The Cauchy problem for the linearised Einstein equation and the Goursat problem for wave equations In this thesis we study two initial value problems arising in general relativity. The first is the Cauchy problem for the linearised Einstein equation on general globally hyperbolic spacetimes with smooth and distributional initial data. We extend well known results by showing that given a solution to the linearised constraint equations of arbitrary real Sobolev regularity there is a globally defined solution which is unique up to addition of gauge solutions. Two solutions are considered equivalent if they differ by a gauge solution. Our main result is that the equivalence class of solutions depends continuously on the corre sponding equivalence class of initial data. We also solve the linearised constraint equations in certain cases and show that there exist arbitrarily irregular (non gauge) solutions to the linearised Einstein equation on Minkowski spacetime and Kasner spacetime. In the second part we study the Goursat problem (the characteristic Cauchy problem) for wave equations. We specify initial data on a smooth compact […]