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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.

Vegetation has been identified to play a significant role in river environments by providing a wide range of ecosystem services. For this reason, the use of plants has become relevant in river restoration projects. However, the presence of plants in channel beds increases the flow resistance and, thus, the water levels during flood conditions. Additionally, river vegetation, whether instream or riparian, influences the morphological evolution of rivers. Observations show that instream vegetation has a strong impact on bedload transport. Yet, there is a scarcity of sediment transport predictors that directly account for the effects of plants, and existing methods, based on re-calculation of roughness coefficients, may present some inconsistencies. Therefore, an approach that extends Einstein’s (1950) parameters to include the effects of vegetation geometry and spatial density on sediment transport is herein proposed. The new formulations of the dimensionless transport parameter Φ and the flow intensity parameter Ψ were derived for their implementation in existing bedload predictors of the form Φ = (Ψ). The applicability of this new approach considers the presence of submerged and emergent vegetation, but reduces to the original Einstein’s model if vegetation is absent. The research methodology was carried out in four phases. First, a comprehensive literature review for the identification of, mainly, the different effects of vegetation on river morphodynamics, the state-of-the-art knowledge on the flow-sediment-vegetation interactions, and the current approaches to bedload estimation in channels with vegetated beds. Second, the derivation of the extended Einstein’s parameters, starting from a momentum balance for a control volume of a generic channel with instream submerged vegetation (as proposed by Petryk and Bosmajian, 1975). Third, an extensive experimental program carried out on a tilting flume with a mobile bed and with plants being represented by series of aluminum cylinders. Different scenarios of vegetation spatial density were tested while measurements of bedload rate, water level, bed level and flow velocity were periodically performed in order to assess conditions of stationarity and morphodynamic equilibrium. Last, a deep analysis of experimental results allowed for the calibration of the new approach, whereas external datasets from the literature were used to assess its performance in a wide variety of conditions. A study based on four statistical measures showed that the extended Einstein’s parameters are significantly more suitable for bedload rate estimation when compared to the original ones, since predicted and measured values have, on average, the same order of magnitude. Additionally, the new approach outperformed the widely-adopted method of Baptist (2005), which consists of the re-calculation of bed roughness in vegetated settings. Finally, the experimental observations suggest that the submergence ratio and the stem spatial density are the most important traits of river plants to display influence on bedload transport, channel bed stability, and bed form dimensions and patterns. A better understanding of these traits might lead to better prediction capabilities of river evolution. / La vegetazione svolge un ruolo fondamentale negli ambienti fluviali, poiché fornisce un ampio spettro di servizi ecosistemici; per questo essa è una componente rilevante dei progetti di riqualificazione fluviale. Tuttavia, la presenza di piante in alveo aumenta la resistenza al moto e di conseguenza anche il tirante idrico durante gli eventi di piena. Inoltre, la copertura vegetale in alveo e nelle zone riparie influenza l'evoluzione morfologica dei corsi d'acqua. Nonostante le evidenze sperimentali mostrino che la vegetazione in alveo ha un forte impatto sul trasporto dei sedimenti, sono poche le formule di trasporto che tengono conto in modo esplicito dell'effetto della vegetazione e i metodi esistenti, basati sulla determinazione di un coefficiente di scabrezza, possono dare luogo a incongruenze. Per questa ragione, in questa tesi si propone un approccio che estende la formulazione di Einstein (1950) e include l'effetto della geometria e della densità spaziale della vegetazione sul trasporto solido. Sono state derivate nuove espressioni per il parametro di trasporto adimensionale Φ e il parametro di intensità del trasporto Ψ, che possono essere introdotte in modelli di trasporto esistenti del tipo Φ = f(Ψ). Questo nuovo approccio consente di considerare l'effetto della presenza di vegetazione sommersa ed emergente e si riduce al modello originale di Einstein in assenza di vegetazione. L'attività di ricerca si è svolta in quattro fasi. Nella prima fase si è svolta un'analisi approfondita della letteratura mirata soprattutto a identificare gli effetti della vegetazione sulla morfodinamica fluviale, definire lo stato dell'arte relativo alle interazioni fra flusso liquido, sedimenti e vegetazione, ed analizzare gli approcci esistenti per la stima del trasporto di fondo in alvei vegetati. Nella seconda fase si sono derivati i parametri della formulazione di Einstein estesa a partire dal bilancio di quantità di moto per un volume di controllo di un canale generico con vegetazione sommersa (come proposto da Petryk e Bosmajian, 1975). Nella terza