Quadratic curvature terms are commonly introduced in the action as first-order corrections of General Relativity, and, in this thesis, we investigated their impact on the most simple isolated objects, that are the static and spherically symmetric ones. Most of the work has been done in the context of Stelle's theory of gravity, in which the most general quadratic contractions of curvature tensors are added to the action of General Relativity without a cosmological constant. We studied this theory's possible static, spherically symmetric and asymptotically flat solutions with both analytical approximations and numerical methods. We found black holes with Schwarzschild and non-Schwarzschild nature, naked singularities which can have either an attractive or repulsive gravitational potential in the origin, non-symmetric wormholes which connects an asymptotically flat spacetime with an asymptotically singular one, and non-vacuum solutions modeled by perfect fluids with different equations of state. We described the general geometrical properties of these solutions and linked these short-scale behaviors to the values of the parameters which characterize the gravitational field at large distances. We studied linear perturbations of these solutions, finding that most are unstable, and presented a first attempt to picture the parameter space of stable solutions. We also studied the Thermodynamics of black holes and described their evaporation process: we found that either evaporation leads black holes to unstable configurations, or the predictions of quadratic gravity are unphysical. We also considered the possibility of generalizing Stelle's theory by removing the dependence from the only mass-scale present by including a new dynamical scalar field, making the theory scale invariant. Having a more complex theory, we did not investigate exotic solutions but limited ourselves to the impact of the new additional degrees of freedom on known analytical solutions. It was already known that in a cosmological setting this theory admits a transition between two de Sitter configurations; we analyzed the same problem in the context of static and spherically symmetric solutions and found a transition between two Schwarzschild-de Sitter configurations. In order to do that, we studied both linear perturbations and the semiclassical approximation of the path integral formulation of Euclidean quantum gravity. At last, we tried to extract some phenomenological signatures of the exotic solutions. In particular, we investigated the shadow of an object on background free-falling light, and a possible way of determining the behavior close to the origin using mass measurements that rely on different physical processes. We show that, whenever these measurements are applied to the case of compact stars, in principle it could be possible to distinguish solutions where different equations of state describe the fluid.

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Made available in DSpace on 2014-12-17T15:14:58Z (GMT). No. of bitstreams: 1 CalistroSCN.pdf: 698756 bytes, checksum: 2ee4fe2f9e66e82eba6613dc8d66afde (MD5) Previous issue date: 2006-07-07 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In this work we obtain the cosmological solutions and investigate the thermodynamics of matter creation in two diferent contexts. In the first we propose a cosmological model with a time varying speed of light c. We consider two diferent time dependence of c for a at Friedmann-Robertson- Walker (FRW) universe. We write the energy conservation law arising from Einstein equations and study how particles are created as c decreases with cosmic epoch. The variation of c is coupled to a cosmological Λ term and both singular and non-singular solutions are possible. We calculate the "adiabatic" particle creation rate and the total number of particles as a function of time and find the constrains imposed by the second law of thermodynamics upon the models. In the second scenario, we study the nonlinearity of the electrodynamics as a source of matter creation in the cosmological models with at FRW geometry. We write the energy conservation law arising from Einstein field equations with cosmological term Λ, solve the field equations and study how particles are created as the magnetic field B changes with cosmic epoch. We obtain solutions for the adiabatic particle creation rate, the total number of particles and the scale factor as a function of time in three cases: Λ = 0, Λ = constant and Λ α H2 (cosmological term proportional to the Hubble parameter). In all cases, the second law of thermodynamics demands that the universe is not contracting (H ≥ 0). The first two solutions are non-singular and exhibit in ationary periods. The third case studied allows an always in ationary universe for a suficiently large cosmological term / Neste trabalho, n?s obtemos as solu??es cosmol?gicas e investigamos a termodin?mica da cria??o de mat?ria em dois contextos diferentes. No primeiro, n?s propomos um modelo cosmol?gico com a velocidade da luz c variando com o tempo. N?s consideramos duas depend?ncias temporais diferentes para c em um universo plano de Friedmann-Robertson-Walker (FRW). N?s