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.

Submitted by Automa??o e Estat?stica (sst@bczm.ufrn.br) on 2017-07-17T13:08:15Z No. of bitstreams: 1 CrislaneDeSouzaSantos_TESE.pdf: 859033 bytes, checksum: 6c4933c54ee1e77f0ea6be3839fc13c1 (MD5) / Approved for entry into archive by Arlan Eloi Leite Silva (eloihistoriador@yahoo.com.br) on 2017-07-18T14:28:54Z (GMT) No. of bitstreams: 1 CrislaneDeSouzaSantos_TESE.pdf: 859033 bytes, checksum: 6c4933c54ee1e77f0ea6be3839fc13c1 (MD5) / Made available in DSpace on 2017-07-18T14:28:54Z (GMT). No. of bitstreams: 1 CrislaneDeSouzaSantos_TESE.pdf: 859033 bytes, checksum: 6c4933c54ee1e77f0ea6be3839fc13c1 (MD5) Previous issue date: 2017-03-24 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior (CAPES) / A evid?ncia observacional da expans?o acelerada do Universo tem sido a principal raz?o para uma revis?o da evolu??o cosmol?gica como previsto pela Relatividade Geral (RG). Atualmente existe duas principais abordagens para resolver este problema: pela introdu??o nas equa??es de Einstein de um termo o qual representa um novo tipo de fluido (a chamada energia escura) possuindo caracter?sticas ex?ticas ou pela modifica??o da teoria de gravita??o. Nesta tese n?s focamos na segunda abordagem, particularmente, as teorias conhecidas como teorias f(R) de gravidade as quais t?m recebido muita aten??o nos ?ltimos anos. Neste contexto, a equa??o de Raychaudhuri permite examinar a estrutura do espa?o-tempo como um todo sem solu??es espec?ficas das equa??es de Einstein, desempenhando assim um papel central para a compreens?o da atra??o gravitacional em Astrof?sica e Cosmologia. Na teoria da Relatividade Geral sem uma constante cosmol?gica, uma contribui??o n?o-positiva da geometria do espa?o-tempo a equa??o de Raychaudhuri ? usualmente interpretada como a manifesta??o do car?ter atrativo da gravidade. Neste caso, condi??es de energia espec?ficas - de fato a condi??o de energia forte - deve ser assumida, a fim de garantir o car?cter atrativo. No contexto das teorias f(R) de gravidade, no entanto, mesmo assumindo as condi??es de energia usuais pode-se ter uma contribui??o positiva para a equa??o de Raychaudhuri. Al?m de nos fornecer uma maneira simples de explicar a observada expans?o acelerada do Universo, este fato abre a possibilidade de um car?ter repulsivo deste tipo de gravidade. Nesta tese n?s abordamos o car?cter atrativo/n?o-atrativo da gravidade f(R) ? luz da equa??o de Raychaudhuri e fazemos uso da condi??o de energia forte, juntamente com estimativas recentes dos par?metros cosmogr?ficos, para colocar limites em uma classe paradigm?tica de teorias f(R) de gravidade. / The observational evidence of the accelerated expansion of the Universe has been the main reason for a revision of the cosmological evolution as predicted by General Relativity (GR). Currently there are two main approaches to solving this problem: by introducing in the Einstein?s equations a term which represent a new kind of fluid (the so-called dark energy possessing exotic features) or by the modification of the gravitation theory. In this thesis we focus on the second approach, particularly the theories know as f(R) theories of gravity, which have received many attention in the last years. In this framework, the Raychaudhuri equation makes possible to examine the whole of spacetime structures without specific solutions of Einstein?s equations, playing so a central role to the understanding of gravitational attraction in Astrophysics and Cosmology. In the general relativity theory of gravity without a cosmological constant, a non-positive contribution from the spacetime to Raychaudhuri?s equation is usually interpreted as manifestation of the attractive character of gravity. In this case, particular energy conditions - indeed the strong energy condition - must be assumed in order to guarantee this attractive character. In the context of f(R) theories of gravity however, even assuming the usual energy conditions we may have a positive contribution to Raychaudhuri?s equation. Besides giving us a simple way to explain the observed accelerated expansion of the Universe, this fact opens the possibility of a repulsive character of this kind of gravity. In this thesis we address the attractive/non-attractive character of f(R) theories of gravity at the light of Raychaudhuri?s equation and make use of the strong energy condition, jointly with recent estimated values for the cosmographic parameters, in order to put bounds on a paradigmatic class of f(R) theories of gravity.