Made available in DSpace on 2015-05-14T12:14:18Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1268026 bytes, checksum: 26540918af151a01bb663415557d121e (MD5) Previous issue date: 2010-02-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper deals with the influence of the gravitational field produced by a rotating black hole on quantum systems. More specifically, are considered scalar quantum particles, which are described by the Klein-Gordon equation. Initially, it was shown a way by which is possible to obtain the Kerr metric, which characterize a rotating black hole. Still on the Kerr metric, it was studied some important properties of this spacetime. Was then obtained the exact solution of the Klein-Gordon equation in the Kerr spacetime, which is given in terms of the confluent Heun s functions and, in the particular case of extreme Kerr, was obtained that the solution of the Klein-Gordon equation in this spacetime is given by the doubly confluent Heun s functions. For the Klein-Gordon equation in the Kerr spacetime, it was verified that the solution is consistent with results already known in the literature for regions near the event horizon and at infinity. Moreover, due to the difficulties inherent in the Kerr metric, was considered the limit where the black hole has low rotational speed, resulting in the metric of Lense-Thirring. In this situation, using an asymptotic method and a method in series, were obtained approximate solutions that describe the behavior of scalar quantum particles in the presence of the gravitational field produced by the body. Finally, some physical effects in Kerr spacetime were considered. / Este trabalho trata sobre a influência do campo gravitacional produzido por um buraco negro com rotação sobre sistemas quânticos. Mais especificamente, são consideradas partículas quânticas escalares, que são descritas através da equação de Klein-Gordon. Inicialmente, é mostrado uma maneira através da qual é possíıvel obter a métrica de Kerr, a qual caracteriza um buraco negro com rotação. Ainda sobre a métrica de Kerr, são estudadas algumas propriedades importantes deste espaço-tempo. Em seguida, foi obtida a solução exata da equação de Klein- Gordon no espaço-tempo de Kerr, sendo esta dada em termos das funções confluentes de Heun e, no caso particular de Kerr extremo, foi obtido que a soluçã o da equação de Klein-Gordon neste espaço-tempo é dada pelas funções duplamente confluente de Heun. Para a equação de Klein-Gordon no espaço-tempo de Kerr, verificou-se que a soluçãoo obtida é compatível com resultados já conhecidos na literatura para regiões próximo ao horizonte de eventos e no infinito. Por outro lado, devido às dificuldades inerentes á métrica de Kerr, foi considerado o limite em que o buraco negro possui baixas velocidades de rotação, resultando na métrica de Lense-Thirring. Nesta situação, usando um método assintótico e um outro método em série, foram obtidas soluções aproximadas que descrevem o comportamento de partículas quânticas escalares na presença do campo gravitacional produzido por este corpo. Por fim, alguns efeitos físicos no espaço-tempo de Kerr foram considerados.

Nesta dissertação apresentamos um método de quantização matemática e conceitualmente rigoroso para o campo escalar livre de interações. Trazemos de início alguns aspéctos importantes da Teoria de Distribuições e colocamos alguns pontos de geometria Lorentziana. O restante do trabalho é dividido em duas partes: na primeira, estudamos equações de onda em variedades Lorentzianas globalmente hiperbólicas e apresentamos o conceito de soluções fundamentais no contexto de equações locais. Em seguida, progressivamente construímos soluções fundamentais para o operador de onda a partir da distribuição de Riesz. Uma vez estabelecida uma solução para a equação de onda em uma vizinhança de um ponto da variedade, tratamos de construir uma solução global a partir da extensão do problema de Cauchy a toda a variedade, donde as soluções fundamentais dão lugar aos operadores de Green a partir da introdução de uma condição de contorno. Na última parte do trabalho, apresentamos um mínimo da Teoria de Categorias e Funtores para utilizar esse formalismo na contrução de um funtor de segunda quantização entre a categoria de variedades Lorentzianas globalmente hiperbólicas e a categoria de redes de álgebras C* satisfazendo os axiomas de Haag-Kastler. Ao fim, retomamos o caso particular do campo escalar quântico livre. / In this thesis we present a both mathematical and conceptually rigorous quantization method for the neutral scalar field free of interactions. Initially, we introduce some aspects of the Theory of Distributions and we establish some points of Lorentzian geometry. The rest of the work is divided in two parts: in the first one, we study wave equations on globally hyperbolic Lorentzian manifolds, hence presenting the concept of fundamental solutions within the context of locally defined wave equations. Next, we progressively construct fundamental solutions for the wave operator from the Riesz distribution. Once established a solution to the wave equation in a neighbourhood of a point of the manifold, we move forward to produce a global solution from the extension of the Cauchy problem to the whole manifold. At this stage, fundamental solutions are replaced by Green\'s operators by the imposition of appropriate boundary conditions. In the last part, we present a minimum on the Theory of Categories and Functors. This is followed by the use of this formalism in the development of a second-quantization functor between the category of Lorentzian globally hyperbolic manifolds and the category of nets of C*-algebras obeying Haag-Kastler axioms. Finally, we turn our attention to the particular case of the quantum free scalar field.