escrevemos a lei da conserva??o da energia que surge das equa??es de Einstein e estudamos como as part?culas s?o criadas quando c decresce com o tempo c?smico. A varia??o de c ? acoplada a um termo cosmol?gico Λ e solu??es singulares e n?o-singulares s?o poss?veis. N?s calculamos a taxa de cria?o adiab?tica de part?culas e o n?mero total de partculas como fun??o do tempo e encontramos os v?nculos impostos pela segunda lei da termodin?mica sobre esses modelos. No segundo cen?rio, n?s estudamos a n?o-linearidade da eletrodin?mica como uma fonte de cria??o de mat?ria em modelos cosmol?gicos com geometria de FRW. N?s escrevemos a lei de conserva??o da energia obtida a partir das equa??es de Einstein com termo cosmol?gico A, resolvemos as equa??es de campo e estudamos como as part?culas s?o criadas quando o campo magn?tico B muda com a ?poca c?smica. N?s obtemos solu??es para a taxa de cria??o adiab?tica de part?culas, o n?mero total de part?culas e o fator de escala como uma fun??o do tempo em tr?s casos: Λ = 0, Λ = constante e Λ / H2 (termo cosmol?gico proporcional ao par?metro de Hubble). Em todos os casos, a segunda lei da termodin?mica imp?e que o universo n?o est? em contra??o (H ≥ 0). As primeiras duas solu??es s?o n?o-singulares e exibem per?odos in acion?rios. O terceiro caso permite universos sempre in acion?rios para um termo cosmol?gico suficientemente grande

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Elaboramos um estudo detalhado de alguns aspectos d(e uma versão d)a correspondência AdS/CFT, conjeturada por Maldacena e Witten, entre teorias quânticas de campo num fundo gravitacional dado por um espaço-tempo assintoticamente anti-de Sitter (AAdS), e teorias quânticas de campos conformalmente covariantes no infinito conforme (no sentido de Penrose) deste espaço-tempo, aspectos estes: (a) independentes d(o par d)e modelos específicos em Teoria Quântica de Campos, e (b) suscetíveis a uma reformulação em moldes matematicamente rigorosos. Adotamos como ponto de partida o teorema demonstrado por Rehren no contexto da Física Quântica Local (também conhecida como Teoria Quântica de Campos Algébrica) em espaços-tempos anti-de Sitter (AdS), denominado holografia algébrica ou dualidade de Rehren. O corpo do presente trabalho consiste em estender o resultado de Rehren para uma classe razoavelmente geral de espaços-tempos AAdS d-dimensionais (d>3), escrutinar como as propriedades desta extensão são enfraquecidas e/ou modificadas em relação ao espaço-tempo AdS, e como efeitos gravitacionais não-triviais se manifestam na teoria quântica no infinito conforme. Dentre os resultados obtidos, citamos: condições razoavelmente gerais sobre geodésicas nulas no interior (cuja plausibilidade justificamos por meio de resultados de rigidez geométrica) não só garantem que a nossa generalização é geometricamente consistente com causalidade, como também permite uma reconstrução ``holográfica\'\' da topologia do interior na ausência de horizontes e singularidades; a implementação das simetrias conformes na fronteira, que associamos explicitamente a uma família de isometrias assintóticas do interior construída de maneira intrínseca, ocorre num caráter puramente assintótico e é atingida dinamicamente por um processo de retorno ao equilíbrio, mediante condições de contorno adequadas no infinito; efeitos gravitacionais podem eventualmente causar obstruções à reconstrução da teoria quântica no interior, ou por torná-la trivial em regiões suficientemente pequenas ou devido à existência de múltiplos vácuos inequivalentes, que por sua vez levam à existência de excitações solitônicas localizadas ao redor de paredes de domínio no interior, similares a D-branas. As demonstrações fazem uso extensivo de geometria Lorentziana global. A linguagem empregada para as teorias quânticas relevantes para nossa generalização da dualidade de Rehren segue a formulação funtorial de Brunetti, Fredenhagen e Verch para a Física Quântica Local, estendida posteriormente por Sommer para incorporar condições de contorno. / We elaborate a detailed study of certain aspects of (a version of) the AdS/CFT correspondence, conjectured by Maldacena and Witten, between quantum field theories in a gravitational background given by an asymptotically anti-de Sitter (AAdS) spacetime, and conformally covariant quantum field theories in the latter\'s conformal infinity (in the sense of Penrose), aspects such that: (a) are independent from (the pair of) specific models in Quantum Field Theory, and (b) susceptible to a recast in a mathematically rigorous mould. We adopt as a starting point the theorem demonstrated by Rehren in the context of Local Quantum Physics (also known as Algebraic Quantum Field Theory) in anti-de Sitter (AdS) spacetimes, called algebraic holography or Rehren duality. The main body of the present work consists in extending Rehren\'s result to a reasonably general class of d-dimensional AAdS spacetimes (d>3), scrutinizing how the properties of such an extension are weakened and/or modified as compared to AdS spacetime, and probing how non-trivial gravitational effects manifest themselves in the conformal infinity\'s quantum theory. Among the obtained results, we quote: not only does the