Dissertations / Theses on the topic 'Mathematical aspects of general relativity'

This thesis contains two main themes. The first is Einstein's theory of general relativity in higher dimensions, while the second is M-theory. The first part of the thesis concerns the use of classification techniques based on the Weyl curvature in an attempt to systematically study higher dimensional general relativity and its solutions. After a review of the various classification schemes, the application of these schemes to the study of higher dimensional solutions is explained. The first application of the tensor approach that is discussed is the systematic classification of higher dimensional axisymmetric solutions. A complete classification of all algebraically special axisymmetric solutions to the vacuum Einstein equation in higher dimensions is presented. Next, the study of perturbations of higher dimensional solutions within this framework and the possibility of decoupling equations for black hole solutions of interest, as has been successfully done in four dimensions, is considered. In the case where such a decoupling of the perturbations is possible, a map for constructing solutions of the perturbation equation is presented and is applied to the Kerr/CFT correspondence. Also, the property of gravitational radiation emitted from an isolated source in higher dimensions is considered and the tensor classification scheme is used to derive the peeling property of the Weyl tensor in higher dimensions. This is shown to be different to that which occurs in four dimensions. Finally, after an in-depth exposition of the spinor classification scheme and its relation to the tensor approach, solutions belonging to the most special type in the spinor classification are classified. In addition, the classification of the black ring in this scheme is discussed. The second part of the thesis explores the use of generalised geometry as a tool for better understanding M-theory. After briefly reviewing the curious phenomenon of M-theory dualities, it is explained how generalised geometry can be used to show that these symmetries are not exclusive to compactifications of the theory, but can be made manifest without recourse to compactification. Finally, results regarding the local symmetries of M-theory in the generalised geometry framework for a particular symmetry group are presented.

The purpose of this thesis is to study in depth the relationship between the curvature of space-time and the other geometrical objects which naturally arise in general relativity. Most of the results obtained apply to the generic case. Chapter 1 contains a discussion of certain aspects of fibre bundle theory required in later chapters which may be unfamiliar to many relativists, while chapter 2 contains preliminary material on curvature in relativity and proves a continuity property of the algebraic classification of the Weyl and energy-momentum tensors. Chapter 3 describes the generic behaviour of the Riemann, Weyl and energy-momentum tensors, and chapter 5 goes on to use this description to investigate the relationship of the Riemann tensor to the metric, conformal class and connection of space-time in the generic case. In particular it is proved that the Riemann tensor uniquely and continuously determines the connections. The information obtained in chapter 3 on the algebraic type of curvature in the general case is related in chapter 4 to the topology of the underlying manifold. In chapter 6 a topology is defined on the set of sectional curvatures of all Lorentz metrics on a given manifold. The remainder of the chapter attempts to do for the sectional curvature what was done for the Riemann tensor in chapter 5 but, because sectional curvature is more difficult to handle, the results obtained are necessarily more modest.

The first part of this thesis is concerned with the question of global uniqueness of solutions to the initial value problem in general relativity. In 1969, Choquet-Bruhat and Geroch proved, that in the class of globally hyperbolic Cauchy developments, there is a unique maximal Cauchy development. The original proof, however, has the peculiar feature that it appeals to Zorn’s lemma in order to guarantee the existence of this maximal development; in particular, the proof is not constructive. In the first part of this thesis we give a proof of the above mentioned theorem that avoids the use of Zorn’s lemma. The second part of this thesis investigates the behaviour of so-called Gaussian beam solutions of the wave equation - highly oscillatory and localised solutions which travel, for some time, along null geodesics. The main result of this part of the thesis is a characterisation of the temporal behaviour of the energy of such Gaussian beams in terms of the underlying null geodesic. We conclude by giving applications of this result to black hole spacetimes. Recalling that the wave equation can be considered a “poor man’s” linearisation of the Einstein equations, these applications are of interest for a better understanding of the black hole stability conjecture, which states that the exterior of our explicit black hole solutions is stable to small perturbations, while the interior is expected to be unstable. The last part of the thesis is concerned with the wave equation in the interior of a black hole. In particular, we show that under certain conditions on the black hole parameters, waves that are compactly supported on the event horizon, have finite energy near the Cauchy horizon. This result is again motivated by the investigation of the conjectured instability of the interior of our explicit black hole solutions.

This thesis considers various extensions to the fluid/gravity correspondence as well as problems fundamental to the study of fluid dynamics. The fluid/gravity correspondence is a map between the solutions of the Navier-Stokes equations of fluid dynamics and the solutions of the Einstein equations in one higher spatial dimension. This map arose within the context of string theory and holography and is a specific realisation of a much wider class of dualities known as the Anti de Sitter/Conformal Field Theory (AdS/CFT) correspondence. The first chapter is an introduction; the second chapter reviews the fluid/gravity correspondence. The next two chapters extend existing work on the fluid/gravity map. Our first result concerns the fluid/gravity map for forced fluid dynamics in arbitrary spacetime dimensions. Forced fluid flows are of particular interest as they are known to demonstrate turbulent behaviour. For the case of a fluid with a dilaton-dependent forcing term, we present explicit expressions for the dual bulk metric, the fluid dynamical stress tensor and Lagrangian to second order in boundary spacetime derivatives. Our second result concerns fluid flows with multiple anomalous currents in the presence of external electromagnetic fields. It has recently been shown using thermodynamic arguments that the entropy current for such anomalous fluids contains additional first order terms proportional to the vorticity and magnetic field. Using the fluid/gravity map, we replicate this result using gravitational methods. The final two chapters consider questions related to the equations of fluid dynamics themselves; these chapters do not involve the fluid/gravity correspondence. The first of these chapters is a review of the various constraints that must be satisfied by the transport coefficients. In the final chapter, we derive the constraints obtained by requiring that the equilibrium fluid configurations are linearly stable to small perturbations. The inequalities that we obtain here are slightly weaker than those found by demanding that the divergence of the entropy current is non-negative.

Orientador: Prof. Dr. Roldão da Rocha Jr.
Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2017.
Neste trabalho foram investigadas algumas conseguências da física de buracos negros em teorias cujo domínio está além do domínio da relatividade geral, em especial em teorias efetivos com dimensões extras. A investigação foi em substancialmente conduzida baseando-se em três efeitos gravitacionais, a saber, a radiação Hawking, o regime de deflexão forte de lentes gravitacionais e a formação de buracos negros quânticos. Uma solução de modelo cosmológico imerso em uma brana espessa foi também investigada. Modelos e teorias efetivas fornecem meios para testar os limites de validade de teorias conhecidas e indicam o que deveríamos esperar além desses limites. Baseado nessa ideia foram usados alguns modelos efetivos para estudar efeitos não previstos pela relatividade geral, associados a cada um dos fenômenos mencionados.
This work is devoted to investigate some consequences of black holes physics beyond the domain of general relativity, mainly in effective extra dimensional models. The investigation is carried along three gravitational effects, namely the Hawking radiation, the strong deflection of gravitational lensing and the formation of quantum black holes. A cosmological thick brane solution is also investigated. Effective theories and models provide a prominent approach for testing the limits of known theories and show what would be expected beyond that. Based on such idea we have used effective models for finding deviations of general relativity associated to each of the mentioned phenomena.

Conformal holonomy theory is the holonomy theory of the tractor connection on a conformal manifold. In this thesis, we present the first application of conformal holonomy theory to theoretical physics and determine the conformal holonomy groups/algebras of physically relevant spaces. After recalling some necessary background on conformal structures, tractor bundles and conformal holonomy theory in chapter 1, we begin in chapter 2 by discussing the role of conformal holonomy in the gauge-theoretic MacDowell-Mansouri formulation of general relativity. We show that the gauge algebra of this formulation is uniquely determined by the conformal structure of spacetime itself, in both Lorentzian and Riemannian metric signatures, through the conformal holonomy algebra. We then show that one may construct a MacDowell-Mansouri action functional for scale-invariant gravity, and we discuss a geometric interpretation for the scalar field therein. In chapter 3 we study a class of spacetimes relevant to Maldacena's AdS5=CFT4 correspondence in quantum gravity. It is well known that a Lie group coincidence lies at the heart of this correspondence: the proper isometry group of the bulk precisely matches the conformal group of the boundary. It has previously been proposed that the AdS5=CFT4 correspondence be extended to so-called Poincar e-Einstein spacetimes, which need not be as symmetric as anti-de Sitter space. We show that the conformal holonomy groups of the boundary and bulk furnish such a Lie group coincidence for 5-dimensional Poincar e-Einstein spacetimes in general. We completely characterise this boundary-bulk conformal holonomy matching for the Riemannian theory and present partial results for the Lorentzian theory. In chapter 4 we use the tools developed in the preceding chapters to further the classiification of the conformal holonomy groups of conformally Einstein spaces. Specifically, we determine the conformal holonomy groups of generic neutral signature conformally Einstein 4-manifolds subject to a condition on the conformal holonomy representation. Lastly, in chapter 5, we investigate the conformal holonomy reduction of the Fefferman conformal structures of residual twistor CR manifolds. A sufficient condition for reducible conformal holonomy is that the (Fefferman conformal structure of a) residual twistor CR manifold admit a parallel tractor. We show that this occurs if and only if the residual twistor CR manifold admits a Sasakian structure.

