Orientador: George Emanuel Avraam Matsas / Banca: Patricio Anibal Letelier Sotomayor / Banca: Vitório Alberto de Lorenci / Resumo: Nesta dissertação estudamos aspectos clássicos dos modelos análogos à Relatividade Geral em matéria condensada visando sobretudo criar uma nova percepção dos efeitos gravitacionais semi-clássicos, tais como a radiação Hawking, afim de melhor compreendê-los. Neste sentido, demonstramos que as ondas sonoras se propagando em um fluido ideal, barotrópico e irrotacional sobre uma esfera 'S POT. 2' de raio r se comportam como um campo escalar de Klein-Gordon não massivo em um espaço tempo curvo. Analisamos ao longo desta dissertação diversas propriedades deste espaço-tempo efetivo sentido pelo som, cuja geometria é descrita por uma métrica lorentziana dependente das variáveis hidrodinâmicas do fluxo, como a velocidade do fluido, sua densidade e a velocidade local do som, sempre buscando estabelecer correlações entre os conceitos clássicos da dinâmica dos fluidos e conceitos puramente relativísticos. Feita uma análise mais geral destes espaços-tempos, que denominamos de espaços-tempos acústicos, nos propomos a encontrar soluções das variáveis dinâmicas do fluido, uma vez que são elas que determinam a geometria acústica, capazes de modelar espaços-tempos efetivos dotados de horizontes de eventos e singularidades, criando portanto um buraco mudo/surdo, ou seja, um análogo de um buraco negro e de buraco branco da Relatividade Geral. Discutimos ainda alguns pontos da estrutura causal dos espaços-tempos acústicos construindo assim um diagrama de Carter-Penrose do buraco mudo/surdo com o intuito de evidenciar as possíveis trajetórias nulas deste espaço-tempo. Ademais, mostramos que na aproximação da acústica geométrica, ou também aproximação eikonal, os raios de som seguem geodésicas tipo luz do espaço-tempo acústico. Por fim, calculamos a curvatura escalar deste espaço-tempo verificando a presença... / Abstract: In this dissertation we study the classical aspects of analogue models of General Relativity in condensed matter seeking mainly to create a new perception about semi-classical gravitational effects, such as Hawking radiation, in order to better comprehend them. We demonstrate that sound waves propagating in an ideal barotropic fluid with a non-homogeneous irrotacional flow, over a sphere 'S POT. 2' with radius r behave as a Klein-Gordon massless scalar field in a curved spacetime. Through this dissertation, we analyze several properties of this effective spacetime governing the propagation of sound, whose geometry is described by a Lorentzian metric that depends on the hydrodynamic variables of the flow such as the flow velocity, the density and the local speed of sound, always trying to establish correlations between classical concepts of fluid dynamics and purely relativistic concepts. Once a general analysis of these spacetimes is made, which we denominate acoustic spacetimes, we find solutions of the dynamic variables of the fluid, since they determine the acoustic geometry, capable of modeling effective spacetimes endowed with event horizons and singularities, creating therefore a dumb/deaf hole, i.e., an analogue of a black hole and white hole of the General Relativity. We further discuss some points of the causal structure of the acoustic spacetimes, so constructing a Carter-Penrose diagram of the dumb/deaf hole with the aim of bringing to evidence the possible null trajectories of this spacetime. Furthermore, we show that in the approximation of the acoustic geometry, also called eikonal approximation, the sound rays follow lightlike geodesics of the acoustic spacetime. Finally we calculate the scalar curvature of this spacetime verifying the presence of the non flat structure of the 'S POT. 2' sphere, over which the fluid moves / Mestre