imposition of reasonably general conditions on bulk null geodesics (whose plausibility we justify through geometrical rigidity techniques) guarantee that our generalization is geometrically consistent with causality, but it also allows a ``holographic\'\' reconstruction of the bulk topology in the absence of horizons and singularities; the implementation of conformal symmetries in the boundary, which we explicitly associate to an intrinsically constructed family of bulk asymptotic isometries, have a purely asymptotic character and is dynamically attained through a process of return to equilibrium, given suitable boundary conditions at infinity; gravitational effects may cause obstructions to the reconstruction of the bulk quantum theory, either by making the latter trivial in sufficiently small regions or due to the existence of multiple inequivalent vacua, which on their turn lead to the existence of solitonic excitations localized around domain walls, similar to D-branes. The proofs make extensive use of global Lorentzian geometry. The language employed for the quantum theories relevant for our generalization of Rehren duality follows the functorial formulation of Local Quantum Physics due to Brunetti, Fredenhagen and Verch, extended afterwards by Sommer in order to incorporate boundary conditions. (An English translation of the full text can be found at arXiv:0712.0401)

Nous entendons confronter pratique des mathématiques et réflexions sur les mathématiques dans l'œuvre de Weyl. Nous étudierons : (a) ses monographies en analyse complexe, en relativité générale et en mécanique quantique, (b) les articles en lien avec ces ouvrages, (c) certains de ses cours, (d) sa correspondance avec divers scientifiques, principalement A. Einstein, E. Cartan, J. von Neumann. Nous voulons savoir si les théories mathématiques qu'il investit conditionnent ses positions sur les fondements des mathématiques. Inversement, nous montrerons que les philosophies auxquelles il se réfère – essentiellement le criticisme kantien, l'idéalisme fichtéen et la phénoménologie de Husserl – conditionnent ses recherches. Tout d'abord, nous reviendrons sur Die Idee der Riemannschen Fläche (première éd. 1913). Nous montrerons qu'il opte alors pour un formalisme mitigé. Il se revendique de deux traditions incarnées par Klein et par Hilbert. Ensuite, nous étudierons les éditions successives de Raum, Zeit, Materie (1918-1923). Nous aborderons le projet d'une géométrie purement infinitésimale qui permet à Weyl de proposer une théorie unifiée des champs, cette dernière étant réfutée par Einstein, Pauli, Reichenbach, Hilbert and Eddington. Nous décrirons aussi la construction et la résolution de son « problème de l'espace » (1921-1923). Nous indiquerons comment la référence aux philosophies de Fichte et de Husserl permet d'éclairer ces deux projets. Enfin, nous commenterons l'article de Weyl sur les groupes de Lie (1925-1926) ainsi que son ouvrage Gruppentheorie und Quantenmechanik (1928, 1931). Son article sur les groupes de Lie manifeste la voie moyenne entre formalisme et intuitionnisme qu'il adopte en 1924. Son ouvrage en mécanique quantique incarne quant à lui un « tournant empirique » dans son épistémologie qu'il conviendra de comparer \`a l'« empirisme logique ». / Our purpose consists in comparing Weyl's mathematical practice with his philosophical reflections on mathematics. We will study (a) his monographs on complex analysis, general relativity and quantum mechanics, (b) the articles which are linked to these books, (c) some of his lecture courses, (d) his correspondence with different scientists, mainly A. Einstein, E. Cartan, J. von Neumann. We will show that his mathematical research has a strong influence on the different stands he successively takes regarding the foundations of mathematics. Conversely, we will show that the philosophical systems he refers to (mainly kantian criticism, fichtean idealism and husserlian phenomenology) have a real impact on his investigations in mathematics. We will first analyse Die Idee der Riemannschen Fläche (first edition 1913). In this book, Weyl seems to take up a formalist point of view, but this is partly true. In fact, he is influenced by two traditions respectively embodied by Hilbert and Klein. Then, we will study the successive editions of Raum, Zeit, Materie (1918-1923). We will describe Weyl's project of a “purely infinitesimal geometry”. Thanks to this geometrical framework, he builds a unified fields theory, which will be disproved by Einstein, Pauli, Reichenbach, Hilbert and Eddington. During this short period, Weyl also constructs and solves the so-called space problem (1921-1923). Weyl's references to Fichte and Husserl have a significant impact on these two projects. Finally, we will comment Weyl's main article on Lie groups (1925-1926) and his monograph on quantum mechanics, i.e. Gruppentheorie und Quantenmechanik (1rst ed. 1928, 2nd ed. 1931). Weyl's article on Lie groups is in accordance with his compromise between intuitionism and formalism (1924). On the other hand, Weyl's book on quantum mechanics encapsulates an “empirical turn” in his epistemology, which will be compared with the so-called empirical logicism.

