Master of Science / Department of Physics / Bharat Ratra / In this report, we discuss the four standard energy conditions of General Relativity (null, weak, dominant, and strong) and investigate their cosmological consequences. We note that these energy conditions can be compatible with cosmic acceleration provided that a repulsive cosmological constant exists and the acceleration stays within certain bounds. Scalar fields and dark energy, and their relationships to the energy conditions, are also discussed. Special attention is paid to the 1988 Ratra-Peebles scalar field model, which is notable in that it provides a physical self-consistent framework for the phenomenology of dark energy. Appendix B, which is part of joint-research with Anatoly Pavlov, Khaled Saaidi, and Bharat Ratra, reports on the existence of the Ratra-Peebles scalar field tracker solution in a curvature-dominated universe, and discusses the problem of investigating the evolution of long-wavelength inhomogeneities in this solution while taking into account the gravitational back-reaction (in the linear perturbative approximation).

Nous analysons des bulles d'espace-temps d'épaisseur finie en relativité générale. Les conditions d'énergie sont utilisées afin d'obtenir un ensemble de critères permettant de restreindre la structure du bord de la bulle. Dans le cas des bulles statiques et à symétrie sphérique, nous obtenons quatre inégalités différentielles équivalentes aux trois conditions d'énergie les plus communes. Nous montrons qu'elles sont équivalentes à un ensemble de deux inégalités différentielles simples lorsque le potentiel gravitationnel effectif a une forme particulière. Nous paramétrons alors l'espace-temps de manière à rendre la vérification de ces inégalités plus simple lorsqu'il sera question de bulles d'espace-temps. Nous traitons en particulier quatre formes de bulles, toutes caractérisées par un extérieur de type Schwarzschild de Sitter. Nous montrons que notre méthode donne les bons résultats lorsque la limite où l'épaisseur de la bulle tend vers zéro est prise. Nous terminons par un traitement succinct du problème d'une onde gravitationnelle se propageant dans un nuage de bulles d'espace-temps. / We analyze space-time bubbles of finite thickness in general relativity. We use the energy conditions to restrict their structures. In the case of static, spherically symmetric bubbles, we get a set of four differential inequalities. If the effective gravitational potential is taken of a particular form, we show that they can be further reduced to a set of two differential inequalities. We then parameterize the bubble's wall in a particular way, simplifying the inequalities, and easing the application of boundary conditions on our solutions. We then treat four different cases of bubbles that all have a Schwarzschild de Sitter exterior. We show that in the limit where the thickness of the bubble's wall goes to zero, we recover the standard results. Lastly, we treat gravitational waves propagating in a dilute gas of non-interacting space-time bubbles.

Nous analysons des bulles d'espace-temps d'épaisseur finie en relativité générale. Les conditions d'énergie sont utilisées afin d'obtenir un ensemble de critères permettant de restreindre la structure du bord de la bulle. Dans le cas des bulles statiques et à symétrie sphérique, nous obtenons quatre inégalités différentielles équivalentes aux trois conditions d'énergie les plus communes. Nous montrons qu'elles sont équivalentes à un ensemble de deux inégalités différentielles simples lorsque le potentiel gravitationnel effectif a une forme particulière. Nous paramétrons alors l'espace-temps de manière à rendre la vérification de ces inégalités plus simple lorsqu'il sera question de bulles d'espace-temps. Nous traitons en particulier quatre formes de bulles, toutes caractérisées par un extérieur de type Schwarzschild de Sitter. Nous montrons que notre méthode donne les bons résultats lorsque la limite où l'épaisseur de la bulle tend vers zéro est prise. Nous terminons par un traitement succinct du problème d'une onde gravitationnelle se propageant dans un nuage de bulles d'espace-temps. / We analyze space-time bubbles of finite thickness in general relativity. We use the energy conditions to restrict their structures. In the case of static, spherically symmetric bubbles, we get a set of four differential inequalities. If the effective gravitational potential is taken of a particular form, we show that they can be further reduced to a set of two differential inequalities. We then parameterize the bubble's wall in a particular way, simplifying the inequalities, and easing the application of boundary conditions on our solutions. We then treat four different cases of bubbles that all have a Schwarzschild de Sitter exterior. We show that in the limit where the thickness of the bubble's wall goes to zero, we recover the standard results. Lastly, we treat gravitational waves propagating in a dilute gas of non-interacting space-time bubbles.

