A Global Existence Result for the Cauchy Problem in General Relativity

Specialeforsvar ved Ali J H Mohammad

Titel: A Global Existence Result for the Cauchy Problem in General Relativity

  

Abstract: In this thesis we study a global existence result for the Cauchy problem in general relativity which is due to Lindblad and Rodnianski. In this work the global stability of Minkowski space for restricted initial data is established in harmonic gauge as a small data global existence for quasilinear wave equations. The initial data is assumed to coincide with the initial data of the Schwarzschild solution outside a ball of radius one.
This result generalizes to a new proof of the famous stability problem of Minkowski space originally proven by Klainerman and Christodoulou. The proof relies on establishing energy estimates using the Killing and conformal Killing vector fields of Minkowski spacetime together with decay estimates and Klainerman-Sobolev inequality. As a background to the global existence result we provide a rather detailed account of the Cauchy problem in general relativity. This will take us through the local theory for quaslinear wave equations which is a necessary first step for understanding the global theory. After proving the necessary results from the local theory we touch on two results in Lorentzian geometry: the characterization of global hyperbolicity in terms of Cauchy hypersurfaces and a uniqueness result of the geometric wave equation on a Lorentz manifold. We next show how these analytic and geometric results provide general relativity with an initial value formulation. We discuss the concept of fixing a gauge and define the initial data to the equations. We then show that given initial data there exists a globally hyperbolic development. The final step for proving well-posedness of Einstein’s equations is to show that for each initial data there exists a maximal globally hyperbolic development which is unique up to isometry, settling the problem of global uniquenss. This seminal result is due to Choquet-Bruhat and Geroch and we provide the details of the proof.

  

Vejledere:  Niels Martin Møller,
                    Jan Ambjørn, NBI
Censor:      Poul Hjorth, DTU