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.

Cette thèse se consacre premièrement à certains aspects de la définition de charges conservées en relativité générale pour les espaces asymptotiquement plats à l’infini spatial. À l’aide de la dualité gravitationnelle, présente au niveau linéarisé, on étudie également l’existence de charges topologiques, magnétiques, ainsi que leurs contributions aux superalgèbres dans les théories de supergravité N = 1 et N = 2 à quatre dimensions. La thèse est divisée en trois parties.

Dans la première partie, les espaces asymptotiquement plats à l’infini spatial sont décrits à l’aide d’une généralisation de la métrique de type Beig-Schmidt. La construction de charges à partir de l’étude des équations du mouvement et de la classification de tenseurs symétriques et de divergences nulles nous permet de démontrer l’unicité des charges de Poincaré pour l’ansatz non-généralisé en présence de conditions de parité. L’équivalence des charges de Ashtekar- Hansen et Mann-Marolf est ainsi revisitée. Dans le cas d’un ansatz généralisé, une régulation de la forme symplectique divergente, à l’aide de contre-termes rajoutés à l’action de Mann-Marolf, nous donne la possibilité de considérer un espace des phases sans conditions de parité, tout en gardant un principe variationnel bien défini. Le groupe asymptotique comprend alors, en plus des charges de Poincaré où les charges de Lorentz ne sont plus asymptotiquement linéaires, des charges non-triviales associées aux supertranslations et aux transformations logarithmiques.

Dans la deuxième partie, on étudie la dualité gravitationnelle et la définition de charges magnétiques en gravitation linéarisée. On revisite la dualité et on montre qu’une dualisation sur les indices de Lorentz facilite la compréhension de celle-ci. Les dix charges de Poincaré ainsi que leurs duales magnétiques sont alors exprimées en termes d’intégrales de surface. Nous illustrons ensuite nos résultats à travers l’étude des sources de certaines solutions électriques et de leur duales magnétiques. Les solutions électriques envisagées sont :les trous noirs de type Schwarzschild et de type Kerr ainsi que les ondes de chocs de type pp.

Dans la dernière partie, on établit la supersymétrie des espaces de type Taub-NUT lorentzien chargés électriquement et magnétiquement dans la supergravité N = 2. Motivé par l’existence d’une égalité BPS, on entreprend alors une recherche sur l’inclusion de la charge NUT dans l’algèbre de supersymétrie. Grâce à une complexification de la forme de Witten-Nester, cette contribution de la charge NUT à la superalgèbre est comprise comme une déformation topologique, symétrique, au crochet antisymétrique des super-charges. Ce résultat est alors appliqué à la superalgèbre N = 1 à travers l’étude des ondes de chocs de type pp.

Doctorat en Sciences
info:eu-repo/semantics/nonPublished

ix, 111 p.
This dissertation contains the results obtained from a study of two subjects in mathematical general relativity. The first part of this dissertation is about the existence of Killing symmetries in spacetimes containing a compact Cauchy horizon. We prove the existence of a nontrivial Killing symmetry in a large class of analytic cosmological spacetimes with a compact Cauchy horizon for any spacetime dimension. In doing so, we also remove the restrictive analyticity condition and obtain a generalization to the smooth case. The second part of the dissertation presents our results on the global stability problem for a class of cosmological models. We investigate the power law inflating cosmological models in the presence of electromagnetic fields. A stability result for such cosmological spacetimes is proved. This dissertation includes unpublished co-authored material.
Committee in charge: James Brau, Chair; James Isenberg, Advisor; Paul Csonka, Member; John Toner, Member; Peng Lu, Outside Member

In recent years, more and more efforts have been expended on the study of $n$-dimensional asymptotically anti-de Sitter spacetimes $(\mathcal{M},g)$ as solutions to the Einstein vacuum equations \begin{align*} \mathrm{Ric}(g)=\frac{2}{n-2}\Lambda\, g \end{align*} with negative cosmological constant $\Lambda$. This has been motivated mainly by the conjectured instability of these solutions. The author of this thesis joins these efforts with two contributions, which are themselves independent of each other. In the first part, we are concerned with a superradiant instability for $n=4$. For any cosmological constant $\Lambda=-3/\ell^2$ and any $\alpha < 9/4$, we find a Kerr-AdS spacetime $(\mathcal{M},g_{\mathrm{KAdS}})$, in which the Klein-Gordon equation \begin{align*} \Box_g\psi+\frac{\alpha}{\ell^2}\psi=0 \end{align*} has an exponentially growing mode solution satisfying a Dirichlet boundary condition at infinity. The spacetime violates the Hawking-Reall bound $r_+^2 > |a|\ell$. We obtain an analogous result for Neumann boundary conditions if $5/4 < \alpha < 9/4$. Moreover, in the Dirichlet case, one can prove that, for any Kerr-AdS spacetime violating the Hawking-Reall bound, there exists an open family of masses $\alpha$ such that the corresponding Klein-Gordon equation permits exponentially growing mode solutions. Our result provides the first rigorous construction of a superradiant instability for a negative cosmological constant. In the second part, we study perturbations of five-dimensional Eguchi-Hanson-AdS spacetimes exhibiting biaxial Bianchi IX symmetry. Within this symmetry class, the Einstein vacuum equations are equivalent to a system of non-linear partial differential equations for the radius $r$ of the spheres, the Hawking mass $m$ and $B$, a quantity measuring the squashing of the spheres, which satisfies a non-linear wave equation. First we prove that the system is well-posed as an initial-boundary value problem around infinity $\mathcal{I}$ with $B$ satisfying a Dirichlet boundary condition. Second, we show that initial data in the biaxial Bianchi IX symmetry class around Eguchi-Hanson-AdS spacetimes cannot form horizons in the dynamical evolution.

Orientador: Samuel Rocha de Oliveira
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
Made available in DSpace on 2018-08-08T09:44:11Z (GMT). No. of bitstreams: 1 Garcia_RaphaeldeOliveira_M.pdf: 2899892 bytes, checksum: 6befb1d82e7b4effd083097342934f7d (MD5) Previous issue date: 2007
Resumo: Esta dissertação trata de um problema de colapso gravitacional do campo escalar sem massa com simetria esférica, de acordo com as equações de Einstein com acoplamento mínimo. Desenvolvemos algoritmos simples mas com ajuste automático de refinamento para obter soluções numéricas. O refinamento é necessário para descrever os fenômenos do colapso. Utilizamos rotinas comuns para resolver equações diferenciais ordinárias advindas do método das linhas aplicado às equações diferenciais parciais do modelo matemático, a saber, rotinas de Runge-Kutta e de splines cúbicas. Mostramos que é possível obter soluções tão próximas quanto se queria à solução de Buracos Negros e da solução crítica com os algoritmos desenvolvidos e simples computadores de mesa.
Abstract: This dissertation is about a problem of gravitational collapse of the spherically symmetric massless scalar field in accord to the Einstein equations with minimal coupling. We developed simple but with adaptative mesh refinement algorithm to obtain numerical solutions. The mesh refinement is required to describe the collapse phenomena. We used common routines to solve ordinary differential equations that come from the method of lines applied to the partial differential equations of the mathematical model, namely, Runge-Kutta and cubic splines routines. We showed that it is possible to get solutions as close as we want to the Black Holes solution and to the critical solution using our developed algorithms and simple desktop computers
Mestrado
Fisica-Matematica
Mestre em Matemática Aplicada