fase è stato condotto un esteso set di esperimenti, utilizzando un modello fisico costituito da una canaletta di laboratorio a pendenza variabile e fondo mobile, in cui le piante sono state simulate tramite cilindri in alluminio. Sono stati riprodotti diversi scenari di densità spaziale della vegetazione e sono stati misurati periodicamente la portata solida, la quota della superficie libera e del fondo e la velocità della corrente per valutare le condizioni di stazionarietà ed equilibrio morfodinamico. Infine, il nuovo approccio è stato calibrato sulla base di un'analisi approfondita dei risultati sperimentali e quindi applicato a set di dati di letteratura per valutarne l'accuratezza in un ampio intervallo di condizioni. Un'analisi statistica basata su quattro indicatori ha mostrato che i parametri della formulazione di Einstein estesa producono stime di trasporto solido sensibilmente più accurate rispetto ai parametri originali, in quanto i valori calcolati sono, in generale, dello stesso ordine di grandezza dei valori misurati. Inoltre, il nuovo approccio dà risultati migliori rispetto al metodo di Baptist (2005), ampiamente adottato, che consiste nel ricalcolo della scabrezza per gli alvei vegetati. Infine, le osservazioni sperimentali suggeriscono che il rapporto di sommergenza e la densità spaziale delle piante sono i parametri che influenzano in modo più significativo il trasporto solido, la stabilità del fondo dell'alveo, la scala delle forme di fondo e la loro organizzazione spaziale. Una conoscenza più approfondita di questi aspetti può contribuire a una maggiore capacità di prevedere l'evoluzione dei corsi d'acqua. / Se ha identificado a la vegetación como un actor importante en ambientes fluviales al proporcionar una amplia gama de servicios ecosistémicos. Por esta razón, el uso de plantas se ha vuelto cada vez más relevante en proyectos de restauración de ríos. Sin embargo, su presencia en lechos fluviales impacta la resistencia al flujo, aumentando los niveles del agua en condiciones de inundación. Además, este tipo de vegetación, ya sea que esté en el lecho o en las márgenes, influye en la evolución morfológica de los ríos. Diversas observaciones han mostrado que la vegetación fluvial tiene un fuerte impacto en las tasas de transporte sólido de fondo. A pesar de ello, hay una escasez de métodos confiables para la estimación de este tipo de sedimentos que tome en consideración el efecto de las plantas y, aquéllos que existen, los cuales se basan en la corrección del coeficiente de rugosidad del canal, suelen presentar resultados inconsistentes. Por tanto, se propone aquí un método que extiende las definiciones fundamentales de Einstein (1950) en modo que se incluyan los efectos de la geometría y la densidad espacial de las plantas sobre el transporte sólido. Las nuevas ecuaciones del parámtero de transporte, Φ, y el parámetro de movilidad, Ψ, fueron obtenidas para su implementación en métodos predictores de transporte de fondo de la forma Φ = (Ψ). La aplicabilidad de este nuevo enfoque considera la posibilidad de vegtación fluvial tanto emergente como sumergida, y se reduce a las ecuaciones originales de Einstein si ésta fuera inexistente. La metodología de investigación se llevó a cabo en cuatro fases. Primero, una revisión exhaustiva de la literatura para la identificación, principalmente, de los diferentes efectos de la vegetación en la morfodinámica de ríos, los avances más recientes en el conocimiento sobre las interacciones flujo-sedimento-vegetación, y los métodos actualmente existentes para la estimación del transporte sólido de fondo en canales naturales vegetados. En segundo lugar, la obtención de los parámetros de Einstein extendidos a partir de un balance de momentum para el volumen de control de un canal genérico con vegetación sumergida (según lo propuesto por Petryk y Bosmajian, 1975). En tercer lugar, un extenso programa experimental realizado en un canal de fondo móvil y pendiente variable, con las plantas siendo representadas por series de cilindros metálicos. Se probaron diferentes escenarios de densidad espacial de vegetación, mientras que periódicamente se realizaron mediciones transporte sólido, niveles del agua, topografía del fondo y velocidad del flujo con el objeto de evaluar las condiciones de flujo uniforme y equilibrio morfodinámico. Por último, un análisis profundo de los resultados experimentales permitió la calibración del nuevo método, mientras que se utilizaron datos externos disponibles en la literatura para evaluar su desempeño bajo diversas condiciones. Un estudio basado en cuatro medidas estadísticas mostró que los parámetros extendidos de Einstein son mucho más adecuados para la estimación del transporte de fondo en comparación con los originales, ya que los valores estimados y los medidos muestran, en promedio, el mismo orden de magnitud. Además, el nuevo método superó al propuesto por Baptist (2005), ampliamente utilizado, el cual consiste en la corrección de la rugosidad del canal en presencia de vegetación. Finalmente, las observaciones experimentales sugieren que la sumergencia de las plantas y la densidad espacial de los tallos son las variables más influyentes en el transporte sedimentos de fondo, la estabilidad del lecho, y las dimensiones y patrones de la forma de fondo. Una mejor comprensión de estas variables puede significar una mejor capacidad para predecir la evolución de un río.