Dans cette thèse, on s'est attaché d'une part à d'écrire le défaut de compacité de l'injection de Sobolev critique dans les différentes classes d'espaces d'Orlicz, et d'autre part à étudier l'équation de Klein-Gordon avec une non-linéarité exponentielle. Ce travail se divise en trois parties. L'objectif de la première partie est de caractériser le défaut de compacité de l'injection de Sobolev de $H^2_{rad}(R^4)$ dans l'espace d'Orlicz $mathcal{L}(R^4)$.Le but de la deuxième partie est double : tout d'abord, on a décrit le défaut de compacité de l'injection de Sobolev de $H^1(R^2)$ dans les différentes classes d'espaces d'Orlicz, ensuite on a étudié une famille d'équations de Klein-Gordon non linéaires à croissance exponentielle. Cette étude inclut à la fois les problèmes d'existence globale, de complétude asymptotique et d'étude qualitative pour le problème de Cauchy associé. La troisième partie est dédiée à l'analyse des solutions de l'équation de Klein-Gordon 2D issues d'une suite de données de Cauchy bornée dans $H^1_{rad}(R^2)times L^2_{rad}(R^2)$. Basée sur les décompositions en profils, cette analyse a été conduite dans le cadre de la norme d'Orlicz / In this thesis, we focused on the one hand on the description of the lack of compactness of the critical Sobolev embedding into different classes of Orlicz spaces, and on the other hand on the study of the nonlinear Klein-Gordon equation with exponential nonlinearity. This work is divided into three parts. The aim of the first part is to characterize the lack of compactness of the Sobolev embedding of $H^2_{rad}(R^4)$ into the Orlicz space $mathcal{L}(R^4)$.The aim of the second part is twofold: firstly, we describe the lack of compactness of the Sobolev embedding of $H^1(R^2)$ into different classes of Orlicz spaces, secondly we investigate a family of nonlinear Klein-Gordon equations with exponential nonlinearity. This study includes both the global existence problem, the asymptotic completeness and the qualitative study for the associated Cauchy problem. The third part is dedicated to the analysis of the solutions to the 2D Klein-Gordon equation associated to a sequence of bounded Cauchy data in $H^1_{rad}(R^2)times L^2_{rad}(R^2)$. Based on the profile decompositions, this analysis was conducted in the framework of Orlicz norm

An investigation of information loss in black hole spacetimes is performed. We demon- strate that the definition of particles as energy levels of the Harmonic oscillator will not have physical significance in general and is thus not a good instrument to study the ra- diation of black holes. This is due to the ambiguity of the choice of coordinates on the phase space of the quantum field. We demonstrate how to identify quantum states in the functional Schr ̈dinger picture. o We demonstrate that information is truly lost in the case of a Vaidya black hole (a black hole formed from null dust) if we neglect back reaction. This is done by quantizing the constrained classical system of a Klein-Gordon field in a Vaidya background. The interaction picture of quantum mechanics can be applied to this system. We find a physically well motivated vacuum state for a spherically symmetric space- time with an extra conformal Killing vector. We also demonstrate how to calculate the response of a particle detector in the a LeMaitre-Tolman-Bondi spacetime with a self- similarity. Finally, some of the claims and confusion surrounding Unruh radiation, Hawking radiation and the equivalence principle are investigated and shown to be false.