The inspiral and merger of two black holes is among the most exciting and extreme events in our universe. Being one of the loudest sources of gravitational waves, they provide a unique dynamical probe of strong-field general relativity and a fertile ground for the observation of fundamental physics. While the detection of gravitational waves alone will allow us to observe our universe through an entirely new window, combining the information obtained from both gravitational wave and electro-magnetic observations will allow us to gain even greater insight in some of the most exciting astrophysical phenomena. In addition, binary black-hole mergers serve as an intriguing tool to study the geometry of space-time itself. In this dissertation we study the merger process of binary black-holes in a variety of conditions. Our results show that asymmetries in the curvature distribution on the common apparent horizon are correlated to the linear momentum acquired by the merger remnant. We propose useful tools for the analysis of black holes in the dynamical and isolated horizon frameworks and shed light on how the final merger of apparent horizons proceeds after a common horizon has already formed. We connect mathematical theorems with data obtained from numerical simulations and provide a first glimpse on the behavior of these surfaces in situations not accessible to analytical tools. We study electro-magnetic counterparts of super-massive binary black-hole mergers with fully 3D general relativistic simulations of binary black-holes immersed both in a uniform magnetic field in vacuum and in a tenuous plasma. We find that while a direct detection of merger signatures with current electro-magnetic telescopes is unlikely, secondary emission, either by altering the accretion rate of the circumbinary disk or by synchrotron radiation from accelerated charges, may be detectable. We propose a novel approach to measure the electro-magnetic radiation in these simulations and find a non-collimated emission that dominates over the collimated one appearing in the form of dual jets associated with each of the black holes. Finally, we provide an optimized gravitational wave detection pipeline using phenomenological waveforms for signals from compact binary coalescence and show that by including spin effects in the waveform templates, the detection efficiency is drastically improved as well as the bias on recovered source parameters reduced. On the whole, this disseration provides evidence that a multi-messenger approach to binary black-hole merger observations provides an exciting prospect to understand these sources and, ultimately, our universe.
Schwarze Löcher gehören zu den extremsten und faszinierensten Objekten in unserem Universum. Elektromagnetische Strahlung kann nicht aus ihrem Inneren entkommen, und sie bilden die kompaktesten Objekte, die wir kennen. Wir wissen heute, dass in den Zentren der meisten Galaxien sehr massereiche schwarze Löcher vorhanden sind. Im Fall unserer eigenen Galaxie, der Milchstrasse, ist dieses schwarze Loch ungefähr vier Millionen mal so schwer wie unsere Sonne. Wenn zwei Galaxien miteinander kollidieren, führt dies auch dazu, dass ihre beiden schwarzen Löcher kollidieren und zu einem einzelnen schwarzen Loch verschmelzen. Das Simulieren einer solchen Kollision von zwei schwarzen Löchern, die Vorhersage sowie Analyse der von ihnen abgestrahlten Energie in Form von Gravitations- und elektromagnetischen Wellen, bildet das Thema der vorliegenden Dissertation. Im ersten Teil dieser Arbeit untersuchen wir die Verschmelzung von zwei schwarzen Löchern unter verschiedenen Gesichtspunkten. Wir zeigen, dass Ungleichmässigkeiten in der Geometrie des aus einer Kollision entstehenden schwarzen Loches dazu führen, dass es zuerst beschleunigt und dann abgebremst wird, bis diese Ungleichmässigkeiten in Form von Gravitationswellen abgetrahlt sind. Weiterhin untersuchen wir, wie der genaue Verschmelzungsprozess aus einer geometrischen Sicht abläuft und schlagen neue Methoden zur Analyse der Raumzeitgeometrie in Systemen vor, die schwarze Löcher enthalten. Im zweiten Teil dieser Arbeit beschäftigen wir uns mit den Gravitationswellen und elektromagnetischer Strahlung, die bei einer Kollision von zwei schwarzen Löchern freigesetzt wird. Gravitationswellen sind Wellen, die Raum und Zeit dehnen und komprimieren. Durchläuft uns eine Gravitationswelle, werden wir in einer Richtung minimal gestreckt, während wir in einer anderen Richtung minimal zusammengedrückt werden. Diese Effekte sind allerdings so klein, dass wir sie weder spüren, noch auf einfache Weise messen können. Bei einer Kollision von zwei schwarzen Löchern wird eine grosse Menge Energie in Form von Gravitationswellen und elektromagnetischen Wellen abgestrahlt. Wir zeigen, dass beide Signale in ihrer Struktur sehr ähnlich sind, dass aber die abgestrahlte Energie in Gravitationswellen um ein Vielfaches grösser ist als in elektromagnetischer Strahlung. Wir führen eine neue Methode ein, um die elektromagnetische Strahlung in unseren Simulationen zu messen und zeigen, dass diese dazu führt, dass sich die räumliche Struktur der Strahlung verändert. Abschliessend folgern wir, dass in der Kombination der Signale aus Gravitationswellen und elektromagnetischer Strahlung eine grosse Chance liegt, ein System aus zwei schwarzen Löchern zu detektieren und in einem weiteren Schritt zu analysieren. Im dritten und letzen Teil dieser Dissertation entwickeln wir ein verbessertes Suchverfahren für Gravitationswellen, dass in modernen Laser-Interferometerexperimenten genutzt werden kann. Wir zeigen, wie dieses Verfahren die Chancen für die Detektion eines Gravitationswellensignals deutlich erhöht, und auch, dass im Falle einer erfolgreichen Detektion eines solchen Signals, seine Parameter besser bestimmt werden können. Wir schliessen die Arbeit mit dem Fazit, dass die Kollision von zwei schwarzen Löchern ein hochinteressantes Phenomenon darstellt, das uns neue Möglichkeiten bietet die Gravitation sowie eine Vielzahl anderer fundamentaler Vorgänge in unserem Universum besser zu verstehen.

[Truncated abstract] As gravitational wave detectors approach sensitivities that will allow observations to become routine, astrophysics lies on the cusp of an exciting new era. Potential sources will include transients such as merging neutron stars and black holes, supernova explosions or the engines that power gamma-ray bursts. This thesis will be devoted to the astrophysical gravitational wave background signal produced by cosmological populations of such transient signals. Particular attention will be devoted to the observation-time dependence imposed on the individual sources that accumulate to produce a gravitational wave background signal. The ultimate aim is to determine what information is encoded in the temporal evolution of such a signal. To lay the foundations for further investigation, the stochastic gravitational wave background signal from neutron star birth throughout the Universe has been calculated. In view of the uncertainties in both the single-source emissions and source rate histories, several models of each are employed. The results show that that the resulting signals are only weakly dependent on the source-rate evolution model and that prominent features in the single-source spectra can be related to the background spectra. In comparison with previous studies, the use of relativistic single-source gravitational wave waveforms rather than Newtonian models and a more slowly evolving source-rate density results in a 1 { 2 order of magnitude reduction in signal. ... A comparison with the more commonly used brightness distribution of events shows that when applying both methods to a data stream containing a background of Gaussian distributed false alarms, the brightness distribution yielded lower standard errors, but was biased by the false alarms. In comparison, a fitting procedure based on the time evolution of events was less prone to errors resulting from false alarms, but as fewer events contributed to the data, had a lower resolution. In further support of the time dependent signature of transient events, an alternative technique is fiapplied to the same source population. In this case, the local rate density is probed by measuring the statistical compatibility of the filtered data against synthetic time dependent data. Although this method is not as compact as the fitting procedure, the rate estimates are compatible. To further investigate how the observation time dependence of transient populations can be used to constrain global parameters, the method is applied to Swift long gamma-ray burst data. By considering a distribution in peak °ux rather than a gravitational wave amplitude, gamma-ray bursts can be considered as a surrogate for resolved gravitational wave transients. For this application a peak °ux{observation time relation is described that takes the form of a power law that is invariant to the luminosity distribution of the sources. Additionally, the method is enhanced by invoking time reversal invariance and the temporal cosmological principle. Results are presented to show that the peak °ux{observation time relation is in good agreement with recent estimates of source parameters. Additionally, to show that the intrinsic time dependence allows the method to be used as a predictive tool, projections are made to determine the upper limits in peak °ux of future gamma-ray burst detections for Swift.

In the first part of this thesis, we discuss some aspects of the four-dimensional N = 2 superconformal index of theories of class S. We first consider a generalized index on S¹ × S³/Z_r, and prove S-duality in a particular fugacity slice. We then go on to study the (round) superconformal index in the presence of surface defects. We develop a systematic prescription to compute surface defects labeled by arbitrary irreducible representations of the gauge group and subject those defects to various tests in several different limits. Each of these limits is interesting in its own right, and we go on to explore them in some depth. In the second part of this thesis, we construct the gravity duals of large N supersymmetric gauge theories defined on squashed five-spheres with SU(3) × U(1) symmetry. The gravity duals are constructed in Euclidean Romans F(4) gauged supergravity in six- dimensions, and uplift to massive type IIA supergravity. We compute the partition function and Wilson loop in the large N limit of the gauge theory and compare them to their corresponding supergravity dual quantities. As expected from AdS/CFT, both sides agree perfectly. Based on these results, we conjecture a general formula for the partition function and Wilson loop on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector. We then go on to construct rigid supersymmetric gauge theories on more general Riemannian five-manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of the six-dimensional bulk supergravity solution. This leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals.

Depuis sa formulation au début du 20ème siècle, la théorie de la Relativité Générale a été vérifiée avec une précision sans cesse croissante. Cette théorie prédit, entre autre, l'existence d'ondes gravitationnelles qui restent à ce jour inobservées, et ce malgré de nombreuses tentatives de détections. Ces ondes sont caractérisées par leur absence de masse. Une des questions qui se pose alors est de savoir si cette absence de masse est une condition nécessaire pour que théorie et observations concordent. Pour répondre à cette question, il est indispensable d'étudier les différents aspects des théories décrivant des ondes gravitationnelles massives. Au-delà de cet intérêt purement théorique, l'étude de ces théories est, entre autre, motivée par de récentes observations cosmologiques. Celles-ci indiquent que l'accord entre la Relativité Générale et les observations n'est possible que si on suppose l'existence de matière et d'énergie noires.

Cette thèse est dédiée à une classe de théories décrivant des ondes gravitationnelles massives. Dans un premier temps, nous résumons les différents problèmes qui surgissent lorsqu'on tente de donner une masse aux ondes gravitationnelles. Ensuite, nous introduisons une classe de modèles et étudions certaines de leurs caractéristiques.

Le premier aspect étudié concerne l'existence d'une interaction de type instantanée. De telles interactions sont possibles étant donné que l'invariance de Lorentz est spontanément brisée dans les modèles considérés. Celles-ci sont dès lors discutées et un exemple concret est fourni.