Made available in DSpace on 2014-12-17T15:14:52Z (GMT). No. of bitstreams: 1 CrislaneSS_DISSERT.pdf: 1091298 bytes, checksum: 831e6bef52e8fad49a4683ec16886d4d (MD5) Previous issue date: 2011-04-14 / In the Einstein s theory of General Relativity the field equations relate the geometry of space-time with the content of matter and energy, sources of the gravitational field. This content is described by a second order tensor, known as energy-momentum tensor. On the other hand, the energy-momentum tensors that have physical meaning are not specified by this theory. In the 700s, Hawking and Ellis set a couple of conditions, considered feasible from a physical point of view, in order to limit the arbitrariness of these tensors. These conditions, which became known as Hawking-Ellis energy conditions, play important roles in the gravitation scenario. They are widely used as powerful tools for analysis; from the demonstration of important theorems concerning to the behavior of gravitational fields and geometries associated, the gravity quantum behavior, to the analysis of cosmological models. In this dissertation we present a rigorous deduction of the several energy conditions currently in vogue in the scientific literature, such as: the Null Energy Condition (NEC), Weak Energy Condition (WEC), the Strong Energy Condition (SEC), the Dominant Energy Condition (DEC) and Null Dominant Energy Condition (NDEC). Bearing in mind the most trivial applications in Cosmology and Gravitation, the deductions were initially made for an energy-momentum tensor of a generalized perfect fluid and then extended to scalar fields with minimal and non-minimal coupling to the gravitational field. We also present a study about the possible violations of some of these energy conditions. Aiming the study of the single nature of some exact solutions of Einstein s General Relativity, in 1955 the Indian physicist Raychaudhuri derived an equation that is today considered fundamental to the study of the gravitational attraction of matter, which became known as the Raychaudhuri equation. This famous equation is fundamental for to understanding of gravitational attraction in Astrophysics and Cosmology and for the comprehension of the singularity theorems, such as, the Hawking and Penrose theorem about the singularity of the gravitational collapse. In this dissertation we derive the Raychaudhuri equation, the Frobenius theorem and the Focusing theorem for congruences time-like and null congruences of a pseudo-riemannian manifold. We discuss the geometric and physical meaning of this equation, its connections with the energy conditions, and some of its several aplications. / Na teoria da Relatividade Geral de Einstein as equa??es de campo relacionam a geometria do espa?o-tempo com o conte?do de mat?ria e de energia, fontes do campo gravitacional. Esse conte?do ? descrito por um tensor de segunda ordem, conhecido como tensor energia-momento. Por outro lado, os tensores energia-momento que possuem significado f?sico n?o s?o especificados por essa teoria. Na d?cada de 70, Hawking e Ellis estabeleceram algumas condi??es, consideradas plaus?veis do ponto de vista f?sico, com o intuito de limitar as arbitrariedades desses tensores. Essas condi??es ficaram conhecidas como condi??es de energia de Hawking-Ellis, desempenham pap?is importantes no cen?rio da gravita??o. Elas s?o largamente usadas como poderosas ferramentas de an?lise, desde a demonstra??o de importantes teoremas relativos ao comportamento de campos gravitacionais e geometrias associadas, comportamento qu?ntico da gravita??o, at? as an?lises de modelos cosmol?gicos. Nesta disserta??o apresentamos uma dedu??o rigorosa das v?rias condi??es de energia em voga atualmente na literatura cient?fica, tais como: Condi??o de Energia Nula (NEC), Condi??o de Energia Fraca (WEC), Condi??o de Energia Forte (SEC), Condi??o de Energia Dominante (DEC) e Condi??o de Energia Dominante Nula (NDEC). Tendo em mente as aplica??es mais corriqueiras em Gravita??o e Cosmologia, as dedu??es foram feitas inicialmente para um tensor energia-momento de um fluido perfeito generalizado e depois estendidas aos campos escalares com acoplamento m?nimo e n?o-m?nimo ao campo gravitacional. Apresentamos tamb?m um estudo sobre as poss?veis viola??es de algumas dessas condi??es de energia, visando o estudo da natureza singular de algumas solu??es exatas da Relatividade Geral de Einstein, em 1955, o f?sico indiano Raychaudhuri derivou uma equa??o que hoje ? considerada fundamental para o estudo da atra??o gravitacional da mat?ria, a qual ficou conhecida como equa??o de Raychaudhuri. Essa c?lebre equa??o ? considerada o alicerce da compreens?o da atra??o gravitacional em Astrof?sica e Cosmologia e dos teoremas de Singularidades, como por exemplo, o teorema de Hawking e Penrose sobre a singularidade do colapso gravitacional. Nesta disserta??o derivamos a equa??o de Raychaudhuri, o teorema de Frobenius e o teorema da Focaliza??o para congru?ncias tipo-tempo e tipo-nulas de uma variedade pseudo-riemanniana. Discutimos o significado geom?trico e f?sico dessa equa??o, sua conex?o com as condi??es de energia, e algumas de suas in?meras aplica??es.

The aim of this thesis is to construct exact, axially symmetric solutions of Einstein- Maxwell(-dilaton) equations, which possess a discrete translational symmetry along an axis. We present two possible approaches to their construction. The first one is to solve Einstein-Maxwell equations, the second one relies on a dimensional reduction from a higher dimension. We examine the geometry of the solutions, their horizons and singu- larities, motions of charged test particles and compare them. 1

Nous étudions différentes situations de distribution de la matière d’une bulle de masse négative. En effet, pour les bulles statiques et à symétrie sphérique, nous commençons par l’hypothèse qui dit que cette bulle, étant une solution des équations d’Einstein, est une déformation au niveau d’un champ scalaire. Nous montrons que cette idée est à rejeter et à remplacer par celle qui dit que la bulle est formée d’un fluide parfait. Nous réussissons à démontrer que ceci est la bonne distribution de matière dans une géométrie Schwarzschild-de Sitter, qu’elle satisfait toutes les conditions et que nous sommes capables de résoudre numériquement ses paramètres de pression et de densité. / We study different situations of matter distribution of a negative mass bubble. For the case of static and spherically symmetric bubbles, we start with the hypothesis saying that this kind of bubble, being a solution of Einstein equations, is a deformation of scalar field. We show that this idea must be rejected and replaced by another saying that the bubble is formed by a perfect fluid. We succeed to demonstrate that this is the proper matter distribution within Schwarzschild-De Sitter geometry, that it satisfies all conditions and that we’re capable of resolving numerically its parameters of pressure and density.

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