A presente dissertação tem como objeto de estudo sistemas da física da matéria condensada que sejam capazes de simular sistemas gravitacionais, tais como buracos negros e universos em expansão, onde processos quânticos tomam parte. Neste estudo nos debruçamos principalmente sobre o modelo do fluido e condensados de Bose-Einstein. No modelo do fluido exploramos a geometria efetiva que surge e os problemas de back-reaction e dos modos trans-planckianos de campos quânticos. No modelo baseado em condensados exploramos sua faceta cosmológica e a possibilidade de campos maciços. Além destes dois modelos de grande relevância na literatura, ainda expomos os análogos em cordas elásticas e os baseados em ondas na superfícies de fluidos e uma análise geral baseada no formalismo lagrangeano para campos. / This dissertation has as object of study systems of condensate-matter physics which can simulate gravitational systems like black holes and expanding universes where quantum processes take place. In this study we lay attention mainly on the fluid model and on Bose-Einstein-condensate-based models. In the fluid model we explore the features of the emergent geometry and other problems like the back-reaction and the trans-planckian modes of quantum fields. In the condensate-based models we explore their cosmological aspects and the possibility for massive fields. Moreover, we shall present two other models, the elastic string and the surface-wave-based models in fluids, and a very general analysis based on the Lagrangean formalism for fields.

Dans le modèle standard de la cosmologie, la dynamique globale de l'Univers est modélisée par l'intermédiaire d'un espace-temps de référence (ou de fond) fortement symétrique, admettant des sections spatiales homogènes et isotropes. Le couplage entre les sources fluides, homogènes, et l'expansion globale, y est déterminé par les équations d'Einstein de la Relativité Générale. La formation de structures inhomogènes de matière peut également être décrite dans ce modèle. Selon l'époque et l'échelle considérées, cette description est effectuée soit à l'aide d'un schéma perturbatif relativiste supposant une faible déviation de chaque grandeur par rapport au fond homogène imposé, soit au moyen d'une approche newtonienne au sein du même fond en expansion. L'interprétation des observations dans ce modèle suggère cependant une accélération inattendue de l'expansion, qui requiert une nouvelle composante énergétique mal comprise, l' «Énergie Noire», en plus de la Matière Noire. La cosmologie inhomogène a pour but de lever les restrictions imposées par ces modèles sur la géométrie et sur les sources sans sortir du cadre de la Relativité Générale. Cela peut notamment permettre d'améliorer le modèle de formation des structures pour prendre en compte de fortes déviations par rapport à l'homogénéité dans la distribution de matière et dans la géométrie. Cela permet également d'étudier les conséquences dynamiques, appelées effets de rétroaction («backreaction»), du développement local de telles inhomogénéités sur l'expansion à de plus grandes échelles. De telles rétroactions peuvent alors reproduire, au moins partiellement, les comportements attribués à l'Énergie Noire ou à la Matière Noire. Au cours de mon travail de thèse sous la direction de Thomas Buchert, j'ai étudié plusieurs aspects analytiques de la cosmologie inhomogène en Relativité Générale. Je présente ci-dessous les résultats de travaux au sein de collaborations, auxquels j'ai apporté des contributions majeures dans le cadre de la thèse. Je me suis tout d'abord concentré sur l'écriture d'un schéma d'approximation relativiste lagrangien, pour décrire la dynamique locale des structures jusqu'à un régime non-linéaire, dans des fluides parfaits barotropes irrotationnels. Je me suis ensuite intéressé à la description effective de fluides inhomogènes admettant un tenseur d'énergie-impulsion général ainsi que de la vorticité, au moyen de deux schémas possibles de moyenne spatiale. Ces schémas s'appliquent à un choix quelconque des hypersurfaces spatiales sur lesquelles moyenner, et fournissent pour chacun de ces choix un système d'équations d'évolution effectives, présentant plusieurs termes de rétroaction, pour un domaine d'intégration suivant la propagation des sources. Cela permet une discussion qualitative de la dépendance au choix du feuilletage des équations moyennes et des rétroactions. J'ai également étudié la réécriture de ces schémas de moyennes et équations d'évolution, et d'autres obtenus de façon similaire, sous une forme unifiée et manifestement 4-covariante. Ce dernier résultat permettra une étude plus explicite de la dépendance au feuilletage / In the standard model of cosmology, the global dynamics of the Universe is modelled via a highly symmetric background spacetime with homogeneous and isotropic spatial sections. The coupling of the homogeneous fluid sources to the overall expansion is then determined by the Einstein equations of General Relativity. In addition, the formation of inhomogeneous matter structures is described either via a relativistic perturbation scheme assuming small deviations of all fields to the prescribed homogeneous background, or using Newtonian dynamics