La présence d'une interaction instantanée dans ces modèles a une conséquence directe sur les solutions "trous noirs" des équations du champ. En effet, on s'attend à ce que l'interaction instantanée puisse propager de l'information à l'extérieur d'un trou noir, ce qui entraînerait une modification de ces solutions par rapport à celles de la Relativité Générale. Cette supposition est confirmée par les solutions "trous noirs" obtenues dans cette thèse. Celles-ci peuvent soit imiter une certaine quantité de matière noire, soit conduire à un champ gravitationnel répulsif.

Finalement, les mécanismes de formation des grandes structures de l'Univers (galaxies, amas de galaxies, ) sont étudiés pour les théories considérées. Cette dernière discussion démontre que ces modèles reproduisent le comportement prévu par la Relativité Générale et sont, par conséquent, en accord avec les observations.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

Orientador: Ricardo Antonio Mosna
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
Made available in DSpace on 2018-08-16T04:10:22Z (GMT). No. of bitstreams: 1 Nascimento_DaniloBorimdo_M.pdf: 2195612 bytes, checksum: 9db76fa3dc3fca1c9128da411fe240b5 (MD5) Previous issue date: 2010
Resumo: No espaço euclidiano, deformações de corpos quase-rígidos podem gerar rotações globais líquidas que obedecem, em cada instante, a lei de conservação do momento angular (o problema do gato caindo é um exemplo). Em espaços curvos, um ciclo de deformações de um corpo pode gerar não só rotações, mas também translações globais. Este fenômeno é conhecido como efeito swimming, ou natação. Avron e Kenneth apresentaram recentemente um modelo físico para descrever este fenômeno [Avron JE, Kenneth O, New J. Phys. 8, 68 (2006)]. Os autores tratam de corpos compostos por um conjunto de massas puntiformes em variedades estáticas (no contexto não-relativístico) e calculam o deslocamento obtido por um ciclo de deformações infinitesimais. Tal deslocamento é então relacionado, no caso de corpos pequenos, à curvatura do espaço ambiente. Nesta dissertação, propomos uma nova formulação para o efeito swimming utilizando formalismo de fibrados e conexões. O espaço de configurações do sistema é descrito como o espaço total de um fibrado principal, cujo espaço base é dado pelo espaço dos formatos do sistema e o grupo estrutural é (essencialmente) dado pelas isometrias da variedade ambiente. Dotando o fibrado de uma conexão que carrega consigo a informação sobre as leis físicas de conservação, expressamos o ciclo de deformações como uma curva fechada no espaço base, o movimento do corpo como o levantamento horizontal desta curva e o deslocamento resultante como a holonomia da mesma. Por meio deste formalismo, sistematizamos o cálculo do deslocamento gerado por ciclos de deformações arbitrárias, além de obter, em cada instante e analiticamente, a evolução temporal do sistema em questão
Abstract: In Euclidean space, cyclic deformations of quasi-rigid bodies can lead to net global rotations even though they satisfy, at each moment, the angular momentum conservation law (the falling cat problem is an example). In curved spaces, cyclic changes in the body shape can also lead to rotations, but also to global translations. This phenomenon is known as the swimming effect. In a recent work, Avron and Kenneth developed a formalism to describe this phenomenon in the non-relativistic context [Avron JE, Kenneth O, New J. Phys. 8, 68 (2006)], which may be used to calculate the net displacement caused by an infinitesimal cycle of deformations of a given body. This displacement is then related, for small swimmers, to the curvature of the ambient space. In the present work, we propose a new formulation for the swimming effect in terms of principal bundles and connections. The configuration space of the system is described by the total space of a principal bundle, whose base space is given by the space of shapes of the body and whose structural group is (essentially) given by the isometries of the ambient manifold. A given deformation cycleof the body then corresponds to a loop in the base space. By defining a connection in this bundle which conveys the physical conservation laws of the system, the corresponding physical motion of the body is then given by the horizontal lift of this curve in the base space, while the net displacement of the body is given by the holonomy associated with this loop. As a result we obtain, in a systematical way, the displacement generated by arbitrary deformation cycles and we get, for each instant of time, the time evolution of the system analytically
Mestrado
Geometria e Topologia
Mestre em Matemática

L'objet de cette thèse est l'étude du modèle plat, l'ingrédient principal du programme de quantification de la gravité par les mousses de spins, avec un accent particulier sur ses divergences. Outre une introduction personnelle au problème de la gravité quantique, le manuscrit se compose de deux parties. Dans la première, nous obtenons une formule exacte pour le comptage de puissances des divergences de bulles dans le modèle plat, notamment grâce à des outils de théorie de jauge discrète et de cohomologie tordue. Dans la seconde partie, nous considérons le problème de la limite continue des mousses de spins, tant du point de vue des théorie de jauge sur réseau que du point de vue de la "group field theory". Nous avançons en particulier une nouvelle preuve de la sommabilité de Borel du modèle de Boulatov-Freidel-Louapre, permettant un contrôle accru du comportement d'échelle dans la limite de grands spins. Nous concluons par une discussion prospective du programme de renormalisation pour les mousses de spins.

Le problème des singularités en relativité générale remonte à la première solution exacte de la théorie obtenue en 1915, à savoir celle du trou noir de Schwarzschild. Qu'elles soient de coordonnée ou de courbure, ces singularités ont longtemps questionné les physiciens qui parvinrent à mieux les caractériser à la fin des années 1960. Cela conduisit aux fameux théorèmes sur les singularités, s'appliquant à la fois aux trous noirs et en cosmologie, basés sur un comportement classique du contenu en matière de l'espace-temps résumé par des conditions d'énergie. La violation de ces conditions dans les processus quantiques pourrait indiquer que les singularités doivent être vues comme des limitations de la relativité générale, pouvant ainsi disparaître dans une théorie plus générale de la gravité quantique.Dans l'attente d'une telle théorie, nous avons pour objectif dans cette thèse d'étudier les espaces-temps de trous noirs dépourvus de toute singularité ainsi que leurs conséquences observationnelles. A cette fin, nous considérons à la fois des modifications de la relativité générale et le couplage de la théorie à des contenus en matière exotiques. Dans le premier cas nous montrons qu'il est possible de retrouver des trous noirs réguliers à symétrie sphérique connus, tout d'abord en principe avec la théorie tenseur-scalaire de gravité mimétique, puis implicitement par le biais d'une déformation de la contrainte hamiltonienne en relativité générale inspirée des techniques de gravitation quantique à boucles. Dans le second cas nous restons dans le cadre de la relativité générale, et considérons des tenseurs énergie-impulsion effectifs. Ils sont en premier lieu associés à un modèle régulier à la Hayward en rotation fournissant dans un certain régime un premier exemple de trou noir en rotation exempt de toute singularité, puis à un espace-temps dynamique décrivant la formation et l'évaporation d'un trou noir sans singularité. Pour ce dernier, nous montrons que tout modèle basé sur l'effondrement gravitationnel de coquilles de genre lumière visant à décrire l'évaporation de Hawking est voué à violer les conditions sur l'énergie dans une région non compacte de l'espace-temps. Enfin, l'étude théorique de la métrique de Hayward en rotation est accompagnée de simulations numériques d'un tel objet au centre de la Voie Lactée, obtenues à l'aide du code de calcul de trajectoires de particules Gyoto en reproduisant les propriétés connues de la structure d'accrétion du trou noir présumé Sgr A*. Ces simulations permettent d'illustrer deux régimes très différents de la métrique, avec ou sans horizon, et soulignent la difficulté d'affirmer avec certitude la présence d'un horizon à partir d'images en champ fort telles que celles obtenues par l'instrument Event Horizon Telescope
The issue of singularities in General Relativity dates back to the very first solution to the equations of the theory, namely Schwarzschild's 1915 black hole. Whether they be of coordinate or curvature nature, these singularities have long puzzled physicists, who managed to better characterize them in the late 60's. This led to the famous singularity theorems applying both to cosmology and black holes, and which assume a classical behaviour of the matter content of spacetime summarized in the so-called energy conditions. The violation of these conditions by quantum phenomena supports the idea that singularities are to be seen as a limitation of General Relativity, and would be cured in a more general theory of quantum gravity. In this thesis, pending for such a theory, we aim at investigating black hole spacetimes deprived of any singularity as well as their observational consequences. To that purpose, we consider both modifications of General Relativity and the coupling of Einstein's theory to exotic matter contents. In the first case, we show that one can recover the static spherically symmetric non-singular black holes of Bardeen and Hayward in principle in mimetic gravity, and implicitly by a deformation of General Relativity's hamiltonian constraint in an approach based on loop quantum gravity techniques. In the second case, we stay inside the framework of General Relativity and consider effective energy-momentum tensors associated with a fully regular rotating Hayward metric and with a dynamical spacetime describing the formation and evaporation of a non-singular black hole. For the latter, we show that all models based on the collapse of ingoing null shells and willing to describe Hawking’s evaporation are doomed to violate the energy conditions in a non-compact region of spacetime. Lastly, the theoretical study of the rotating Hayward metric comes with numerical simulations of such an object at the center of the Milky Way, using the ray-tracing code Gyoto and mimicking the known properties of the accretion structure of Sgr A*. These simulations allow exhibiting the two very different regimes of the metric, with or without horizon, and emphasize the difficulty of asserting the presence of a horizon from strong-field images as the ones provided by the Event Horizon Telescope