An investigation of information loss in black hole spacetimes is performed. We demon- strate that the definition of particles as energy levels of the Harmonic oscillator will not have physical significance in general and is thus not a good instrument to study the ra- diation of black holes. This is due to the ambiguity of the choice of coordinates on the phase space of the quantum field. We demonstrate how to identify quantum states in the functional Schr ̈dinger picture. o We demonstrate that information is truly lost in the case of a Vaidya black hole (a black hole formed from null dust) if we neglect back reaction. This is done by quantizing the constrained classical system of a Klein-Gordon field in a Vaidya background. The interaction picture of quantum mechanics can be applied to this system. We find a physically well motivated vacuum state for a spherically symmetric space- time with an extra conformal Killing vector. We also demonstrate how to calculate the response of a particle detector in the a LeMaitre-Tolman-Bondi spacetime with a self- similarity. Finally, some of the claims and confusion surrounding Unruh radiation, Hawking radiation and the equivalence principle are investigated and shown to be false.

Made available in DSpace on 2014-06-11T19:30:27Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-01-28Bitstream added on 2014-06-13T19:40:05Z : No. of bitstreams: 1 nunes_rso_dr_sjrp_parcial.pdf: 159616 bytes, checksum: e530de9f87ecb29201522bba4142a965 (MD5) / Neste trabalho resolvemos o problema de controlabilidade exata na fronteira para a equação linear de Klein-Gordon em domínios limitadosΩ deR N , N≥2, com fronteira suave por partes e sem cuspides. Para dados iniciais emH 1 (Ω)×L 2 (Ω) obtemos controle do tipo Neuman, de quadrado integr´avel, atuando em toda a fronteira do domínio em tempo próximo ao diâmetro de Ω. Inicialmente provamos que a energia da solução do problema de Cauchy para a referida equação decai localmente numa taxa polinomial. Em seguida, estendendo a solução do problema de Cauchy para tempo complexo provamos que o operador solução associado ao problema de Cauchy é analítico num setor adequado do plano complexo. Utilizando o decaimento de energia, a analitidade do operador solução e argumentos introduzidos por D. L. Russell e J. Lagnese nos anos setenta do século passado obtemos o resultado desejado / In this work we solve the problem of exact controllability on the boundary for the linear Klein-Gordon equation in limited domains ΩofR N , N≥2, with piecewise smooth boundary without cusps. For initial data inH 1 (Ω)×L 2 (Ω)we get square integrable control of Neuman type, acting on the entire boundary, in a time near the diameter ofΩ. Initially we prove that the energy of the solution of the Cauchy problem for this equation locally decays at a polynomial rate. Then extending the solution of the Cauchy problem for complex time we prove that the solution operator associated with the Cauchy problem is analytic in a suitable sector of the complex plane. Using the local decay of energy, the analiticity of the solution operator and arguments introduced by D. L. Russell and J. Lagnese in the seventies we obtain the desired result