within the same expanding background, depending on the scale and epoch. However, the interpretation of observations within this model calls for an unexpectedly accelerated expansion requiring a poorly-understood `Dark Energy' component, in addition to Dark Matter. Inhomogeneous cosmology aims at relaxing the restrictions of these models on the geometry and sources while staying within the framework of General Relativity. It can allow, in particular, for an improved modelling of the formation of structures accounting for strong deviations from homogeneity in the matter distribution and the geometry. It can also study the dynamical consequences, or backreaction effects, of the development of such inhomogeneities on the expansion of larger scales. Such a backreaction may then reproduce, at least partially, the behaviours attributed to Dark Energy or Dark Matter. During my PhD under the direction of Thomas Buchert, I have been working on several analytical aspects of general-relativistic inhomogeneous cosmology. I present below the results of collaborations in which I played a major role in the context of the PhD. I first focussed on the expression of a relativistic Lagrangian approximation scheme for the description of the local dynamics of structures up to a nonlinear regime in irrotational perfect barotropic fluids. I then considered the effective description of inhomogeneous fluids with vorticity and a general energy-momentum tensor in terms of two possible schemes of spatial averaging. These schemes are applicable to any choice of spatial hypersurfaces of averaging, providing for each choice a set of effective evolution equations, featuring several backreaction terms, for an averaging region comoving with the sources. This allows for a qualitative discussion of the dependence of the average equations and backreactions on the foliation choice. I also studied the rewriting of such averaging schemes and evolution equations under a unified and manifestly 4-covariant form. This latter result will allow for a more explicit investigation of foliation dependence

La Relativité Générale n'est probablement pas la théorie ultime de la gravitation. Cette affirmation est motivée par des considérations théoriques liées à la recherche d'une théorie quantique de la gravitation et aux tentatives d'unification des interactions fondamentales. Une autre motivation provient des observations cosmologiques expliquées par l'introduction de matière noire et d'énergie sombre non observées directement à ce jour. Cette thèse se propose de déduire des signatures observables expérimentalement provenant de théories alternatives de la gravitation. Dans cette optique, nous avons développé des outils qui permettent de simuler des observations dans des théories alternatives de la gravitation pour deux échelles différentes: dans le cadre du Système Solaire et dans le cadre de la cosmologie. Dans le cadre du Système Solaire, nous avons développé des logiciels qui permettent de simuler des observations spatiales de Range, Doppler et de type astrométrique. Au niveau cosmologique, nous avons développé un logiciel qui permet de simuler des observations de distance-luminosité de Supernovae Ia. Ces outils permettent de dériver les signatures caractéristiques des théories alternatives. Nous avons appliqué ces différents outils à quelques théories alternatives de la gravitation et nous avons pu répondre à la question : quel est l'impact d'une modification de la théorie de la gravitation sur des observations et est-il possible de la détecter ?

Le premier objectif de cette thèse est de répondre au manque en cosmologie de description sur le cône de lumière passé: hypersurface nulle où se propagent tous les signaux observés. Son deuxième est d'évaluer l'importance des inhomogénéités dans la détermination des paramètres de l'énergie sombre, de nature encore inconnue et pour laquelle l'influence des structures de matière a été proposée comme une alternative cosmologique. Nous définissons une jauge "géodésique sur le cône de lumière" qui simplifie grandement l'étude de la propagation lumineuse dans l'Univers. Nous établissons ensuite une moyenne, invariante de jauge, pour les quantités scalaires sur le cône et calculons l'effet des inhomogénéités sur le flux lumineux en considérant leur spectre de puissance jusque son régime non-linéaire. Leur effet sur le flux est négligeable tandis que des observables comme la distance de luminosité, évaluée jusqu'au second ordre en perturbations, sont bien plus affectées. Nous étudions aussi le module de distance à la base de la découverte de l'énergie sombre par les supernovæ (SNe) Ia. La moyenne de ce module est peu affectée par les inhomogénéités mais sa variance l'est sensiblement. Ces résultats montrent, dans leur cadre, qu'une alternative à l'énergie sombre par un effet des structures est impossible mais, en même temps, soulignent leur importance sur la dispersion des données du diagramme de Hubble. Les effets physiques dominants sont la vitesse des SNe et l'effet de lentille faible. Nos prédictions sur la dispersion se révèlent en très bon accord avec les premières analyses de SNe visant à détecter un signal de lentilles et seront vérifiables dans les années futures.