Le but à long terme de mon travail de recherche est de trouver une alternative viable à la méthode conforme, qui nous permettrait de mieux comprendre la structure géométrique de l'espace des solutions des équations de contrainte. L'avantage du modèle de Maxwell (the drift model) par rapport aux modèles plus classiques est la présence des paramètres supplémentaires. Le prix à payer, par contre, sera que la complexité analytique du système correspondant. Ma thèse a été structuré en deux parties : a. Existence sous la condition de petitesse des données initiales. Nous avons montré que le système de Maxwell est raisonnable dans le sens où nous pouvons le résoudre, malgré sa forte nonliniarité, sous des conditions de petitesse sur ses coefficients, en dimension 3, 4 et 5. Par conséquent, l'ensemble des solutions est non-vide. b. Stabilité Nous montrons la stabilité des solutions du système: ce résultat est obtenu en dimension 3,4 et 5, dans le cas où la métrique est conformément plate, et le drift et petit
The long-term goal of my work is to find a viable alternative to the conformal method, which would allow us to better understand the geometry of the space of solutions of the constraint equations. The advantage of Maxwell's model (the drift model) is the presence of additional parameters. Its downside, however, is that it proves to be much more difficult from an analytic standpoint. My thesis is structued in two parts: a. Existence under suitable smallness conditions. We show that Maxwell's system is sufficiently reasonable: it can be solved even given the presence of focusing non linearities. We prove this under smallness conditions of its coefficients, and in dimensions 3,4 and 5. An immediate consequence is that the set of solutions is non-empty. b. Stability. We verify that the solutions of the system are stable: this result holds in dimensions 3,4 and 5, when the metric is conformally flat and the drift is small

This thesis consists of three articles investigating the asymptotic behaviour of cosmological spacetimes with symmetries arising in Mathematical General Relativity. In Paper A and B, we consider spacetimes with Bianchi symmetry and where the matter model is that of a perfect fluid. We investigate the behaviour of such spacetimes close to the initial singularity ('Big Bang'). In Paper A, we prove that the Strong Cosmic Censorship conjecture holds in non-exceptional Bianchi class B spacetimes. Using expansion-normalised variables, we further show detailed asymptotic estimates. In Paper B, we prove similar estimates in the case of stiff fluids. In Paper C, we consider T2-symmetric spacetimes satisfying the Einstein equations for a non-linear scalar field. To given initial data, we show global existence and uniqueness of solutions to the corresponding differential equations for all future times. In the special case of a constant potential, a setting which is equivalent to a linear scalar field on a background with a positive cosmological constant, we investigate in detail the asymptotic behaviour towards the future. We prove that the Cosmic No-Hair conjecture holds for solutions satisfying an additional a priori estimate, an estimate which we show to hold in T3-Gowdy symmetry.
Denna avhandling består av tre artiklar som undersöker det asymptotiska beteendet hos kosmologiska rumstider med symmetrier som uppstår i Matematisk Allmän Relativitetsteori. I Artikel A och B studerar vi rumstider med Bianchi symmetri och där materiemodellen är en ideal fluid. Vi undersöker beteendet av sådana rumstider nära ursprungssingulariteten ('Big Bang'). I Artikel A bevisar vi att den Starka Kosmiska Censur-förmodan håller för icke-exceptionella Bianchi klass B-rumstider. Med hjälp av expansions-normaliserade variabler visar vi detaljerade asymptotiska uppskattningar. I Artikel B visar vi liknande uppskattningar för stela fluider. I Artikel C betraktar vi T2-symmetriska rumstider som uppfyller Einsteins ekvationer för ett icke-linjärt skalärfält. För givna begynnelsedata visar vi global existens och entydighet av lösningar till motsvarande differentialekvationer för all framtid. I det speciella fallet med en konstant potential, en situation som motsvarar ett linjärt skalärfält på en bakgrund med en positiv kosmologisk konstant, undersöker vi i detalj det asymptotiska beteendet mot framtiden. Vi visar att den Kosmiska Inget-Hår-förmodan håller för lösningar som uppfyller en ytterligare a priori uppskattning, en uppskattning som vi visar gäller i T3-Gowdy-symmetri.

QC 20171220

In this work the extension of the criterion for local thermal equilibrium by Buchholz, Ojima and Roos to curved spacetime as introduced by Schlemmer is investigated. Several problems are identified and especially the instability under time evolution which was already observed by Schlemmer is inspected. An alternative approach to local thermal equilibrium in quantum field theories on curved spacetimes is presented and discussed. In the following the dynamic system of the linear field and matter perturbations in the generic model of inflation is studied in the view of ambiguity of quantisation. In the last part the compatibility of the temperature fluctuations of the cosmic microwave background radiation with local thermal equilibrium is investigated
In dieser Arbeit wird die von Schlemmer eingeführte Erweiterung des Kriteriums für lokales thermisches Gleichgewicht in Quantenfeldtheorien von Buchholz, Ojima und Roos auf gekrümmte Raumzeiten untersucht. Dabei werden verschiedene Probleme identifiziert und insbesondere die bereits von Schlemmer gezeigte Instabilität unter Zeitentwicklung untersucht. Es wird eine alternative Herangehensweise an lokales thermisches Gleichgewicht in Quantenfeldtheorien auf gekrümmten Raumzeiten vorgestellt und deren Probleme diskutiert. Es wird dann eine Untersuchung des dynamischen Systems der linearen Feld- und Metrikstörungen im üblichen Inflationsmodell mit Blick auf Uneindeutigkeit der Quantisierung durchgeführt. Zuletzt werden die Temperaturfluktuationen der kosmischen Hintergrundstrahlung auf Kompatibilität mit lokalem thermalem Gleichgewicht überprüft

Nous étudions les champs de Maxwell à l'extérieur de trous noirs de Reissner-Nordstrom-de Sitter. Nous commençons par étudier la géométrie de ces espaces-temps : nous donnons une condition sous laquelle la métrique admet trois horizons puis dans ce cadre nous construisons l'extension analytique maximale d'un trou noir de Reissner-Nordstrom-de Sitter. Nous donnons ensuite une description générale des champs de Maxwell en espace-temps courbe, de leur décomposition en composantes spinorielle ainsi que de leur énergie. La première étude analytique établit la décroissance ponctuelle de champs de Maxwell à l'extérieur d'un trou noir de Reissner-Nordstrom-de Sitter ainsi que la décroissance uniforme de l'énergie sur un hyperboloïde qui s'éloigne dans le futur. Ce chapitre utilise des méthodes de champs de vecteurs (estimations d'énergie géométriques) dans l'esprit des travaux de Pieter Blue. Enfin nous construisons une théorie du scattering conforme pour les champs de Maxwell à l'extérieur du trou noir. Ceci consiste en la résolution du problème de Goursat pour les champs de Maxwell à la frontière isotrope de l'extérieur du trou noir, constituée des horizons du trou noir et horizons cosmologiques futurs et passés. Les estimations de décroissance uniforme de l'énergie sont cruciales dans cette partie
We study Maxwell fields on the exterior of Reissner-Nordstrom-de Sitter black holes. We start by studying the geometry of these spacetimes: we give the condition under which the metric admits three horizons and in this case we construct the maximal analytic extension of the Reissner-Nordstrom-de Sitter black hole. We then give a general description of Maxwell fields on curves spacetimes, their decomposition into spin components, and their energies. The first result establishes the pointwise decay of the Maxwell field in the exterior of a Reissner-Nordstrom-de Sitter black hole, as well as the uniform decay of the energy flux across a hyperboloid that recedes in the future. This chapter uses the vector fields methods (geometric energy estimates) in the spirit of the work of Pieter Blue. Finally, we construct a conformal scattering theory for Maxwell fields in the exterior of the black hole. This amounts to solving the Goursat problem for Maxwell fields on the null boundary of the exterior region, consisting of the future and past black hole and cosmological horizons. The uniform decay estimates of the energy are crucial to the construction of the conformal scattering theory

[Truncated abstract] The core objective of this thesis is summarised by its title: “Understanding the World Wool Market: Trade, Productivity and Grower Incomes”. Thus, we wish to aid understanding of the economic mechanisms by which the world wool market operates. In doing so, we analyse two issues trade and productivity and their effect on, inter alia, grower incomes. To achieve the objective, we develop a novel analytical framework, or model. The model combines two long and rich modelling traditions: the partial-equilibrium commodity-specific approach and the computable-general-equilibrium approach. The result is a model that represents the world wool market in detail, tracking the production of greasy wool through five off-farm production stages ending in the production of wool garments. Capturing the multistage nature of the wool production system is a key pillar in this part of the model . . . The estimated welfare gain for China is 0.1% of real income; this is a significant welfare gain. For three losing regions Italy, Germany and Japan the results are robust and we can be highly confident that these regions are the largest losers from the complete removal of 2005 wool tariffs. In both wool tariff liberalisation scenarios, regions whose exports are skewed towards wool textiles and garments gain the most as it is these wool products that have the highest initial tariff rates. The overall finding of this work is that a sophisticated analytical framework is necessary for analysing productivity and trade issues in the world wool market. Only a model of this kind can appropriately handle the degree of complexity of interactions between members (domestic and foreign) of the multistage wool production system. Further, including the nonwool economy in the analytical framework allows us to capture the indirect effects of changes in the world wool market and also the effects on the nonwool economy itself.

