Asymptotic behaviour of spacetimes with positive cosmological constant

Resumen

In this thesis we study the asymptotic Cauchy problem of general relativity with positive cosmological constant in arbitrary $(n+1)$-dimensions. Our aim is to provide geometric characterizations of Kerr-de Sitter and related spacetimes by means of their initial data at conformally flat ($n$-dimensional) $\mathscr I$. In our setting, the conformal Killing vector fields (CKVFs) of $\scri$ become very relevant because of their relation with the symmetries of the spacetime. In the first part of the thesis, we study the CKVFs $\Y$ of conformally flat $n$-metrics $\gamma$, as well as their equivalence classes $[\Y]$ up to conformal transformations of $\gamma$. We do that by analyzing in detail $\skwend{\mink{1,n+1}}$, the skew-symmetric endomorphisms of the Minkowski space $\mink{1,n+1}$. The cases $n=2,3$ are worked out in special detail. A canonical form that fits every element in $\skwend{\mink{1,n+1}}$ is obtained along with several applications. Of relevance for the study of asymptotic data is that it gives a canonical form for CKVFs which allows us to determine the conformal classes $[\Y]$ and study the quotient topology associated to these clases. In addition, the canonical form for CKVFs is applied to the $n=3$ case to obtain a set of coordinates adapted to an arbitrary CKVF. With these coordinates we provide the set of asymptotic data which generate all conformally extendable spacetimes solving the $(\Lambda>0)$-vacuum field equations and admitting two commuting symmetries, one of which axial. From this, a characterization of Kerr-de Sitter and related spacetimes follows. Our study provides in principle a good arena to test definitions of mass and angular momentum for positive cosmological constant. In the second part of this thesis we focus in the asymptotic Cauchy problem in arbitrary dimensions. For this we use the Fefferman-Graham formalism. We carry out an study of the asymptotic initial data in this picture and extend an existing geometric characterization of them, in the conformally flat $\scri$ case, to arbitrary signature and cosmological constant. We discuss the validity of this geometric characterization of data beyond the conformally flat $\scri$ case. We provide a KID equation for asymptotic analytic data (which comprise Kerr-de Sitter). This equation being satisfyied by the data amounts to the existence of a Killing vector field in the corresponding spacetime. With the above results in hand we provide a geometric characterization of Kerr-de Sitter by means of its asymptotic initial data, which happen to be determined by the conformally flat class of metrics $[\gamma]$ and one particular conformal class of CKVFs $[\Y]$ of $[\gamma]$. These data admit a generalization, keeping $[\gamma]$ conformally flat, by allowing $[\Y]$ to be an arbitrary conformal class. This extends the so-called Kerr-de Sitter-like class with conformally flat $\scri$, defined in previous works in four spacetime dimensions, to arbitrary dimensions. We study this class and prove that the corresponding spacetimes are contained in the set of $(\Lambda>0)$-vacuum Kerr-Schild spacetimes, which share (conformally flat) $\scri$ with their background metric (de Sitter). We name these Kerr-Schild-de Sitter spacetimes. The proof largely relies on our study of the space of classes of CKVFs and in particular on the properties of its quotient topology. In addition, we prove the converse inclusion, providing a full characterization of the Kerr-de Sitter-like class as the Kerr-Schild-de Sitter spacetimes.