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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.
| Identifer | oai:union.ndltd.org:IBICT/oai:repositorio.ufrn.br:123456789/16573 |
| Date | 14 April 2011 |
| Creators | Santos, Crislane de Souza |
| Contributors | CPF:01999044487, http://buscatextual.cnpq.br/buscatextual/visualizacv.do?id=K4763288H3, Alcaniz, Jailson Souza de, CPF:96817747449, http://lattes.cnpq.br/4351190607357917, Carvalho Filho, Joel C?mara de, CPF:65038274820, http://lattes.cnpq.br/4585335971279853, Medeiros, Deusdedit Monteiro, CPF:35534974320, http://lattes.cnpq.br/2725714861694839, Santos, Janilo |
| Publisher | Universidade Federal do Rio Grande do Norte, Programa de P?s-Gradua??o em F?sica, UFRN, BR, F?sica da Mat?ria Condensada; Astrof?sica e Cosmologia; F?sica da Ionosfera |
| Source Sets | IBICT Brazilian ETDs |
| Language | Portuguese |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/masterThesis |
| Format | application/pdf |
| Source | reponame:Repositório Institucional da UFRN, instname:Universidade Federal do Rio Grande do Norte, instacron:UFRN |
| Rights | info:eu-repo/semantics/openAccess |
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