Geometry of abstract null hypersurfaces and matching of spacetimes

Abstract

[EN] The purpose of this thesis is two-fold. As already mentioned, we are firstly interested in the geometry of null hypersurfaces. In this context the formalism of hypersurface data becomes a powerful mathematical framework. Our second aim (and actually the starting point of the thesis) is the study of the problem of matching two completely general spacetimes across a null hypersurface, which we address in Chapters 7, 8, 9. Concerning the part of the thesis where we expand the formalism of hypersurface data, the motivations described above have lead us to study how to characterize curvature information at the abstract level. Also, they have allowed us to understand how the data is affected by the existence of a privileged vector field. In particular, this has permitted that we construct abstract notions of Killing horizons of order zero and one which do not require of any ambient space and which generalize the concepts of non-expanding, (weakly) isolated and Killing horizons. Finally, we have been able to derive an equation, called generalized master equation, that governs the geometry of null hypersurfaces with an extra null tangent vector field. The analysis of this equation reveals properties about the surface gravity of such vector and about homothetic Killing horizons and Killing horizons of order zero and one. Moreover, it allows us to recover, as particular cases, the well-known near horizon equation of isolated horizons as well as the so-called master equation of multiple Killing horizons. The problem of matching two spacetimes across a null hypersurface constitutes the second part of the thesis. In a spacetime context and by requiring a simple topology of the boundaries, we have been able to encode the whole matching information in a function and a diffeomorphism between the set of null generators of both matching hypersurfaces. We have also derived explicit expressions for the matter-energy content of the shell. Finally, we have exploited the formalism of hypersurface data to address the problem of matching in a completely abstract context and without requiring topological restrictions upon the boundaries. This approach, as we will see, has many advantages that will be discussed later.