This thesis considers the problem of calculating and observing the mixing of modes of positive and negative frequency in inhomogeneous, dispersive media. Scattering of vacuum modes of the electromagnetic field at a moving interface in the refractive index of a dielectric medium is discussed. Kinematics arguments are used to demonstrate that this interface may, in a regime of linear dispersion, act as the analogue of the event horizon of a black hole to modes of the field. Furthermore, a study of the dispersion of the dielectric shows that five distinct configurations of modes of the inhomogeneous medium at the interface exist as a function of frequency. Thus it is shown that the interface is simultaneously a black- and white-hole horizon-like and horizonless emitter. The role, and importance, of negative-frequency modes of the field in mode conversion at the horizon is established and yields a calculation of the spontaneous photonic flux at the interface. An algorithm to calculate the scattering of vacuum modes at the interface is introduced. Spectra of the photonic flux in the moving and laboratory frame, for all modes and all realisable increase in the refractive index at the interface are computed. As a result of the various mode configurations, the spectra are highly structured in intervals with black-hole, white-hole and no horizon. The spectra are dominated by a negative-frequency mode, which is the partner in any Hawking-type emission. An experiment in which an incoming positive-frequency wave is populated with photons is assembled to observe the transfer of energy to outgoing waves of positive and negative frequency at the horizon. The effect of mode conversion at the interface is clearly shown to be a feature of horizon physics. This is a classical version of the quantum experiment that aims at validating the mechanism of Hawking radiation.

La première partie de cette thèse s'inscrit dans le cadre de la modélisation des ondes gravitationnelles en provenance des systèmes binaires coalescents de trous noirs, dans la perspective de leur détection par les antennes gravitationnelles terrestres LIGO/VIRGO et spatiale LISA. Nous étudions la dynamique relativiste de tels systèmes binaires d'objets compacts à l'aide de deux méthodes d'approximation en relativité générale : les développements post-newtoniens, et le formalisme de la force propre, une extension naturelle de la théorie des perturbations d'un trou noir ; nous démontrons la cohérence des résultats ainsi obtenus dans leur domaine de validité commun. Dans un second temps, nous combinons ces deux méthodes perturbatives afin d'estimer l'effet de recul gravitationnel lors de la coalescence de deux trous noirs de Schwarzschild ; nos résultats sont en très bon accord avec ceux obtenus par des simulations en relativité numérique. La seconde partie de cette thèse traite du problème de la matière noire en astrophysique. L'hypothèse de la matière noire rend compte de nombreuses observations indépendantes de l'échelle des amas de galaxies jusqu'aux échelles cosmologiques. Les observations à l'échelle galactique sont toutefois en bien meilleur accord avec la phénoménologie de la dynamique newtonienne modifiée (MOND), qui postule une modification des lois de la gravité en l'absence de matière noire. Nous proposons une troisième alternative : conserver la théorie de la gravitation standard, mais doter la matière noire d'une propriété de polarisabilité dans un champ gravitationnel, de façon à rendre compte de la phénoménologie de MOND à l'échelle des galaxies.

In this thesis we consider spherically symmetric cosmological models when the shear is nonzero and also cases when the shear is vanishing. We investigate the role of the Emden-Fowler equation which governs the behaviour of the gravitational field. The Einstein field equations are derived in comoving coordinates for a spherically symmetric line element and a perfect fluid source for charged and uncharged matter. It is possible to reduce the system of field equations under different assumptions to the solution of a particular Emden-Fowler equation. The situations in which the Emden-Fowler equation arises are identified and studied. We analyse the Emden-Fowler equation via the method of Lie point symmetries. The conditions under which this equation is reduced to quadratures are obtained. The Lie analysis is applied to the particular models of Herlt (1996), Govender (1996) and Maharaj et al (1996) and the role of the Emden-Fowler equation is highlighted. We establish the uniqueness of the solutions of Maharaj et al (1996). Some physical features of the Einstein-Maxwell system are noted which distinguishes charged solutions. A charged analogue of the Maharaj et al (1993) spherically symmetric solution is obtained. The Gutman-Bespal'ko (1967) solution is recovered as a special case within this class of solutions by fixing the parameters and setting the charge to zero. It is also demonstrated that, under the assumptions of vanishing acceleration and proper charge density, the Emden-Fowler equation arises as a governing equation in charged spherically symmetric models.
Thesis (M.Sc.)-University of Natal, Durban, 1998.

Tesis (Doctor en Física)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2017.
En esta tesis se abordan tres temáticas principales: el uso de modelos geométricos para la descripción de sistemas astrofísicos, el desarrollo de un nuevo formalismo en la teoría de lentes gravitacionales que extiende ideas de trabajos previos al contexto cosmológico y, por último, la cuestión de cómo realizar promedios en el marco de la relatividad general. Uno de los motivos principales que persigue este trabajo es el de discutir la forma en que el estudio del contenido de materia en el Universo es abordado. En particular, aquí indagamos nuevas posibilidades de descripción de la fenomenología asociada al problema de la masa faltante presentando nuevas herramientas desde el área de la teoría de lentes gravitacionales débiles sobre un espaciotiempo de fondo cosmológico. Las mismas permiten un modelado mucho más general de la geometría de la lente que aquellas que son habitualmente consideradas y se basan esencialmente en una concepción Newtoniana de la distribución de materia.
This doctorate thesis deals with three main topics: the use of geometric models for the description of astrophysical systems, the development of a new formalism within the theory of gravitational lensing which improves and generalize previous ideas already presented to the cosmological context and, finally the problem of how to perform averages in the conceptual framework of general relativity. One of the main motivation of this work is to discuss the way in which the study of the matter content in the Universe is addressed. In particular, here we investigate new possibilities of description of the phenomenology related to the missing mass problem. We present new tool from the subject of weak gravitational lensing over a cosmological background spacetime. Such tools allow a much more general modelling of the lens geometry than those that are usually considered and which are based essentially on a Newtonian concepton of the matter distribution.

Two different works in mathematical physics are presented: A construction of conformal infinity in null and spatial directions is constructed for the Rainbow-flat space-time corresponding to doubly special relativity. From this construction a definition of asymptotic DSRness is put forward which is com- patible with the correspondence principle of Rainbow gravity. Furthermore a result equating asymptotically flat space-times with asymptotically DSR spacetimes is presented. An overview of microlocality in braided ribbon networks is presented. Follow- ing this, a series of definitions are presented to explore the concept of microlocality and the topology of ribbon networks. Isolated substructure of ribbon networks are introduced, and a theorem is proven that allows them to be relocated. This is fol- lowed by a demonstration of microlocal translations. Additionally, an investigation into macrolocality and the implications of invariants in braided ribbon networks are presented.

This thesis is written within the framework of the abstract boundary (or a-boundary) of Scott and Szekeres. The a-boundary provides a concept of "boundary" for any n-dimensional, paracompact, connected, Hausdorff manifold, defined in such a way that the boundary is independant of the particular embedding used to display the manifold. This makes it possible to define various types of boundary points of space-time such as "singularities" and "points at infinity". The original research that will be presented in this thesis can be roughly divided up into two categories; results relating to the existence of optimal embeddings of solutions to Einstein's Field Equations and a-boundary singularity theorems. In addition, the implications of the "finite connected neighbourhood region property" and the bounded "acceleration" property are explored. It is also shown that not all space-times are maximally extendable.

The body of this thesis consists of four chapters (2 through 5), each of which is a paper that has been published in or submitted for publication to Physical Review D. I am the sole author of chapters 2, 3, and 4; Curt Cutler coauthored chapter 5 with me. These four chapters deal with two topics in general relativity: the formation of horizons in nonspherical gravitational collapse (chapters 2 and 3), and some theoretical aspects of the effort to detect gravitational waves from cosmic sources (chapters 4 and 5). These four chapters were written largely for experts in the topics they cover. As an aid to general readers, I shall give in this introductory chapter some background information about chapters 2 through 5 and a nontechnical overview of their contents.

En este trabajo analizamos la teoría de lentes gravitaciones débiles sobre un background plano. Obtenemos expresiones concretas para los escalares ópticos a segundo orden en la perturbación de la métrica plana de forma general en el sentido que no hacemos mención ni de la extensión de la lente ni de la forma de la misma. Calculamos explícitamente los escalares ópticos en el gauge generalizado de Poisson. Finalmente, considerando lentes axialmente simétricas en la aproximación de lente delgada obtenemos una expresión para el ángulo de deflexión a segundo orden en la perturbación de la métrica plana.

