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Singularities ~n General Relativity are due to incompleteness of space-time. This thesis examines the relationships between some of the different notions of incompleteness of a manifold with connection, together with the stability of geodesic completeness and incompleteness. Some p(parallelisation)-completions of Rand S1 are compared with the b(bundle)-completions, and the applicability of the p-boundary construction to General Relativity is discussed. Relationsips are shown between g(geodesic), b(bundle) and b.a.(bounded acceleration) incompleteness. Further acceleration is dependent notions of incompleteness are defined, and it shown that A they are all equivalent to the existing notions of completeness for a Riemannian manifold. The Whitney C K topologies provide a way of topologising the space of metrics on a manifold, in order to consider stability of geodesic completeness or incompleteness. It is shown how the spaces of connections and sprays may also be topologised, and the continuity of some important mappings is demonstrated. It turns out that for R both geodesic completeness and incompleteness are stable with respect to perturbation of the spray. Incompleteness of st ~s also stable, but the complete sprays are closed. For S1~S1 and R~S1 it is shown that null and time like geodesic completeness and incompleteness all fail to be stable with respect to the space of Lorentz metrics. Given a connection/spray on a pair of manifolds, one can construct a connection/spray on their product, and this is used to show how instability of completeness/incompleteness may arise if the product ~s compact. It is also shown how to induce sprays on the retraction of a manifold.

The motivation for the measurement of refractive index variations in vitreous silica comes from the NASA/Stanford University Gravity Probe-B experiment. This experiment proposes to measure directly some of the predictions of the General Theory of Relativity by observing the extent of precession of a gyroscope in orbit around the Earth. The rotor of the proposed gyroscope will be made of vitreous silica. Screening requirements for the homogeneity of the silica rotor have been indicated in terms of the sensitivity of the mass density distribution measurements as Δ<I>p/p</I> = 3 x 10^-7 (or equivalently, refractive index sensitivity Δ<I>n</I> = 1 x 10 ^-7), with a spatial resolution (to allow control of low order multi-pole moments) to better than 5mm. The thrust of the thesis is towards improvement of existing instrumentation, actual screening of samples, and spectral characterisation of the samples in the spatial frequency domain. After a brief introduction to the instrumentation, a complete matrix analysis of the plane mirror heterodyne interferometer was then developed, using the coherency matrix representation. A detailed analysis of periodic errors (non-linearity) associated with high precision polarisation heterodyne interferometry was carried out. A worst case peak-to-peak nonlinearity of 6.2nm was calculated for the single pass plane mirror heterodyne system. A simple analogue phase meter was then designed and built; its use resulted in an increase of over an order of magnitude in the resolution of the interferometer from λ/300 (2.2nm) to λ/4000 (0.16nm). High precision measurements of refractive index variations in vitreous silica are also reported for the first time. Measurements are repeatable to the Δ<I>n</I> = 10^-7 level, with phase meter errors of ±0.7% (i.e. 10^-8 sensitivity).

This thesis is a study of the validity and application of the on-shell recursion relations within the theory of General Relativity. These relations are also known as the Britto-Cachazo-Feng-Witten (BCFW) relations. They reduce the calculation of a tree-level graviton scattering amplitude into the evaluation of two smaller physical amplitudes and of a propagator. With multiple applications of the recursion relations, amplitudes can be uniquely constructed from fundamental three-graviton amplitudes. The BCFW prescriptions were first applied to gauge theory. We thus provide a self-contained description of their usage in this context. We then generalize the proof of their validity to include gravity. The BCFW recursion relations can then be used to reconstruct the full theory from cubic vertices. We finally describe how these three-graviton vertices can be determined uniquely from Poincare symmetries.

