We firstly numerically recalculate the Ricci tensor of non-stationary axisymmetric space-times (originally calculated by Chandrasekhar) and we find some discrepancies both in the linear and non-linear terms. However, these discrepancies do not affect the results concerning linear perturbations of a Schwarzschild black hole. Secondly, we use these Ricci tensors to derive the Zerilli and Regge-Wheeler equations and use the Newman-Penrose formalism to derive the Bardeen-Press equation. We show the relation between these equations because they describe the same linear perturbations of a Schwarzschild black hole. Thirdly, we illustrate heuristically (when the angular momentum (l) is 2) the relation between the linearized solution of the Einstein vacuum equations obtained from the Bondi-Sachs metric and the Zerilli equation, because they describe the same linear perturbations of a Schwarzschild black hole. Lastly, by means of a coordinate transformation, we extend Chandrasekhar's results on linear perturbations of a Schwarzschild black hole to the Bondi-Sachs framework. / Mathematical Sciences / M. Sc. (Applied Mathematics)

Desde a formulação da Teoria da Relatividade por Einstein no início do século XX, foram várias as tentativas para se unificar as diversas teorias físicas. Kaluza e Klein foram os primeiros a buscarem esta unificação. Atualmente baseado nas idéias de Kaluza e Klein o sonho de uma unificação retorna através das teorias de cordas e dos modelos de grandes dimensões extras. Neste trabalho faremos um estudo de dois desses modelos contendo grandes dimensões extras, mostrando como é possível se explicar o problema da hierarquia existente entre as interações fundamentais através de dimensões extras compactificadas. O foco, contudo, será o estudo da possibilidade de produção de mini buracos negros (MBN's) em laboratório talvez já na próxima geração de colisores, como o LHC. Dentre os possíveis tipos de buracos negros, estudaremos o de Schwarzschild (sem carga e com momento angular nulo) enfatizando o efeito da carga elétrica que dá origem ao buraco negro de Reissner-Nordström (carregado eletricamente e sem momento angular). Finalmente trataremos dos observáveis que os MBN's (se de fato existirem) poderão deixar nos detectores.

From Birkoff's theorem, the geometry in four spacetime dimensions outside a spherically symmetric and static, gravitating source must be given by the Schwarzschild metric. This metric therefore satisfies the Einstein vacuum equations. If the mass which gives rise to the Schwarzschild spacetime geometry is concentrated within a radius of r=2M, a black hole will form. Non-accelerating particles (freely falling) traveling through this geometry will do so along parametrized curves called geodesics, which are curved space generalizations of straight paths. These geodesics can be found by solving the geodesic equation. In this thesis, the geodesic structure in the Schwarzschild geometry is investigated with an attempt to generalize the solution to higher dimensions.

This thesis investigates the dynamics of a set of 8-600 Schwarzschild masses, randomly distributed inside cells which tessellate a 3-sphere. Furthermore the contents of each cell are mirror images of its neighbor. This symmetry give rise to a locally rotationally symmetric (LRS) curve, along which the Einstein field equations governing dynamics can be exactly integrated. The result is an irregular model consisting of discrete matter content, but where the dynamics is easy to calculate. We see that these local inhomogeneities will cause behavior deviating from the spherical dust-filled FLRW model. For instance, there are cases where configurations exhibit acceleration along the LRS curve, even though the content consists solely of ordinary matter with a vacuum filled exterior and no cosmological constant. / Denna avhandling undersöker konfigurationer av 8-600 Schwarzschild-massor, som är slumpmässigt utplacerade inom celler som tessellerar en 3-sfär. Utöver det är även innehållet i varje cell en spegelbild av granncellen. Denna symmetri ger upphov till en lokalt rotationssymmetrisk (LRS) kurva där Einsteins fältekvationer som beskriver dynamiken längs med är exakt integrerbara. Resultatet är en oregelbunden modell som består av diskreta massor, men vars dynamik är enkel att beräkna. Vi ser att dessa lokala inhomogeniteter ger upphov till beteenden som avviker från den sfäriska partikel-fyllda FLRW-modellen. Till exempel uppstår konfigurationer som uppvisar acceleration längs med LRS-kurvan, trots att innehållet består endast av ordinära massor med vakuum utanför och ingen kosmologisk konstant.

We firstly numerically recalculate the Ricci tensor of non-stationary axisymmetric space-times (originally calculated by Chandrasekhar) and we find some discrepancies both in the linear and non-linear terms. However, these discrepancies do not affect the results concerning linear perturbations of a Schwarzschild black hole. Secondly, we use these Ricci tensors to derive the Zerilli and Regge-Wheeler equations and use the Newman-Penrose formalism to derive the Bardeen-Press equation. We show the relation between these equations because they describe the same linear perturbations of a Schwarzschild black hole. Thirdly, we illustrate heuristically (when the angular momentum (l) is 2) the relation between the linearized solution of the Einstein vacuum equations obtained from the Bondi-Sachs metric and the Zerilli equation, because they describe the same linear perturbations of a Schwarzschild black hole. Lastly, by means of a coordinate transformation, we extend Chandrasekhar's results on linear perturbations of a Schwarzschild black hole to the Bondi-Sachs framework. / Mathematical Sciences / M. Sc. (Applied Mathematics)