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

In this thesis, after a general introduction, we first review some differential geometry to provide the mathematical background needed to derive the key equations in cosmology. Then we consider the Robertson-Walker geometry and its relationship to cosmography, i.e., how one makes measurements in cosmology. We finally connect the Robertson-Walker geometry to Einstein's field equation to obtain so-called cosmological Friedmann-Lemaître models. These models are subsequently studied by means of potential diagrams.

This work focuses on three nonlinear relativistic equations: the symmetric Chiral field equation, Einstein's field equation for metrics with two commuting Killing vectors and Einstein's field equation for diagonal metrics that depend on three variables. The symmetric Chiral field equation is studied using the Zakharov-Mikhailov transform, with which its infinitely many local conservation laws are derived and its solitons on diagonal backgrounds are studied. It is also proven that it is equivalent to a novel equation that poses a fascinating similarity to the Sinh-Gordon equation. For the 1+1 Einstein equation the Belinski-Zakharov transformation is explored. It is used to derive explicit formula for N gravitational solitons on arbitrary diagonal background. In particular, the method is used to derive gravitational solitons on the Einstein-Rosen background. The similarities and differences between the attributes of the solitons of the symmetric Chiral field equation and those of the 1+1 Einstein equation are emphasized, and their origin is pointed out. For the 1+2 Einstein equation, new equations describing diagonal metrics are derived and their compatibility is proven. Different gravitational waves are studied that naturally extend the class of Bondi-Pirani-Robinson waves. It is further shown that the Bondi-Pirani-Robinson waves are stable with respect to perturbations of the spacetime. Their stability is closely related to the stability of the Schwarzschild black hole and the relation between the two allows to conjecture about the stability of a wide range of gravitational phenomena. Lastly, a new set of equations that describe weak gravitational waves is derived. This new system of equations is closely and fundamentally connected with the nonlinear Schrödinger equation and can be properly called the nonlinear Schrödinger-Einstein equations. A few preliminary solutions are constructed.

In this thesis, Newman-Penrose techniques are used to obtain some new exact solutions to Einstein's field equations of general relativity and to assist in the physical interpretation of some exact radiative space-times. Attention is restricted to algebraically special space-times with a twist-free, repeated principal null congruence. In particular, the Robinson-Trautman type N solutions, which describe expanding gravitational waves, are investigated for all possible values of the cosmological constant A and the Gaussian curvature parameter E. The wave surfaces are always (hemi-)spherical, with successive surfaces displaced along time-like, space-like or null lines, depending on E. Explicit sandwich waves of this class are studied in Minkowski, de Sitter or anti-de Sitter backgrounds and a particular family of such solutions, which can be used to represent snapping or decaying cosmic strings, is considered in detail. The singularity and global structure of the solutions is also presented. In the remaining part of the thesis, the complete family of space-times with a non-expanding, shear-free, twist-free, geodesic principal null congruence (Kundt waves), that are of algebraic type III and for which the cosmological constant (Ac) is non-zero, is presented. The possible presence of an aligned pure radiation field is also assumed. These space-times generalise the known vacuum solutions of type N with arbitrary Ac and type III with Ac = O. It is shown that there are two, one and three distinct classes of solutions when Ac is respectively zero, positive and negative and, in these cases, the wave surfaces are plane, spherical or hyperboloidal in Minkowski, de Sitter or anti-de Sitter backgrounds respectively. The singularities which occur in these space-times are interpreted in terms of envelopes of these wave surfaces. Again, by considering functions of the retarded time which "cross-over" between canonical types, sandwich waves are also studied. The limiting cases of these, giving rise to shock or impulsive waves, are also considered.

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.