Recentes observações astrofísicas, em especial de supernovas tipo Ia e das anisotropias da radiação cósmica de fundo, indicam que recentemente na história do universo um tipo desconhecido de energia passou a dominar sua evolução induzindo uma expansão acelerada. Esta componente misteriosa ficou conhecida como energia escura. Nosso objetivo é o estudo da energia escura, através de modelos com constante cosmológica, campo escalar canônico e com o campo de Born-Infeld. Para os modelos com campo escalar, usando potenciais tipo lei de potência, estudamos soluções atratoras para a evolução homogênea. Também estudamos, na aproximação de largas escalas, soluções atratoras para as perturbações dos campos escalares. Mostramos que, para modelos de energia escura com campo escalar canônico, as soluções atratoras a nível perturbativo não geram modos de isocurvatura. Para o campo de Born-Infeld, fazemos uma análise de estabilidade de suas soluções atratoras a nível perturbativo, determinamos em que circunstâncias elas podem gerar modos de isocurvatura. Para os modelos de energia escura mais realistas, estes modos tendem a ser pequenos. / Recent astrophysical observations, specially supernova type Ia and cosmic microwave back­ ground anisotropies, indicate that recently in the history of the universe, some unknown type of energy is dominating its evolution and inducing an accelerated expansion. This mysterious component has been named dark energy. Our aim is to study dark energy, by using cosmological constant, canonical scalar field and Born-Infeld scalar field models. In the models with scalar field, using power law potentials, we study attractor solutions for the homogeneous evolution. We also study, in the large scale approximation, attractor solutions for the scalar field perturbations. We show that, for models with canonical sca­ lar field , the attractor solutions for its perturbations do not generate isocurvature modes. For the Born-Infeld scalar field, we analyze the stability of its attractor solutions in the perturbative levei, and we determine in which case they can generate isocurvature modes. For the more realistic dark energy models, these modes tend to be small.

Estudos de perturbações em sistemas gravitacionais no âmbito da Relatividade Geral vêm sofrendo grandes desenvolvimentos nos últimos anos, especialmente em face da evolução dos modernos detectores de ondas gravitacionais. Abordamos neste trabalho as perturbações de diferentes cenários. Principiamos com a métrica de Vaidya, utilizada para descrever espaços-tempos esfericamente simétricos e dependentes do tempo. Nossas simulações mostraram que as freqüências dos modos quasi-normais (MQN\'s) apresentam um novo efeito inercial para variações rápidas da função de massa, retornando depois ao comportamento adiabático. Em seguida, apresentamos um modelo para a evaporação de mini buracos negros por radiação de Hawking inspirado no cenário de criação destes objetos em aceleradores de partículas, previsto pelas novas teorias com dimensões extras. Nosso modelo, baseado na métrica de Vaidya n-dimensional, tornou possível a análise de MQN\'s resultando na possibilidade de se obter os parâmetros relevantes do buraco negro, como a sua massa inicial e o número de dimensões extras, a partir de medições experimentais. Finalmente, realizamos um estudo sobre uma nova solução denominada gravastar, proposta como um modelo alternativo para o estágio final de estrelas com grande massa. Obtivemos limites para os parâmetros da solução e verificamos a sua estabilidade frente a perturbações axiais, concluindo positivamente a respeito da possibilidade de se distinguir entre buracos negros e gravastares com base no seu espectro de MQN\'s. / Perturbative studies of gravitational systems in General Relativity have gone through big developments in the last years, especially due to the evolution of the modern gravitational wave detectors. We consider in this work different perturbations in different scenarios. Firstly we consider the Vaidya metric, mainly used to describe time-dependent spherically symmetric spacetimes. Our simulations show that the frequencies of the quasinormal modes (QNM\'s) present a new inertial effect for rapidly varying mass functions, returning afterwards to the adiabatic behavior. Next we present a model for evaporating mini black holes in particle accelerators, in the context of the new gravity models with extra dimensions. With our model, based on the n-dimensional Vaidya metric, we are able to perform a QNM analysis which results in the possibility of obtaining the parameters of the black hole, such as its initial mass and the number of extra dimensions, from the experimental measurements. Finally, we present a study of a new solution, the gravastar, proposed as an alternative model for the end state of massive stars. We obtain bounds for the parameters of the solution and verify its stability against axial perturbations. Our results indicate that the gravastar\'s QNM spectrum can indeed be used to distinguish a black hole from a gravastar.