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Made available in DSpace on 2011-03-23T21:19:31Z (GMT). No. of bitstreams: 0 / Item created via OAI harvest from source: http://www.bdtd.ufpa.br/tde_oai/oai2.php on 2011-03-23T21:19:31Z (GMT). Item's OAI Record identifier: oai:bdtd.ufpa.br:223 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico / UFPA - Universidade Federal do Pará / In this dissertation we compute the absorption cross section of Schwarzschild black holes for the massless scalar and electromagnetic fields. We also compute the absorption cross section of canonical acoustic holes for sound waves. We use a numerical method to obtain the results in arbitrary frequencies. We also obtain analytic expressions for the low- and high-frequency absorption cross sections. The numerical results are in excellent agreement with the low- and high-frequency absorption cross section values obtained analytically. In the zero-frequency limit the absorption cross section tends to the event horizon area value for both the massless scalar field in Schwarzschild spacetime and the canonical acoustic hole cases. However, as the frequency increases, these two results become very different. This shows that, although the spacetime geometry does not influence the absorption cross section in the zero-frequency limit, it is important for arbitrary frequencies. We also see that massless scalar and electromagnetic absorption cross section values for the Schwarzschild black hole coincide for high enough frequencies and angular momenta. The spin of the scattered particle, in this case, although being very important for low frequencies, becomes less relevant to the absorption cross section value as the frequency and the angular momentum of the incident particle increase. / Na presente dissertação calculou-se a seção de choque de absorção de buracos negros de Schwarzschild para os campos escalar não massivo e eletromagnético. Também calculamos a seção de choque de absorção de buracos acústicos canônicos. Utilizamos um método numérico para obter os resultados em freqüências arbitrárias. Obtemos também expressões analíticas para as seções de choque de absorção nos limites de baixas e altas freqüências. Os resultados numéricos estão em excelente concordância com os valores das seções de choque de absorção em baixas e altas freqüências obtidos analiticamente. No limite em que a freqüência tende a zero, a seção de choque de absorção tende ao valor da área do horizonte de eventos tanto para o caso do campo escalar não massivo em Schwarzschild quanto para o buraco acústico canônico. Entretanto, a medida que a freqüência aumenta, estes resultados se tornam bastante distintos. Isto mostra que, apesar de a forma do espaço-tempo não exercer influência sobre a seção de choque escalar no limite em que a freqüência tende a zero, ela é determinante fora desse limite. Observamos também que os valores das seções de choque de absorção escalar e eletromagnética em Schwarzschild coincidem para freqüências e momentos angulares suficientemente grandes. O spin da partícula espalhada, neste caso, apesar de ter grande influência a baixas energias, é menos importante para o valor da seção de choque de absorção quanto maiores forem a freqüência e o momento angular da onda incidente.

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Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-04-26T18:18:25Z No. of bitstreams: 1 monalisasilvadeoliveira.pdf: 473060 bytes, checksum: 1e394ef35b65c023c130f0dbca9a9d12 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-13T12:04:23Z (GMT) No. of bitstreams: 1 monalisasilvadeoliveira.pdf: 473060 bytes, checksum: 1e394ef35b65c023c130f0dbca9a9d12 (MD5) / Made available in DSpace on 2017-05-13T12:04:23Z (GMT). No. of bitstreams: 1 monalisasilvadeoliveira.pdf: 473060 bytes, checksum: 1e394ef35b65c023c130f0dbca9a9d12 (MD5) Previous issue date: 2013-02-28 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho, fazemos uma pequena revisão sobre tensores e sua utilização na Relatividade Geral, apresentamos o método de transformação conforme e o teorema da fatorização e discutimos as soluções de Schwarzschild com e sem constante cosmológica. Então, a solução de Schwarzschild com constante cosmológica é derivada, a partir das equações de campo de Einstein, utilizando-se os conceitos abordados. / In this work, we make a brief review of tensors and their use in General Relativity, we present the local conformal transformation method and the factorization theorem and we discuss Schwarzschild's solutions with and without cosmological constant. Then, the Schwarzschild's solution with cosmological constant is derived, from the Einstein's field equations, using the concepts addressed.

This work is focused on visualisation of Schwarzschild, Reissner- Nordström and Kerr black hole. The two-dimensional conformal diagram was constructed. In the case of Kerr black hole, the causal structure was visualized by intersection of chronological future of given point in spacetime with hyper- surfaces of constant value of Boyer-Lindquist coordinate t. Conformal diagram for Kerr black hole was constructed only in the neighbourhood of outer event horizon. Then the causal diagram, which is analogous to conformal diagram for Reissner-Nordström black hole was constructed. In all cases two-dimensional spa- celike hypersurfaces were chosen that were embedded into Euclidean space. The interpretation of time evolution of black hole universe was given to a sequence of such embedded hypersurfaces. In the case of Kerr black hole the embedding of outer ergosphere and outer event horizon were also constructed. 1