In this thesis a number of problems concerning proper curvature collineations, proper Weyl collineations and projective vector fields will be considered. The work on the above areas can be summarised as: (i) A study of proper curvature collineations in plane symmetric static, spherically symmetric static and Bianchi type <I>I</I> spacetimes will be presented by considering the rank of their 6 x 6 Riemann tensors and using a theorem which eliminates those space-times where proper curvature collineations can not exist; (ii) A study of proper Weyl collineations is given by using the algebraic classification and associated rank of the Weyl tensor and using a theorem similar to that used in (i); (iii) A technique is developed to study projective vector fields in the Friedmann Robertson-Walker models and plane symmetric static spacetimes; (iv) The situations when conformally flat spacetimes admit proper curvature collineations are fully explored.

Einstein's field equations are extremely difficult to solve, and when solved, the solutions are even harder to understand. In this thesis, two analysis tools are developed to explore and visualize the curvature of spacetimes. The first tool is based on a thorough examination of observer independent curvature invariants constructed from different contractions of the Riemann curvature tensor. These invariants are analyzed through their gradient fields, and attention is given to the resulting flow and critical points. Furthermore, we propose a Newtonian analog to some general relativistic invariants based on the underlying physical meaning of these invariants, where they represent the cumulative tidal and frame-dragging effects of the spacetime. This provides us with a novel and intuitive tool to compare Newtonian gravitational fields to exact solutions of Einstein's field equations on equal footing. We analyze the obscure Curzon-Chazy solution using the new approach, and reveal rich structure that resembles the Newtonian gravitational field of a non-rotating ring, as it has been suspected for decades. Next, we examine the important Kerr solution, which describes the gravitational field of rotating black holes. We discover that the observable part of the geometry outside the black hole's event horizon depends significantly on its angular momentum. The fields representing the cumulative tidal and frame-dragging forces change qualitatively at seven specific values of the dimensionless spin parameter of the black hole. The second tool we develop in this thesis is the accurate construction of the Penrose conformal diagrams. These diagrams are a valuable tool to explore the causal structure of spacetimes, where the entire spacetime is compactified to a finite size, and the coordinate choice is fixed such that light rays are straight lines on the diagram. However, for most spacetimes these diagrams can only be constructed as a qualitative guess, since their null geodesics cannot be solved. We developed an algorithm to construct very accurate Penrose diagrams based on numeric solutions to the null geodesics, and applied it to the McVittie metric. These diagrams confirmed the long held suspicion that this spacetime does indeed describe a black hole embedded in an isotropic universe. / Thesis (Ph.D, Physics, Engineering Physics and Astronomy) -- Queen's University, 2013-09-30 14:02:55.865

Orientador: Samuel Rocha de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T22:23:20Z (GMT). No. of bitstreams: 1 D'Afonseca_LuisAlberto_D.pdf: 4257675 bytes, checksum: 54debc66eff41b6c8b450adbcfc3bab6 (MD5) Previous issue date: 2007 / Resumo: Nesse trabalho delineamos a teoria clássica para o campo de Einstein-Yang-Mills e elaboramos um conjunto particular de equações para obtermos soluções numéricas. Estudamos dois casos com simetria espaço-temporal: Simetria esférica com campo auto-gravitante e simetria axial no espaço-tempo de Minkowski. Utilizamos métodos numéricos das linhas para fazer a evolução temporal dos campos discretizados. No caso com simetria esférica, os campos são discretizados por diferenças finitas e no caso da simetria axial comparamos as discretizações por métodos Pseudo-Espectrais e por diferenças finitas. Para evolução temporal um método auto-adaptativo de Runge-Kutta é empregado. Na simulação dos campos de Yang-Mills auto-gravitantes com simetria esférica mostramos a evolução da implosão e explosão de uma casca energética sem formação de buraco negro nem de corpo estável. No caso com simetria axial além da implosão e explosão de pulsos de cores diferentes dos campos de Yang-Mills, geramos também várias soluções dinâmicas em que vemos o transiente do intercâmbio de energia entre essas cores / Abstract: In this work we outline the classic theory of Einstein-Yang-Mills fields and work out a set of particular equations suited for numerical simulations. We consider two special cases with space-time symmetries: self-gravitating spherical symmetric and axially symmetric field on a Minkowski space-time. We use the numerical method of lines for time evolution of discretized fields. On the spherical symmetric case, the fields are discretized by finite differences and on the axial symmetric case we compare the field discretization by the pseudo-spectral method and finite differences method. For time stepping we use a self-adaptive Runge-Kutta method. In the simulation of Yang-Mills self-gravitating fields with spherical symmetry we show the evolution of implosion and explosion of a energetic shell without black hole or stable body formation. In the axial symmetric case besides implosion and explosion of pulses of different colours of Yang-Mills fields, we also generate several dynamic solutions that display the transient of the energy exchange among these colours / Doutorado / Fisica-Matematica / Doutor em Matemática Aplicada