Orientador: Alberto Vazquez Saa / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T16:45:35Z (GMT). No. of bitstreams: 1 Dadam_Fabio_D.pdf: 1924347 bytes, checksum: 659b242dc84dcd65c774c8d252ca54b4 (MD5) Previous issue date: 2011 / Resumo: O objetivo do presente trabalho foi o de estudar as oscilações de alguns wormholes na tentativa de se encontrar candidatos que apresentassem soluções exatas para modos quase normais. Apresentamos uma nova classe de wormholes estáticos que generaliza os wormholes de Morris-Thorne pela inclusão de dois parâmetros adicionais a fim de distorcer a simetria esférica e alcançar equações de perturbação onde o potencial pode ser dissociado das respectivas auto-frequências. A nova métrica provou ser muito geral no sentido de que a maioria das geometrias de wormhole estudadas atualmente na literatura podem ser expressas como casos particulares dela. As equações de Teukolsky para esta métrica geral foram determinadas por meio do formalismo de Newman-Penrose e, em consequência deste processo, obtivemos um tipo de solução com freqüências de MQN exatas, a menos de uma equação transcendental. Esse tipo especial de solução foi usado para aproximar potenciais de buracos negros de uma forma semelhante às quadraturas. Estudamos também a propagação de ondas eletromagnéticas ao longo das soluções do tipo wormhole através do formalismo de Newman Penrose e, seguindo certos critérios, obtivemos certos tipos de geometrias de wormhole que são capazes de modelar barreiras de Coulomb ou Morse. Esses resultados podem indicar que wormholes poderiam ser usados no futuro como modelos para sistemas físicos, como as supercordas são usadas atualmente, e também como guia nos chamados modelos análogos de grativação. Finalmente, estudamos outros tipos de soluções do tipo "estrelas exóticas", as chamadas dobras espaciais. Esperamos que as equações, e especialmente os princípios, apresentados neste trabalho ajudem futuros pesquisadores a procurar wormholes susceptíveis a fornecer fontes para uma descrição exata das ondas gravitacionais e uma percepção mais profunda do problema das singularidades na Relatividade Geral e na Mecânica Quântica / Abstract: The aim of the present work was to study the oscillations of certain wormholes in an attempt to find candidates for exact solutions of quasinormal modes. We presented a new class of static wormholes which generalizes Morris-Thorne wormholes by adding two additional parameters in order to distort spherical symmetry and achieve perturbation equations where the potential may be decoupled from the frequency. The new metric proved to be very general in the sense that most of the current wormhole geometries studied in the literature can be expressed as particular cases of it. The Teukolsky equations for this class of wormholes were determined via Newman-Penrose formalism and, as a result of this procedure, we constructed one special solution with exact QNM frequencies except for a transcendental equation. This special type of solution is used to approximate black hole potentials in a similar manner than quadratures. We also studied the propagation of electromagnetic waves in wormhole solutions through Newman-Penrose formalism and, following a set of criteria, we obtained certain types of wormhole geometries that are capable of modeling Coulomb or Morse scatterers. These results may indicate that wormholes could be used in the future as models for physical systems just as superstrings are used today. Finally, we studied other kinds of exotic stars, the warp drives. We hope that the equations, and specially the principles, presented in this work will help future researchers to search for wormholes which could provide sources for exact description of gravitational waves and a deeper insight into the problem of singularities in both General Relativity and Quantum Mechanics / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

