Perturbative analysis of the Conformal Constraint Equations in General Relativity

Abstract

The main purpose of this thesis is to develop a perturbative method for the construction of initial data for the Cauchy problem in General Relativity. More precisely, it considers the problem of constructing solutions to the so-called Extended Constraint Equations (ECEs), based on the method of A. Butscher and H. Friedrich. For much of the thesis, attention is restricted to closed initial hypersurfaces —that is to say, initial data for cosmological spacetimes. In doing so, it is possible to study the potential obstructions to the implementation of the “Friedrich–Butscher method” in a more systematic manner. The central result of this thesis is that initial data describing certain spatially-closed analogues of the Friedmann–Lemaître–Robinson–Walker (FLRW) spacetime are suitable background initial data sets on which to apply the Friedrich-Butscher method. That is to say, one can construct solutions of the ECEs as non-linear perturbations of these background geometries, for which certain components of the extrinsic curvature and of the Weyl curvature (of the resulting spacetime development) are prescribed at the outset. Progress is then made towards identifying a broader class of admissible background geometry, and a streamlined version of the method is proposed which overcomes some of the difficulties inherent to the earlier approach. An elliptic reduction of the full Conformal Constraint Equations of H. Friedrich is then described, and the earlier analysis of the spatially-closed FLRW background geometries is generalised in this context. The last part of the thesis concerns the separate (though not altogether unrelated) problem of Killing Initial Data sets, and how they may be generalised to describe a notion of approximate spacetime Killing symmetry, at the level of the initial data. This builds on work of S. Dain, which is extended from the time symmetric case to the generic asymptotically-Euclidean case.