Abstract null hypersurfaces and characteristic initial value problems in general relativity

Abstract

[EN]This thesis is framed within the field of Mathematical Relativity and is organized into six chapters. After an introduction to the topic in Chapter 1, Chapter 2 reviews and further develops the formalism of hypersurface data, which provides the unifying framework for the entire thesis. In Chapter 3 we study the characteristic Cauchy problem from a fully detached perspective. Chapter 4 is devoted to the Killing initial data problem, also analyzed within this detached framework. In Chapter 5 we investigate the transverse (or asymptotic) expansion of the metric at a general null hypersurface. Finally, Chapter 6 addresses the geometry of conformal null infinity. The hypersurface data formalism allows one to describe hypersurfaces of arbitrary causal character without the need of being embedded in any ambient manifold. This detached viewpoint is particularly well-suited for the formulation of initial value problems in General Relativity, as it allows the study of the initial data without a priori reference to the spacetime to be constructed. Within this framework, Chapter 3 provides a geometrization of the characteristic Cauchy problem via the notion of double null data, while Chapter 4 develops a general treatment of Killing and homothetic initial data on hypersurfaces of any causal character, with particular emphasis on null and characteristic settings. When only a single null hypersurface is available, establishing existence and uniqueness of solutions to the Einstein equations becomes a subtle problem. This issue is analyzed in detail in Chapter 5, where general identities governing the transverse expansion of the metric at null hypersurfaces are derived and applied, in particular, to the study of Killing horizons. Finally, Chapter 6 considers the case in which the null hypersurface represents conformal null infinity. In this setting, we analyze the conformal Einstein equations in arbitrary dimension and characterize the free data at null infinity. We also find a conformal completion of the Fefferman–Graham ambient metric and analyze its asymptotic properties.