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Título
Asymptotic behaviour of spacetimes with positive cosmological constant
Autor(es)
Director(es)
Materia
Tesis y disertaciones académicas
Universidad de Salamanca (España)
Tesis Doctoral
Academic dissertations
Teoría de la relatividad
Teoría de la gravitación universal
Geometría diferencial
Ecuaciones diferenciales en derivadas parciales
Clasificación UNESCO
2212.14 Teoría de la Relatividad
2212.05 Gravitación
1204.04 Geometría Diferencial
1206.02 Ecuaciones Diferenciales
Fecha de publicación
2021
Resumen
[EN] 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) I . In our setting, the conformal Killing vector fields
(CKVFs) of I become very relevant because of their relation with the symmetries of
the spacetime.
In the first part of the thesis, we study the CKVFs ξ of conformally flat n-metrics γ,
as well as their equivalence classes [ξ] up to conformal transformations of γ. We do
that by analyzing in detail SkewEnd(M1,n+1), the skew-symmetric endomorphisms of
the Minkowski space M1,n+1. The cases n = 2, 3 are worked out in special detail. A
canonical form that fits every element in SkewEnd(M1,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 [ξ] 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 (Λ > 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 characteri-
zation of them, in the conformally flat I case, to arbitrary signature and cosmological
constant. We discuss the validity of this geometric characterization of data beyond
the conformally flat I 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 [γ] and one particular conformal class of CKVFs [ξ] of [γ]. These data admit
a generalization, keeping [γ] conformally flat, by allowing [ξ] to be an arbitrary confor-
mal class. This extends the so-called Kerr-de Sitter-like class with conformally flat I ,
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 (Λ > 0)-vacuum Kerr-Schild spacetimes, which share (conformally flat) I 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.
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