Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, 2015. / Submitted by Ana Cristina Barbosa da Silva (annabds@hotmail.com) on 2015-07-06T14:30:58Z No. of bitstreams: 1 2015_NewtonMayerSolorzanoChavez.pdf: 713834 bytes, checksum: fa5dcfcc4bcd42f4b02d1ce4b3e3f95b (MD5) / Approved for entry into archive by Raquel Viana(raquelviana@bce.unb.br) on 2015-08-18T12:21:50Z (GMT) No. of bitstreams: 1 2015_NewtonMayerSolorzanoChavez.pdf: 713834 bytes, checksum: fa5dcfcc4bcd42f4b02d1ce4b3e3f95b (MD5) / Made available in DSpace on 2015-08-18T12:21:50Z (GMT). No. of bitstreams: 1 2015_NewtonMayerSolorzanoChavez.pdf: 713834 bytes, checksum: fa5dcfcc4bcd42f4b02d1ce4b3e3f95b (MD5) / Consideramos métricas de Finsler esfericamente simétricas do tipo Douglas. Caracterizamos tais métricas por uma equação diferencial e obtemos a solução geral desta equação em termos de quatro funções arbitrárias. Quando as métricas de Finsler são esfericamente simétricas mostramos que as métricas do tipo Berwald coincidem com as do tipo Landsberg. Provamos que o problema de classificar as métricas esfericamente simétricas do tipo Douglas com S−curvatura nula reduz-se a classificar as métricas esfericmanete simétricas do tipo Berwald ou Landsberg. Obtemos a classificação de tais métricas. Incluímos vários exemplos e classes de novas métricas de Douglas. / We consider spherically symmetric Finsler metrics of Douglas type. We characterize such metrics by a differential equation and we obtain the general solution of this equation in terms of four arbitrary functions. For spherically symmetric Finsler metrics we show that the metrics of Berwald type coincide whit those of Landsberg type. We prove that the problem of classifying the spherically symmetric Douglas metrics whose S−curvature vanishes reduce to classifying the spherical symmetric metrics of Berwald or Landsberg type. We obtain the classification of such metrics. We include several examples and new classes of Douglas metrics.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-08-11T14:47:03Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) DISSERTAÇÃO (2015-03-20) - MARCELO SANTOS CARIELO[30-03-2016].pdf: 1081373 bytes, checksum: d94173b038b7893578b286bd3703e416 (MD5) / Made available in DSpace on 2016-08-11T14:47:03Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) DISSERTAÇÃO (2015-03-20) - MARCELO SANTOS CARIELO[30-03-2016].pdf: 1081373 bytes, checksum: d94173b038b7893578b286bd3703e416 (MD5) Previous issue date: 2015-03-20 / CNPq / Usando a teoria de KCC, estaremos estudando a construção do método da Trofodinâmica Analítica. Apresentaremos os conceitos de estabilidade de Jacobi e estabilidade linear para sistemas de equaçoes diferenciais em nossa modelagem. Veremos como a abordagem da Trofodinâmica Analítica permite um estudo mais adequado, se comparado com os métodos tradicionais, de certos sistemas ecológicos onde a dinâmica populacional dependa da densidade de população, ou das taxas de crescimento destas. Finalizamos com exemplos de aplicação em recifes de corais. / Using KCC theory, we will be studying the construction of the Analytical Trophodynamics method. We will introduce the concepts of Jacobi's stability and linear stability for systems of di erential equations within our modelling. We will show how Analytical Trophodynamics is a more appropriate approach, compared with the traditional methods, for modelling certain ecological systems where the population dynamics depend on population density, or on their growth rates. We conclude with examples of applications in coral reef ecology.

Nesta tese foi introduzido o conceito de folheação Finsleriana singular, que generaliza ação Finsleriana, submersão Finsler e folheação Finsleriana. O primeiro resultado desta tese afirma que qualquer folheação Finsleriana singular sobre uma variedade Randers com data (h,W) é folheação Riemanniana singular com respeito a h e W é um campo folheado. Para obter este resultado provou-se um teorema de redução ao slice, que permite relacionar uma folheação Finsleriana singular com uma folheação Finsleriana singular em um espaço de Minkowski. O terceiro resultado garante a equifocalidade para as fibras regulares de uma submersão singular analítica que na parte regular é uma submersão Finsleriana. No transcurso do trabalho verificou-se propriedades relevantes das folheações Finslerianas singulares e a existência de vizinhanças tubulares Finslerianas, uma propriedade básica que não estava escrita na literatura. / In this thesis we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and Finsler foliations. In the first theorem we show that if F is a singular Finsler foliation on a Randers manifold (M,Z) with Zermelo data (h,W), then F is a singular Riemannian foliation on the Riemannian manifold (M, h). In order to prove this theorem we present in the second theorem a slice reduction that relates local singular Finsler foliations on Finsler manifolds with singular Finsler foliations on Minkowski spaces. Finally in the third result we prove the equifocality of the fibers of an analytic singular submersion that is a Finsler submersion in the regular part. Along the work we stress relevant properties on singular Finsler foliations and we also remark the existence of tubular neigborhoods in Finsler geometry, a basic result that we did not find properly written in the literature.

Indiana University-Purdue University Indianapolis (IUPUI) / In the present work, we consider a class of Finsler metrics using the warped product notion introduced by B. Chen, Z. Shen and L. Zhao (2018), with another “warping”, one that is consistent with the form of metrics modeling static spacetimes and simplified by spherical symmetry over spatial coordinates, which emerged from the Schwarzschild metric in isotropic coordinates. We will give the PDE characterization for the proposed metrics to be Ricci-flat and construct explicit examples. Whenever possible, we describe both positive-definite solutions and solutions with Lorentz signature. For the latter, the 4-dimensional metrics may also be studied as Finsler spacetimes.