O presente trabalho se dedica em apresentar uma nova interpretação do princípio da relatividade dos efeitos contratuais e uma de suas principais conseqüências: a responsabilidade civil do terceiro que contribui com o inadimplemento contratual. Com as transformações ocorridas durante o século XIX, sobretudo em decorrência da Revolução Industrial, o contrato deixou de ser um mero instrumento de circulação de riquezas, para se tornar um instrumento social. Assim, ele não pode mais ser visto como algo que interessa exclusivamente às partes, como determinava a leitura clássica do princípio da relatividade dos efeitos contratuais. A sociedade moderna impõe a necessidade de o contrato ser entendido como um fato social, que produz seus efeitos para além de suas partes. Desse modo, aqueles que são terceiros em relação a ele devem considerar a sua existência, não interferindo negativamente na relação contratual, de forma a causar o inadimplemento da obrigação contratual. Esse dever de respeito é imposto pelo princípio da oponibilidade dos efeitos contratuais, por meio do qual se entende que o direito de crédito decorrente do contrato pode ser oponível a terceiros, que devem respeitá-lo da mesma maneira que respeitam um direito real. Diante disso, defende-se aqui que o princípio da relatividade dos efeitos contratuais deve ser interpretado exclusivamente no sentido de que o contrato apenas gera obrigações para as partes contratuais, o que não significa que terceiros possam ignorar a existência desta relação jurídica. Dessa maneira, se o terceiro celebra com o devedor de um contrato já existente um segundo contrato que impossibilite o cumprimento do primeiro, causando, assim, o inadimplemento contratual, ele deve ser civilmente responsável perante o credor do primeiro contrato. Com o objetivo de alcançar estas conclusões, analisam-se, nessa dissertação, as transformações sofridas pelo direito contratual, a interação entre os seus seis princípios (princípio da liberdade contratual, da obrigatoriedade dos efeitos contratuais, da relatividade dos efeitos contratuais, da boa-fé objetiva, do equilíbrio contratual e da função social do contrato) e os fundamentos da responsabilidade civil que justificam a responsabilidade do terceiro cúmplice do inadimplemento contratual. / This work aims to introduce a new interpretation of the relativity principle of contractual effects and one of its main consequences: third party civil liability that contributes to breach of contract. With the changes that occurred in the 19th century, in particular as a consequence of the Industrial Revolution, the contract was no longer a mere tool for the exchange of wealth it became a social tool. Thus, it no longer can be seen as something of exclusive interest to the parties, as set by the classical interpretation of the relativity principle of contractual effects. Modern society imposes the need for the contract to be understood as a social fact that has its effect beyond the contracting parties. Therefore, third parties must consider its existence, not interfering negatively in the contractual relationship so as to cause a breach of contract. This duty of respect is imposed by the enforceability principle of contractual effects, which provides that the credit right arising from the contract can be enforced against third parties, who have to respect it in the same way that they respect a right in rem. Having said that, this work argues that the relativity principle of contractual effects must be exclusively interpreted in the sense that the contract creates only obligations to the contracting parties, which does not mean that third parties can ignore the existence of this legal relationship. Therefore, if a third party enters into a second contract with a debtor of a pre-existing contract that prevents compliance with the first contract, leading to a breach of contract, this third party should have civil liability before the creditor of the first contract. In order to reach these conclusions, this work analyzes the transformations undergone by contractual law, the interaction between its six principles (contractual freedom, the obligations of contractual effects, relativity of contractual effects, objective good faith, contractual equilibrium and social function of the contract) and the fundamentals of civil liability that justifies the liability of the third party accomplice in a breach of contract.

Emission from X-ray binaries is variable on a wide range of timescales. On long timescales, changes in mass accretion rate drive changes in spectral state. There is also rapid variability, the power spectrum of which consists of a low frequency quasi-periodic oscillation (QPO) superimposed on a broad band noise continuum. Here I investigate a model intended to quantitatively explain the observed spectral and variability properties. I consider a truncated disc geometry whereby the inner regions of an optically thick, geometrically thin accretion disc evaporate to form an optically thin, large scale height accretion flow. The QPO is driven by Lense-Thirring precession of the entire hot flow and the broad band noise is due to fluctuations in mass accretion rate which propagate towards the central object. Mass conservation ties these two processes together, enabling me to define a model for the QPO and broad band noise which uses only one set of parameters. I am thus able fit the model to data. The accretion rate fluctuations drive fluctuations in the precession frequency, giving rise to a quasi-periodic oscillation rather than a pure periodicity. The model thus predicts recent observations which show the QPO frequency to correlate with flux on short timescales. I then investigate a more unique model prediction. As the flow precesses, the patch of the disc preferentially illuminated by the flow rotates such that a non face on observer sees a quasi-periodic shift between blue and red shift in the iron K alpha line. An observation of such an effect would constitute excellent evidence for the model.

In this work, we use the recently developed equilibrium generating functional and systematic derivative expansion approach to hydrodynamics to explore three new regimes of relativistic hydrodynamics. First, we derive the equations of motion and write the constitutive relations to first order in derivatives for relativistic fluids coupled to an external vector field. Next, for relativistic fluids in strong magnetic fields B ~ O(1), we derive the equations of motion and present the constitutive relations to first order in derivatives. From the resulting system of equations, we find the hydrodynamic modes for these systems. We also find the constraints on the transport coefficients due to the entropy production argument and derive the corresponding Kubo formulas. Finally, we repeat the same analysis for relativistic fluids coupled to dynamical electromagnetic fields with <B> ~ O(1). / Graduate