This thesis is a study of the validity and application of the on-shell recursion relations within the theory of General Relativity. These relations are also known as the Britto-Cachazo-Feng-Witten (BCFW) relations. They reduce the calculation of a tree-level graviton scattering amplitude into the evaluation of two smaller physical amplitudes and of a propagator. With multiple applications of the recursion relations, amplitudes can be uniquely constructed from fundamental three-graviton amplitudes. The BCFW prescriptions were first applied to gauge theory. We thus provide a self-contained description of their usage in this context. We then generalize the proof of their validity to include gravity. The BCFW recursion relations can then be used to reconstruct the full theory from cubic vertices. We finally describe how these three-graviton vertices can be determined uniquely from Poincare symmetries.

The prime concern of this paper is with the nature of probability. It is argued that questions concerning the nature of probability are intimately linked to questions about the nature of time. The case study here concerns the single case propensity interpretation of probability. It is argued that while this interpretation of probability has a natural place in the quantum theory, the metaphysical picture of time to be found in relativity theory is incompatible with such a treatment of probability.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Investigamos inicialmente o problema da obtenção de uma gravitação newtoniana modificada no caso da teoria da Relatividade Geral. Na sequência, encontramos uma solução geral com simetria esférica para um campo gravitacional fraco no contexto da Teoria de Brans-Dicke. Esta solução é independente de qualquer condição de coordenada. Então, para uma escolha conveniente da função f(r) na métrica de Brans-Dicke, exibimos uma fórmula modificada para a gravitação newtoniana, a qual pode descrever a gravidade em uma escala de comprimento pequena. Discutimos os resultados em comparação com o caso da Relatividade Geral / We investigate initially the problem of obtaining a modified newtonian gravitation in the case of General Relativity theory. Following, we find a general solution with spherical symmetry for a weak gravitational field in the context of Brans-Dicke theory. This solution is independent of any coordinate condition. Then, for a convenient choice of the function f(r) in Brans-Dicke metric, we exhibit a modified formula to newtonian gravity, which can describe gravity in a small length scale. We discuss the results in comparison with the case of General Relativity

Orientador: Wladimir Seixas / Banca: Manoel Borges Ferreira Neto / Banca: Henrique Lazari / Resumo: Esta dissertação tem a proposta de organizar, discutir e apresentar de maneira precisa os conceitos matemáticos de variedade diferenciável e de tensores envolvidos no estudo da Cosmologia sob o ponto de vista da Teoria da Relatividade Geral para o modelo de Friedmann-Lemaître-Robertson-Walker. Busca-se assim apresentar um texto didático que possa ser utilizado tanto nos cursos de graduação em Matemática como de Física para uma disciplina optativa de Introdução Matemática à Cosmologia / Abstract: The goal of this dissertation is to organize and discuss in a rigorous way the mathematical concepts of manifolds and tensors needed to the study of Cosmology and the Friedmann-Lemaître-Robertson-Walker model under the point of view of the General Relativity. In this way, this dissertation was written as textbook that could be used in an undergraduate course of Physics and Mathematics / Mestre

Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-10-15Bitstream added on 2014-06-13T20:47:45Z : No. of bitstreams: 1 delbem_nf_me_rcla.pdf: 885461 bytes, checksum: 9ba35dff1d53b0378c1e134087c575b7 (MD5) / Universidade Estadual Paulista (UNESP) / Esta dissertação tem a proposta de organizar, discutir e apresentar de maneira precisa os conceitos matemáticos de variedade diferenciável e de tensores envolvidos no estudo da Cosmologia sob o ponto de vista da Teoria da Relatividade Geral para o modelo de Friedmann-Lemaître-Robertson-Walker. Busca-se assim apresentar um texto didático que possa ser utilizado tanto nos cursos de graduação em Matemática como de Física para uma disciplina optativa de Introdução Matemática à Cosmologia / The goal of this dissertation is to organize and discuss in a rigorous way the mathematical concepts of manifolds and tensors needed to the study of Cosmology and the Friedmann-Lemaître-Robertson-Walker model under the point of view of the General Relativity. In this way, this dissertation was written as textbook that could be used in an undergraduate course of Physics and Mathematics