Hyperbolic Equations and General Relativity

Abstract: This is the author Master's Thesis and its main purpose is to demonstrate that it is possible to formulate Einstein's field equations as an initial value problem. The first chapter concerns the hyperbolic equations theory. The definition of hyperbolic equation is given and the concept of wavelike propagation is presented. Hence, once Riemann kernel definition is given, Riemann method to solve an hyperbolic equation in two variables is showed. The second chapter is about the fundamental solution and its relation to Riemann kernel. Therefore, the study of the fundamental solutions proceeds by showing how to build them and by providing some examples of solution with odd and even number of variables. In chapter four, by following Foures-Bruhat, the problem of finding a solution to the Cauchy problem for Einstein field equations in vacuum with non analytic initial data is presented by studying under which assumptions second-order systems of partial differential equations, linear and hyperbolic, with n functions and four variables admit a solution. Thus, in chapter five, it is showed how to turn non-linear systems of partial differential equations into linear systems of the same type for which the previous results hold. Eventually, in chapter six, the existence and uniqueness of the solution to the Cauchy problem for Einstein vacuum field equations is proved. Another part of this work consists in the study of the causal structure of space-time. The definitions of strong causality, stable causality and global hyperbolicity are given and the relation between the property of global hyperbolicity and the existence of Cauchy surfaces is stressed. The last chapter concerns the study of gravitational radiation in axisymmetric black hole collisions at the speed of light as an useful application of Riemann kernel method to solve a second-order hyperbolic equation in two variables.