Introduction: The discovery of new solutions to Einstein's field equations has long been a problem in General Relativity. However due to new techniques of Newman and Penrose [1], Carter [2] and others there has been a considerable proliferation of new solutions in recent times. Consequently a new problem has arisen. How are we to interpret the new solutions physically? The tools available, despite a spate of papers in the past fifteen years, remain inadequate although often sophisticated. Any attempts at physical interpretations of metrics are beset with difficulties. There is always the possibility that two entirely different physical pictures will emerge. For example a direct approach would be to attempt an "infilling" of the metric, that is, an extension of the metric into the region occupied by the gravitating matter. However even for the Kerr [1] metric the infilling is by no means unique, in fact a most natural "infilling" turns out to be unphysical (Israel [1]). Yet few people would doubt the physical significance of the Kerr metric. Viewed in this light our attempt to discuss, among other things, the physical interpretation of type D metrics is slightly ambitious. However the problems with regard to this type of metric are not as formidable as for most of the other metrics, since we have been able to integrate the geodesic equations. Nevertheless it is still not possible to produce complete answers to all the questions posed. After a chapter on Mathematical preliminaries the study divides naturally into four sections. We start with an outline of the Hamilton-Jacobi theory of Rund [1] and then go on to show how this theory can be applied to the Carter [2] metrics. In the process we lay a foundation in the calculus of variations for Carter's work. This leads us to the construction of Killing tensors for all but one of the Kinnersley [1] type D vacuum metrics and the Cartei [2] metrics which are not necessarily vacuum metrics. The geodesic equations, for these metrics, are integrated using the Hamilton-Jacobi procedure. The remaining chapters are devoted to the Kinnersley [1] type D vacuum metrics. We omit his class I metrics since these are the Schwarzschild metrics, and have been studied in detail before. Chapter three is devoted to a general study of his class II a metric, a generalisation of the Kerr [1] and NUT (Newman, Tamburino and Unti [1]) metrics. We integrate the geodesic equations and discuss certain general properties: the question of geodesic completeness, the asymptotic properties, and the existence of Killing horizons. Chapter four is concerned with the interpretation of the new parameter 'l', that arises in the class II a and NUT metrics. This parameter was interpreted by Demianski and Newman [1] as a magnetic monopole of mass. Our work centers on the possibility of obtaining observable effects from the presence of 'l'. We have been able to show that its presence is observable, at least in principle, from a study of the motion of particles in the field. In the first place, if l is comparable to the mass of the gravitating system, a comparatively large perihelion shift is to be expected. The possibility of anomalous behaviour in the orbits of test particles, quite unlike anything that occurs in a Newtonian or Schwarzschild field, also arises. In the fifth chapter the Kinnersley class IV metrics are considered. These metrics, which in their simplest form have been known for some time, present serious problems and no interpretations have been suggested. Our discussion is essentially exploratory and the information that does emerge takes the form of suggestions rather than conclusions. Intrinsically the metrics give the impression that interesting results should be obtainable since they are asymptotically flat in certain directions. However the case that we have dealt with does not appear to represent a radiation metric.

The study of geodesics in space-time is essential to a comprehensive understanding of the physics of the field. Global properties, e.g. the singularity structure and completeness of space-time, can be related to the geodesic properties, thus it is through the solutions of the geodesic equation of motion that many of the global properties of space-time can be obtained in an easily interpretable form. However, it is usually very difficult to integrate the geodesic equations for the particle motion in the presence of a gravitational field (Introduction, p. 1)

I study linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes. In the first part of this thesis two decay results are proven for general finite energy solutions to the linear wave equation on higher dimensional Schwarzschild black holes. I establish uniform energy decay and improved interior first order energy decay in all dimensions with rates in accordance with the 3 + 1-dimensional case. The method of proof departs from earlier work on this problem. I apply and extend the new physical space approach to decay of Dafermos and Rodnianski. An integrated local energy decay estimate for the wave equation on higher dimensional Schwarzschild black holes is proven. In the second part of this thesis the global study of solutions to the linear wave equation on expanding de Sitter and Schwarzschild de Sitter spacetimes is initiated. I show that finite energy solutions to the initial value problem are globally bounded and have a limit on the future boundary that can be viewed as a function on the standard cylinder. Both problems are related to the Cauchy problem in General Relativity.