Applications of conformal methods to the analysis of global properties of solutions to the Einstein field equations
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Although the study of the initial value problem in General Relativity started in the decade of 1950 with the work of Foures-Bruhat, addressing the problem of global non-linear stability of solutions to the Einstein field equations is in general a hard problem. The first non-linear global stability result in General Relativity was obtained for the de-Sitter spacetime by means of the so-called conformal Einstein field equations introduced by H. Friedrich in the decade of 1980. The latter constitutes the main conceptual and technical tool for the results discussed in this thesis. In Chapter 1 the physical and geometrical motivation for these equations is discussed. In Chapter 2 the conformal Einstein equations are presented and first order hyperbolic reduction strategies are discussed. Chapter 3 contains the first result of this work; a second order hyperbolic reduction of the spinorial formulation of the conformal Einstein field equations. Chapter 4 makes use of the latter equations to give a discussion of the non-linear stability of the Milne universe. Chapter 5 is devoted to the analysis of perturbations of the Schwarzschild-de Sitter spacetime via suitably posed asymptotic initial value problems. Chapter 6 gives a partial generalisation of the results of Chapter 5. Finally a result relating the Newman-Penrose constants at future and past null infinity for spin-1 and spin-2 fields propagating on Minkowski spacetime close to spatial infinity is discussed in Chapter 7 exploiting the framework of the cylinder at spatial in nity. Collectively, these results show how the conformal Einstein field equations and more generally conformal methods can be employed for analysing perturbations of spacetimes of interest and extract information about their conformal structure.
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