Corvino, Corvino and Schoen, Chruściel and Delay have shown the existence of a large class of asymptotically flat vacuum initial data for Einstein's field equations which are static or stationary in a neighborhood of space-like infinity, yet quite general in the interior. The proof relies on some abstract, non-constructive arguments which makes it difficult to calculate such data numerically by using similar arguments. A quasilinear elliptic system of equations is presented of which we expect that it can be used to construct vacuum initial data which are asymptotically flat, time-reflection symmetric, and asymptotic to static data up to a prescribed order at space-like infinity. A perturbation argument is used to show the existence of solutions. It is valid when the order at which the solutions approach staticity is restricted to a certain range. Difficulties appear when trying to improve this result to show the existence of solutions that are asymptotically static at higher order. The problems arise from the lack of surjectivity of a certain operator. Some tensor decompositions in asymptotically flat manifolds exhibit some of the difficulties encountered above. The Helmholtz decomposition, which plays a role in the preparation of initial data for the Maxwell equations, is discussed as a model problem. A method to circumvent the difficulties that arise when fast decay rates are required is discussed. This is done in a way that opens the possibility to perform numerical computations. The insights from the analysis of the Helmholtz decomposition are applied to the York decomposition, which is related to that part of the quasilinear system which gives rise to the difficulties. For this decomposition analogous results are obtained. It turns out, however, that in this case the presence of symmetries of the underlying metric leads to certain complications. The question, whether the results obtained so far can be used again to show by a perturbation argument the existence of vacuum initial data which approach static solutions at infinity at any given order, thus remains open. The answer requires further analysis and perhaps new methods. / Corvino, Corvino und Schoen als auch Chruściel und Delay haben die Existenz einer grossen Klasse asymptotisch flacher Anfangsdaten für Einsteins Vakuumfeldgleichungen gezeigt, die in einer Umgebung des raumartig Unendlichen statisch oder stationär aber im Inneren der Anfangshyperfläche sehr allgemein sind. Der Beweis beruht zum Teil auf abstrakten, nicht konstruktiven Argumenten, die Schwierigkeiten bereiten, wenn derartige Daten numerisch berechnet werden sollen. In der Arbeit wird ein quasilineares elliptisches Gleichungssystem vorgestellt, von dem wir annehmen, dass es geeignet ist, asymptotisch flache Vakuumanfangsdaten zu berechnen, die zeitreflektionssymmetrisch sind und im raumartig Unendlichen in einer vorgeschriebenen Ordnung asymptotisch zu statischen Daten sind. Mit einem Störungsargument wird ein Existenzsatz bewiesen, der gilt, solange die Ordnung, in welcher die Lösungen asymptotisch statische Lösungen approximieren, in einem gewissen eingeschränkten Bereich liegt. Versucht man, den Gültigkeitsbereich des Satzes zu erweitern, treten Schwierigkeiten auf. Diese hängen damit zusammen, dass ein gewisser Operator nicht mehr surjektiv ist. In einigen Tensorzerlegungen auf asymptotisch flachen Räumen treten ähnliche Probleme auf, wie die oben erwähnten. Die Helmholtzzerlegung, die bei der Bereitstellung von Anfangsdaten für die Maxwellgleichungen eine Rolle spielt, wird als ein Modellfall diskutiert. Es wird eine Methode angegeben, die es erlaubt, die Schwierigkeiten zu umgehen, die auftreten, wenn ein schnelles Abfallverhalten des gesuchten Vektorfeldes im raumartig Unendlichen gefordert wird. Diese Methode gestattet es, solche Felder auch numerisch zu berechnen. Die Einsichten aus der Analyse der Helmholtzzerlegung werden dann auf die Yorkzerlegung angewandt, die in den Teil des quasilinearen Systems eingeht, der Anlass zu den genannten Schwierigkeiten gibt. Für diese Zerlegung ergeben sich analoge Resultate. Es treten allerdings Schwierigkeiten auf, wenn die zu Grunde liegende Metrik Symmetrien aufweist. Die Frage, ob die Ergebnisse, die soweit erhalten wurden, in einem Störungsargument verwendet werden können um die Existenz von Vakuumdaten zu zeigen, die im räumlich Unendlichen in jeder Ordnung statische Daten approximieren, bleibt daher offen. Die Antwort erfordert eine weitergehende Untersuchung und möglicherweise auch neue Methoden.

The aim of this work is to derive mathematical descriptions of gravitation. Postulating gravitation as a force field, Newton's law of gravitation is heuristically derived by considering linear differential operators invariant under euclidean isometries and by finding the fundamental solution to Helmholtz equation in three dimensions. Thereafter, the theory of differential geometry is introduced, providing a framework for the subsequent review of gravitation as curvature. Lastly, in the light of Einstein's postulates and equivalence principle, Lovelock's proof of uniqueness of Einstein's field equations is presented.

Cosmology is the study of the universe as a whole, and attempts to describe the behaviour of the universe mathematically. The simplest relativistic cosmological models are derived from Einstein's field equations with the assumptions of isotropy and homogeneity. In this thesis, a few simple cosmological models will be derived and evaluated with respect to their description of our universe i.e., how well they match observational data from e.g., the cosmic background radiation and redshift from distant supernovae. The models are derived from Einstein's field equations, which is why a large portion of the thesis will lay the ground work for the field equations by introducing and explaining the language of tensors. / Kosmologi är läran om universum i stort samt dess matematiska beskrivning. De enklaste relativistiska kosmologiska modellerna kan härledas från Einsteins fältekvationer med hjälp av antaganden om isotropi och homogenitet. I denna rapport kommer ett par av de enklaste modellerna att härledas, samt evalueras baserat på hur väl de beskriver vårt universum, det vill säga hur bra de passar de observationer som gjorts på exempelvis den kosmiska bakgrundsstrålningen och rödskifte från avlägsna supernovor. Modellerna härleds utifrån Einsteins fältekvationer, varför en stor del av rapporten består av en introduktion till tensoranalys.