condition was the first coordinate condition which was used to make the equations hyperbolic. The chapter discusses the Christodoulou- Klainerman theorem.