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Two topics in classical general relativity are discussed: a) The clothing of singularities by event horizons, and b) various issues in the evolution of coalescing compact binaries, as sources of gravitational waves to be detected by the LIGO/VIRGO/GEO ground-based detectors and/or the LISA space-based detector. More specifically: We investigate a problem related to an important conjecture of classical relativity, namely the existence of a "cosmic censorship" that forbids the formation of naked singularities, and always clothes them with event horizons that causally hide them from the rest of the Universe. Under consideration is the role of rotation in an infinite cylindrical shell consisting of collisionless dust particles, half of which rotate clockwise and half counterclockwise. We show that, although such a shell without any rotation is known to collapse into a line singularity, the presence of an arbitrarily small amount of rotation is sufficient to halt the collapse. Such a shell, starting from a non equibrium configuration, will "breath" radially, emitting gravitational waves, and will finally settle down to an equilibrium radius at which gravity is balanced by centrifugal forces. This suggests the essential role that rotation might play in halting the gravitational collapse of an elongated distribution of mass and preventing the formation of a naked singularity. However, this is a highly idealized example, and it can, by no means, ensure the validity of the "cosmic censorship" hypothesis. On a separate topic, we explore the details of how gravitational radiation reaction drives the evolution of a slightly eccentric orbit of a small body around nonrotating supermassive black holes. A combination of analytic and numerical results arise from the solution of the Teukolsky perturbation equation. It is shown that in the fully relativistic situation, as in the Newtonian quadrupole approximation, there is a tendency for circularization of the orbit down to an orbital radius [...], where M is the mass of the black hole, and G and c are Newton's gravitation constant and the speed of light. It is further shown that for radii smaller than [...] the eccentricity increases. Finally, an attempt is made to understand and construct analytic expressions that, based on the laws of general relativity, approximately describe the simultaneous precession in rapidly spinning black hole and/or neutron star and inspiral binaries with circular orbits. The precession is produced by general relativistic spin-orbit and spin-spin coupling; the inspiral, by gravitational radiation reaction. We derive the corresponding approximate waveforms to be received by the network of LIGO, VIRGO, and GEO earth-based gravitational-wave detectors. We then go on to investigate the adequateness of various "families of templates," to detect these spin-modulated waveforms by the method of "matching filters," We introduce a "fitting factor" FF as a measure of templates' adequateness, and show the complete inadequateness, for the task of detection, of the "Newtonian template family" (the set of the waveforms derived from the Newtonian, quadrupole approximation formalism). Another template family with an extra parameter is suggested that performs much better.

The abstract boundary construction of Scott and Szekeres provides a 'boundary' for any n-dimensional, paracompact, connected, Hausdorff, smooth space-time manifold. For a space-time, singularities and points at infinity may then be defined as objects within this boundary. The abstract boundary of a manifold is typically very large. Even so, one does not necessarily have to consider every abstract boundary point in order to provide detailed comments on the structure of the abstract boundary. Towards this end, this thesis develops a number of results concerning the topological contact properties of elements of the abstract boundary. For the first time two topologies, referred to as the attached point topology and the strongly attached point topology, are defined for a manifold together with its abstract boundary. We consider properties of these topologies related to the topological separation of abstract boundary points. In particular, it is demonstrated that the attached point topology is Hausdorff but the strongly attached point topology is not. Even so, it is observed that the strongly attached point topology loses Hausdorff separability in a way that provides further insight into the structure of the abstract boundary. We next consider how regular abstract boundary points of different space-time extensions are related to each other. We conjecture that if two regular abstract boundary points are in contact with each other, then, under certain physically reasonable conditions, such as strong causality holding, they must be equivalent. This has major implications when considering how to best present the geometrical features of a space-time within an embedding. In particular, this result says that the structure of the boundary through which space-time extensions are performed does not depend on the extension. Great progress is made in proving this result. It is rigorously shown that a significant amount of topological information is preserved around the boundary between different envelopments. In order to prove this result a number of reasonable restrictions were placed on the manifold. By considering special cases it is possible to weaken some of these restrictions. A form of the result is proven for isolated abstract boundary points using weakened assumptions. In the final component of the thesis we apply results developed in the previous sections to collections of abstract boundary points called partial cross sections. Partial cross sections allow one to simplify the appearance of the abstract boundary. In particular, they can be used to formulate optimal embeddings which, in essence, are a formal definition for an embedding of a manifold which clearly displays the important geometric features of that manifold. Refined definitions of partial cross sections and optimal embeddings are presented which incorporate the insights gained regarding the contact properties of abstract boundary points. In particular, optimal presentations are defined for the first time. These objects generalise the notion of an optimal embedding of a space-time to include those space-times which require more then one embedding in order to reveal its geometric structure.

A l'aune d'une possible unification de la Mecanique Quantique et de la Relativite Generale, le processus d'evaporation des trous noirs est sans nul doute l'un des phenomenes privilegies pour avoir acces a des effets de gravite quantique. Apres une rapide presentation de la theorie de la gravitation d'Einstein et d'une possible extension sous forme de serie de Taylor en courbure scalaire, l'evaporation de Hawking est abordee en se focalisant sur le calcul des facteurs de corps gris, i.e. de la probabilite tunnel qu'une particule issue de la brisure des fluctuations du vide s'echappe definitivement de l'attraction gravitationnelle du trou noir. J'ai developpe une methode analytique semi-classique pour l'etude de la dynamique de champs scalaires se propageant en espace-temps statique et a symetrie spherique, ainsi qu'une methode numerique pour la determination exacte des facteurs de corps gris pour des champs de spin s=0, 1/2 et 1. Ces methodes ont ete appliquees au cas de trous noirs de Schwarzschild en theorie de Lovelock a l'ordre deux (theorie de Gauss-Bonnet) et avec dimensions supplementaires, pour ainsi aboutir a une evaluation exacte des spectres rayonnes, etape necessaire pour la verification des quatres principes de la thermodynamique des trous noirs. Dans le cadre des scenarii branaires de type Arkani-Hamed, Dimopoulos & Dvali (ADD), abaissant l'echelle de Planck a l'ordre du TeV et ouvrant ainsi la possibilite de production de trous noirs par collision de particules dans le cosmos ou aupres des futurs collisionneurs, les possibilites de contraindre le Lagrangien gravitationnel a l'aide du mecanisme d'evaporation des trous noirs ont ete abordees: les modeles ADD ne sont pas en contradiction avec les donnees cosmologiques et de rayonnement cosmique et il serait possible de distinguer la presence ou non d'un terme de Gauss-Bonnet dans le Lagrangien d'Einstein-Hilbert. La question des inhomogeneites primordiales aux petites echelles de longueur ainsi que celle de la matiere noire sont aussi abordees dans cette these, respectivement a la lumiere des trous noirs primordiaux et des theories a dimensions supplementaires.

Includes bibliographical references (leaves 168-179).
xiii, 179 leaves ; 30 cm.
Title page, contents and abstract only. The complete thesis in print form is available from the University Library.
Finds that the post-Newtonian approximation seems to be a better approximation of the general relativistic theory than the standard Newtonian theory.
Thesis (Ph.D.)--Adelaide University, Dept. of Physics and Mathematical Physics, 2001

Includes bibliographical references (leaves 168-179).
xiii, 179 leaves ; 30 cm.
Finds that the post-Newtonian approximation seems to be a better approximation of the general relativistic theory than the standard Newtonian theory.
Thesis (Ph.D.)--Adelaide University, Dept. of Physics and Mathematical Physics, 2001

The influence of matter described by the Vlasov equation, on the evolution of anisotropy in the spatially-homogeneous universes, called the Bianchi cosmologies, is studied. Due to the spatial-homogeneity, the Einstein equations for each Bianchi Type are reduced to a set of coupled ordinary differential equations, which has Hamiltonian form with the metric components being the canonical coordinates. In the vacuum Bianchi cosmologies, it is known that a curvature potential, which comes from the symmetries of the three-dimensional Lie groups, determines the basic properties of the evolution of anisotropy. In this work, matter potentials are constructed for Vlasov matter. They are obtained by first introducing a new matter action principle for the Vlasov equation, in terms of a conjugate pair of functions, and then enforcing the symmetry to obtain a reduction. This yields an expression for the matter potential in terms of the phase space density, which is further reduced by assuming cold streaming matter. Some vacuum Bianchi cosmologies and Type I with Vlasov matter are compared. It is shown that the Vlasov-matter potential for cold streaming matter results in qualitatively distinct dynamics from the well-known vacuum Bianchi cosmologies.
text

Tesis (Doctorado en Física)--Universidad Nacional de Córdoba, 2006.
Estudiamos el concepto de distancia entre distribuciones de probabilidad y sus aplicaciones en problemas relevantes de la Física y la Biología. El eje central de este trabajo de tesis lo constituye la divergencia de Jensen-Shannon, una medida de distancia proveniente de la Teoría de la Información. Proponemos dos extensiones de esta cantidad: una en el contexto de la Mecánica Estadística no Extensiva y la otra en el marco de la Mecánica Cuántica.
Introducción --La divergencia de Jensen-Shannon -- La q-divergencia de Jensen-Shannon -- Complejidad de secuencias simbólicas -- Distancias y Filogénesis --La divergencia de Jensen-Shannon cuántica -- ap. a. Palabras en secuencia --ap. b.Complejidad máxima.
Fil: Majtey, Ana Paula. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.