Submitted by Livia Mello (liviacmello@yahoo.com.br) on 2016-09-23T20:38:51Z No. of bitstreams: 1 TeseWNN.pdf: 1247691 bytes, checksum: 63f743255181169a9bb4ca1dfd2312c2 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:35:27Z (GMT) No. of bitstreams: 1 TeseWNN.pdf: 1247691 bytes, checksum: 63f743255181169a9bb4ca1dfd2312c2 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-26T20:35:33Z (GMT) No. of bitstreams: 1 TeseWNN.pdf: 1247691 bytes, checksum: 63f743255181169a9bb4ca1dfd2312c2 (MD5) / Made available in DSpace on 2016-09-26T20:35:40Z (GMT). No. of bitstreams: 1 TeseWNN.pdf: 1247691 bytes, checksum: 63f743255181169a9bb4ca1dfd2312c2 (MD5) Previous issue date: 2016-02-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / In this thesis we study the asymptotic properties for the solution of the Cauchy problem for the Klein-Gordon equation with non-effective time-dependent potential. The main goal was define a suitable energy related to the Cauchy problem and derive decay estimates for such energy. Strichartz’ estimates and results of scattering and modified scattering was established. The C m theory and the stabilization condition was applied to treat the case where the coefficient of the potential term has very fast oscillations. Moreover, we consider a semi-linear wave model scale-invariant time- dependent with mass and dissipation, in this step we used linear estimates related with the semi-linear model to prove global existence (in time) of energy solutions for small data and we show a blow-up result for a suitable choice of the coefficients. / Nesta tese estudamos as propriedades assintóticas para a solução do problema de Cauchy para a equação de Klein-Gordon com potencial não efetivo dependente do tempo. O principal objetivo foi definir uma energia adequada relacionada ao problema de Cauchy e derivar estimativas para tal energia. Estimativas de Strichartz e resultados de scatering e scatering modificados também foram estabelecidos. A teoria C m e a condição de estabilização foram aplicados para tratar o caso em que o coeficiente da massa oscila muito rápido. Além disso, consideramos um mod- elo de onda semi-linear scale-invariante com massa e dissipação dependentes do tempo, nesta etapa usamos as estimativas lineares de tal modelo para provar ex- istência global (no tempo) de solução de energia para dados iniciais suficientemente pequenos e demonstramos um resultado de blow-up para uma escolha adequada dos coeficientes.

No description available.

Au cours des années récentes, plusieurs auteurs ont prouvé des résultats d'existence en temps grand pour des solutions d'équations de Klein-Gordon non-linéaires sur certaines variétés compactes, telles les sphères, lorsque les données initiales sont assez régulières et assez petites, et qu'un certain paramètre de masse évite un sous-ensemble de mesure nulle de la droite réelle. L'une des hypothèses fondamentales dans ces travaux est une propriété de séparation des valeurs propres du laplacien sur les variétés considérées. L'objet des deux premiers articles constituant cette thèse est d'examiner quels résultats peuvent être obtenus lorsqu'une telle hypothèse de séparation n'est plus vérifiée. Nous étudions le cas d'un opérateur de Klein-Gordon associé à l'oscillateur harmonique sur l'espace euclidien, et celui de l'opérateur de Klein-Gordon usuel sur le tore. Nous obtenons, par des méthodes de formes normales, des solutions existant sur des intervalles plus longs que ceux fournis par la théorie locale. Le dernier article de cette thèse s'intéresse au problème de l'estimation en temps grand des normes Sobolev de solutions d'une équation de Schrödinger linéaire sur le tore, à potentiel dépendant du temps. Nous prouvons des bornes logarithmiques, lorsque le potentiel est Gevrey, généralisant des résultats antérieurs de Bourgain et Wang.

L'objet de cette thèse est de présenter une nouvelle mesure de la masse du pion en utilisant la spectroscopie X de l'hydrogène pionique et des résultats de spectroscopie de l'argon et du soufre héliumoïdes. La nouvelle masse du pion a été mesurée avec une précision 30% supérieure à la moyenne mondiale actuelle, c'est-à-dire égale à 1.7 ppm. Elle a été obtenue par spectroscopie de Bragg des transitions 5 -> 4 de l'azote pionique en utilisant les prédictions théoriques de QED. Je présente le calcul de la structure hyperfine et celui de la correction de recul du noyau pour les atomes pioniques au moyen d'une nouvelle méthode de perturbation de l'équation Klein-Gordon.Le spectromètre utilisé pour cette mesure a été caractérisé grâce aux transitions relativistes des atomes héliumoïdes produits dans un nouveau type de source d'ions à résonance cyclotronique des électrons. Les spectres haute statistique de ces ions permettent de mesurer les énergies de transition avec une précision de quelques ppm, ce qui permet de tester, avec un degré de précision jamais atteint, les prédictions théoriques. L'émission de rayons-X des atomes pioniques et des ions multichargés peut ainsi être utilisée pour la définition de nouveaux étalons de rayons-X de quelques keV.