Honors Project--Smith College, Northampton, Mass., 2010. / Includes bibliographical references (p. 42).

Made available in DSpace on 2016-05-17T16:50:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-05-04. Added 1 bitstream(s) on 2016-05-17T16:54:22Z : No. of bitstreams: 1 000855801.pdf: 618196 bytes, checksum: 88ac5f6edd9a6e08f839d677db4ca1f7 (MD5) / Nesta dissertação estudamos aspectos clássicos dos modelos análogos à Relatividade Geral em matéria condensada visando sobretudo criar uma nova percepção dos efeitos gravitacionais semi-clássicos, tais como a radiação Hawking, afim de melhor compreendê-los. Neste sentido, demonstramos que as ondas sonoras se propagando em um fluido ideal, barotrópico e irrotacional sobre uma esfera 'S POT. 2' de raio r se comportam como um campo escalar de Klein-Gordon não massivo em um espaço tempo curvo. Analisamos ao longo desta dissertação diversas propriedades deste espaço-tempo efetivo sentido pelo som, cuja geometria é descrita por uma métrica lorentziana dependente das variáveis hidrodinâmicas do fluxo, como a velocidade do fluido, sua densidade e a velocidade local do som, sempre buscando estabelecer correlações entre os conceitos clássicos da dinâmica dos fluidos e conceitos puramente relativísticos. Feita uma análise mais geral destes espaços-tempos, que denominamos de espaços-tempos acústicos, nos propomos a encontrar soluções das variáveis dinâmicas do fluido, uma vez que são elas que determinam a geometria acústica, capazes de modelar espaços-tempos efetivos dotados de horizontes de eventos e singularidades, criando portanto um buraco mudo/surdo, ou seja, um análogo de um buraco negro e de buraco branco da Relatividade Geral. Discutimos ainda alguns pontos da estrutura causal dos espaços-tempos acústicos construindo assim um diagrama de Carter-Penrose do buraco mudo/surdo com o intuito de evidenciar as possíveis trajetórias nulas deste espaço-tempo. Ademais, mostramos que na aproximação da acústica geométrica, ou também aproximação eikonal, os raios de som seguem geodésicas tipo luz do espaço-tempo acústico. Por fim, calculamos a curvatura escalar deste espaço-tempo verificando a presença... / In this dissertation we study the classical aspects of analogue models of General Relativity in condensed matter seeking mainly to create a new perception about semi-classical gravitational effects, such as Hawking radiation, in order to better comprehend them. We demonstrate that sound waves propagating in an ideal barotropic fluid with a non-homogeneous irrotacional flow, over a sphere 'S POT. 2' with radius r behave as a Klein-Gordon massless scalar field in a curved spacetime. Through this dissertation, we analyze several properties of this effective spacetime governing the propagation of sound, whose geometry is described by a Lorentzian metric that depends on the hydrodynamic variables of the flow such as the flow velocity, the density and the local speed of sound, always trying to establish correlations between classical concepts of fluid dynamics and purely relativistic concepts. Once a general analysis of these spacetimes is made, which we denominate acoustic spacetimes, we find solutions of the dynamic variables of the fluid, since they determine the acoustic geometry, capable of modeling effective spacetimes endowed with event horizons and singularities, creating therefore a dumb/deaf hole, i.e., an analogue of a black hole and white hole of the General Relativity. We further discuss some points of the causal structure of the acoustic spacetimes, so constructing a Carter-Penrose diagram of the dumb/deaf hole with the aim of bringing to evidence the possible null trajectories of this spacetime. Furthermore, we show that in the approximation of the acoustic geometry, also called eikonal approximation, the sound rays follow lightlike geodesics of the acoustic spacetime. Finally we calculate the scalar curvature of this spacetime verifying the presence of the non flat structure of the 'S POT. 2' sphere, over which the fluid moves

Orientador: Bruto Max Pimentel Escobar / Banca: Ana Lúcia Barbosa / Banca: Júlio César Fabris / Resumo: Neste trabalho estudamos a estrutura de vínculos da Relatividade Geral (RG) e do Equivalente Teleparalelo da Relatividade Geral (ETRG), utilizando o formalismo de Hamilton-Jacobi para sistemas singulares. Fazemos uma revisão destas duas teorias de gravitação e de suas formulações ADM, tendo em mente que ambas são construídas sobre variedades que são casos particulares da variedade de Riemann-Cartan. Revemos também o formalismo de Hamilton-Jacobi para o tratamento de sistemas singulares, fazendo em seguida a sua aplicação para as duas teorias supracitadas. Nesta análise constatamos que a invariância do ETRG por transformações de Lorentz no espaço tangente das tetradas faz com que a álgebra do vínculos seja diferente daquela obtida para a RG / Abstract: In this work we study the constraint structure of General Relativity (GR) and Teleparallel Equivalent of General Relativity (TEGR), using the Hamilton-Jacobi formalism for singular systems. We make a review of these two theories of gravitation and their ADM formulation, having in mind that both theories are built over manifolds that are particular cases of the Riemann-Cartan manifold. We also review the Hamilton-Jacobi formalism for singular systems, making its application to the cited theories. In this analysis we testify that the invariance of the TEGR under Lorentz transformations in the tangent space of the tetrads implies in a different constraint algebra than that obtained in GR / Mestre

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

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

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

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

What is an event? What sort of object are they? How is a given event distinguished from other events and other objects? This thesis on science oriented metaphysics will take Davidson's account of events as its starting point to answer the above questions. It will develop this conception of events into one that is consistent with the special theory of relativity by updating its notions of change, cause and property. The new concept of a proper property, a generalization of the notion of an invariant, is introduced to solve some of these metascientific problems. Other features of the work include an analysis of the Lorentz force equation as it applies to one family of cases of causation, showing that a use of cause and effect to help individuate events cannot be complete until relativistic features are built into it. I propose that the conception of a proper property will also solve this worry over the nature of causation as it affects the issues of events above. In particular, it will attempt to solve a charge of circularity which has been leveled at Davidson's account. This property analysis also has the feature that it makes the account of events which started with Davidsonian inspiration (i.e. causes and effects are intimately connected to events) more like Kim's. Kim's account of events is modified on the grounds it does not do justice to our intuitions about changes and events. / Arts, Faculty of / Philosophy, Department of / Graduate

Numerical relativity is concerned with the generation of solutions to Einstein's equations by numerical means. In general, the construction of such a spacetime is accomplished in two stages: 1) the determination of initial data which is specified on a single spacelike hypersurface and satisfies four initial value equations, and 2) the evolution of the initial data to generate the spacetime or some portion of it. One of the key problems is the development of efficient algorithms for the solutions of these equations, as they are sufficiently complex to tax the fastest present computers. This thesis presents a comparison of various algorithms for the solution of the initial value equations, concentrating on the recently developed multi-grid method. The specific problem examined has been previously studied by Bowen, Piran and York. Their initial data has been interpreted as representing "snapshots" of three new families of black holes. Three of the four initial value equations possess analytic solutions. The remaining 2-dimensional nonlinear partial differential equation is solved numerically in this thesis using finite difference techniques. The performance of the multi-grid method, with respect to three more well-known methods, is evaluated through numerical experiments. The speed and reliability of the multi-grid algorithm are found to be very good. In addition, the results which had been previously calculated numerically by Piran are essentially reproduced, with the correction of some errors in that work. Possible extensions of the work to more complex initial value problems are also discussed. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: See fulltext for abstract. / AFRIKAANSE OPSOMMING: Sien volteks vir opsomming.