In this work the extension of the criterion for local thermal equilibrium by Buchholz, Ojima and Roos to curved spacetime as introduced by Schlemmer is investigated. Several problems are identified and especially the instability under time evolution which was already observed by Schlemmer is inspected. An alternative approach to local thermal equilibrium in quantum field theories on curved spacetimes is presented and discussed. In the following the dynamic system of the linear field and matter perturbations in the generic model of inflation is studied in the view of ambiguity of quantisation. In the last part the compatibility of the temperature fluctuations of the cosmic microwave background radiation with local thermal equilibrium is investigated.:1. Introduction 5 2. Technical Background 10 2.1. The Free Scalar Field on a Globally Hyperbolic Spacetime . . . . . . 10 2.1.1. Construction of the Scalar Field . . . . . . . . . . . . . . . . . 10 2.1.2. Algebra of Wick Products . . . . . . . . . . . . . . . . . . . . 13 2.1.3. Local Covariance Principle . . . . . . . . . . . . . . . . . . . . 17 2.2. Local Thermal Equilibirum . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1. Global Thermodynamic Equilibrium - KMS States . . . . . . 21 2.2.2. Local Thermal Observables . . . . . . . . . . . . . . . . . . . 24 2.2.3. LTE on Flat Spacetime . . . . . . . . . . . . . . . . . . . . . . 29 2.2.4. LTE in Cosmological Spacetimes . . . . . . . . . . . . . . . . 32 2.3. Linear Scalar Cosmological Perturbations . . . . . . . . . . . . . . . . 34 2.3.1. Robertson-Walker Cosmology . . . . . . . . . . . . . . . . . . 35 2.3.2. Mathematical Background . . . . . . . . . . . . . . . . . . . . 38 2.3.3. Technical Framework and Formulae . . . . . . . . . . . . . . . 40 2.3.4. The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 46 2.3.5. The Sachs-Wolfe Effect for Adiabatic Perturbations . . . . . . 49 3. Towards a Refinement of the LTE Condition on Curved Spacetimes 54 3.1. Non-Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1.1. Commutator Distribution . . . . . . . . . . . . . . . . . . . . 55 3.1.2. KMS Two-Point Function . . . . . . . . . . . . . . . . . . . . 57 3.1.3. Balanced Derivatives . . . . . . . . . . . . . . . . . . . . . . . 61 3.2. Conformally Static Spacetimes . . . . . . . . . . . . . . . . . . . . . . 65 3.2.1. Conformal KMS States . . . . . . . . . . . . . . . . . . . . . . 66 3.2.2. Extrinsic LTE in de Sitter Spacetime . . . . . . . . . . . . . . 71 3.3. Massive Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.1. Properties of the Model . . . . . . . . . . . . . . . . . . . . . 78 3.3.2. Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . 80 3.3.3. Thermal Observables . . . . . . . . . . . . . . . . . . . . . . . 82 3.4. Towards an Alternative Concept . . . . . . . . . . . . . . . . . . . . . 91 3.4.1. Problems and Open Questions Concerning LTE . . . . . . . . 92 3.4.2. Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . 94 3.4.3. Positivity Inequalities . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.4. Macroobservable Interpretation . . . . . . . . . . . . . . . . . 100 3.5. An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4. Cosmological Perturbation Theory 105 4.1. Dynamics of Perturbations in Inflation . . . . . . . . . . . . . . . . . 106 4.1.1. CCR Quantisation is Ambiguous . . . . . . . . . . . . . . . . 106 4.1.2. Canonical Symplectic Form . . . . . . . . . . . . . . . . . . . 111 4.1.3. The Algebraic Point of View . . . . . . . . . . . . . . . . . . . 117 4.2. LTE States in Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 120 4.2.1. The Link to Fluid Dynamics . . . . . . . . . . . . . . . . . . . 120 4.2.2. Incompatibility of LTE with Sachs-Wolfe Effect . . . . . . . . 125 5. Conclusion and Outlook 131 A. Technical proofs 136 A.1. Proof of Lemma 3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 A.2. Proof of Lemma 3.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 A.3. Proof of Lemma 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.4. Idea of Proof for Conjecture 3.4.3 . . . . . . . . . . . . . . . . . . . . 144 B. Introduction to Probability Theory 146 Bibliography 150 Correction of Lemma 3.1.2 155
In dieser Arbeit wird die von Schlemmer eingeführte Erweiterung des Kriteriums für lokales thermisches Gleichgewicht in Quantenfeldtheorien von Buchholz, Ojima und Roos auf gekrümmte Raumzeiten untersucht. Dabei werden verschiedene Probleme identifiziert und insbesondere die bereits von Schlemmer gezeigte Instabilität unter Zeitentwicklung untersucht. Es wird eine alternative Herangehensweise an lokales thermisches Gleichgewicht in Quantenfeldtheorien auf gekrümmten Raumzeiten vorgestellt und deren Probleme diskutiert. Es wird dann eine Untersuchung des dynamischen Systems der linearen Feld- und Metrikstörungen im üblichen Inflationsmodell mit Blick auf Uneindeutigkeit der Quantisierung durchgeführt. Zuletzt werden die Temperaturfluktuationen der kosmischen Hintergrundstrahlung auf Kompatibilität mit lokalem thermalem Gleichgewicht überprüft.:1. Introduction 5 2. Technical Background 10 2.1. The Free Scalar Field on a Globally Hyperbolic Spacetime . . . . . . 10 2.1.1. Construction of the Scalar Field . . . . . . . . . . . . . . . . . 10 2.1.2. Algebra of Wick Products . . . . . . . . . . . . . . . . . . . . 13 2.1.3. Local Covariance Principle . . . . . . . . . . . . . . . . . . . . 17 2.2. Local Thermal Equilibirum . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1. Global Thermodynamic Equilibrium - KMS States . . . . . . 21 2.2.2. Local Thermal Observables . . . . . . . . . . . . . . . . . . . 24 2.2.3. LTE on Flat Spacetime . . . . . . . . . . . . . . . . . . . . . . 29 2.2.4. LTE in Cosmological Spacetimes . . . . . . . . . . . . . . . . 32 2.3. Linear Scalar Cosmological Perturbations . . . . . . . . . . . . . . . . 34 2.3.1. Robertson-Walker Cosmology . . . . . . . . . . . . . . . . . . 35 2.3.2. Mathematical Background . . . . . . . . . . . . . . . . . . . . 38 2.3.3. Technical Framework and Formulae . . . . . . . . . . . . . . . 40 2.3.4. The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 46 2.3.5. The Sachs-Wolfe Effect for Adiabatic Perturbations . . . . . . 49 3. Towards a Refinement of the LTE Condition on Curved Spacetimes 54 3.1. Non-Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1.1. Commutator Distribution . . . . . . . . . . . . . . . . . . . . 55 3.1.2. KMS Two-Point Function . . . . . . . . . . . . . . . . . . . . 57 3.1.3. Balanced Derivatives . . . . . . . . . . . . . . . . . . . . . . . 61 3.2. Conformally Static Spacetimes . . . . . . . . . . . . . . . . . . . . . . 65 3.2.1. Conformal KMS States . . . . . . . . . . . . . . . . . . . . . . 66 3.2.2. Extrinsic LTE in de Sitter Spacetime . . . . . . . . . . . . . . 71 3.3. Massive Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.1. Properties of the Model . . . . . . . . . . . . . . . . . . . . . 78 3.3.2. Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . 80 3.3.3. Thermal Observables . . . . . . . . . . . . . . . . . . . . . . . 82 3.4. Towards an Alternative Concept . . . . . . . . . . . . . . . . . . . . . 91 3.4.1. Problems and Open Questions Concerning LTE . . . . . . . . 92 3.4.2. Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . 94 3.4.3. Positivity Inequalities . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.4. Macroobservable Interpretation . . . . . . . . . . . . . . . . . 100 3.5. An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4. Cosmological Perturbation Theory 105 4.1. Dynamics of Perturbations in Inflation . . . . . . . . . . . . . . . . . 106 4.1.1. CCR Quantisation is Ambiguous . . . . . . . . . . . . . . . . 106 4.1.2. Canonical Symplectic Form . . . . . . . . . . . . . . . . . . . 111 4.1.3. The Algebraic Point of View . . . . . . . . . . . . . . . . . . . 117 4.2. LTE States in Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 120 4.2.1. The Link to Fluid Dynamics . . . . . . . . . . . . . . . . . . . 120 4.2.2. Incompatibility of LTE with Sachs-Wolfe Effect . . . . . . . . 125 5. Conclusion and Outlook 131 A. Technical proofs 136 A.1. Proof of Lemma 3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 A.2. Proof of Lemma 3.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 A.3. Proof of Lemma 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.4. Idea of Proof for Conjecture 3.4.3 . . . . . . . . . . . . . . . . . . . . 144 B. Introduction to Probability Theory 146 Bibliography 150 Correction of Lemma 3.1.2 155