Orientador: Paulo Regis Caron Ruffino / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T02:49:45Z (GMT). No. of bitstreams: 1 SilvaJunior_RinaldoVieirada_M.pdf: 1586291 bytes, checksum: 8d01bdf434ecba2fb62a57725c46dd4a (MD5) Previous issue date: 2005 / Resumo: Nesta dissertação fizemos um estudo da teoria de difusão em variedades Finsler, onde abor-damos o transporte paralelo estocástico, desenvolvimento estocástico de Cartan e Movimento Browniano. O objetivo principal é obter uma descrição mais geométrica dos objetos citados acima ainda que por enquanto em coordenadas locais e assim termos um paralelo entre o cálculo estocástico em variedades Riemannianas e variedades Finsler / Abstract: In this work we study diffusion theory in Finsler manifolds. It includes the stochastic par-allel transport, stochastic Cartan development and Brownian motion. The main objective is to provide a geometric description of the objects mentioned and 50 to draw a compari-50n between stochastic calculus in Riemannian manifolds and stochastic calculus in Finsler manifolds / Mestrado / Matematica / Mestre em Matemática

SILVA, José Euclides Gomes da. Sobre modificações na estrutura geométrica em cenários de branas. 2013. 130 f. Tese (Doutorado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2013. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2014-05-16T21:35:18Z No. of bitstreams: 1 2013_tese_jegsilva.pdf: 836100 bytes, checksum: c4765585f192ce0d02aa423186d47ae3 (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2014-05-16T21:38:09Z (GMT) No. of bitstreams: 1 2013_tese_jegsilva.pdf: 836100 bytes, checksum: c4765585f192ce0d02aa423186d47ae3 (MD5) / Made available in DSpace on 2014-05-16T21:38:10Z (GMT). No. of bitstreams: 1 2013_tese_jegsilva.pdf: 836100 bytes, checksum: c4765585f192ce0d02aa423186d47ae3 (MD5) Previous issue date: 2013 / This thesis presents our proposals for new braneworlds models. Some of the main open issues in high energy physics have interesting solutions assuming the space-time has more than four dimensions. For instance, the hierarchy problem between the eletroweak and the Planck scales, and the origin of the cosmological constant, find some solutions in the brane scenarios. Since these models are rather sensible on the geometrical structure of the multidimensional space time where the brane is embedded, our main goal is to analyze how the geometrical and physical properties of the braneworld and of fields living on it evolve under a geometrical flow in the transverse manifold. The first step was propose an smoothed string-like braneworld with a transverse resolved conifold. The resolution parameter changes the width of the well and the high of the barrier of the Kaluza-Klein modes. Further, the source of this warped solution has different phases depending on the resolution parameter. The massless modes for the scalar, gauge and spinor fields are only well-behaved on the brane for non singular configurations. Another smooth geometrical flow studied was the so-called Ricci flow. This flux posses diffeomorphic invariant solutions called Ricci solitons which are extremals of the energy and entropy functionals. An important two-dimensional Ricci soliton with axial symmetry is the cigar soliton. A warped product between a 3-brane and the cigar soliton turns to be an interior and exterior string-like solution satisfying the dominant energy condition and that supports a massless gravitational mode trapped to the brane. The last geometric modification proposed was the locally Lorentz symmetry violation through a Finsler geometry approach. This anisotropic differential geometry has been intensely studied in last years. We have chosen the so-called bipartite space where the length of the events is measure using the metric and another symmetric tensor called bipartite tensor. We have shown the bipartite space deforms the causal surface to an elliptic cone and provides an anisotropy into the inertia of a particle. By means of an extended Einstein-Hilbert action we have shown an analogy between the bipartite space and the bumblebee and bipartite models which are effective Lorentz violating models in curved space times. / A presente tese apresenta nossas propostas de estensões dos modelos de mundo Branas. Alguns dos principais problemas em aberto em física de partículas, como o problema da hieraquia entre as escalas de Planck e eletrofraca, e da cosmologia como a origem da matéria escura e o valor da constante cosmológica, encontram soluções nos cenários de branas. Uma vez que tais modelos são extremamente sensíveis à estrutura geométrica do espaço-tempo ambiente multidimensional no qual a brana está imersa, noss ideia básica é analisar como as propriedades da brana e dos campos que vivem no seu entorno mudam quando alteramos a estrutura geométrica do espaço ambiente. Nosso primeiro passo foi uma estensão do cenário de de brana tipo-corda em seis dimensões onde a variedade transversa é uma seção do cone resolvido. O parâmetro de resolução do cone, que controla a singularidade na origem, também altera a largura dos modos sem massa de um campo escalar e do potencial confinante dos modos Kaluza-Klein. Também analisamos as condições de energia da fonte que passa por diferentes fases durante o fluxo de resolução. Estudamos ainda como este fluxo modifica as propriedades dos campos vetoriais e espinoriais neste cenário. Em seguida, propusemos um novo fluxo geométrico para a variedade transversa. O chamado fluxo de Ricci possui soluções invariantes por difeomorfismos chamadas sólitons de Ricci. Tais soluções têm a propriedade de extremizar grandezas durante esse fluxo, como os funcionais energia e entropia. Uma solução particularmente importante e estacionária deste fluxo é o chamado sóliton charuto de Hamilton que possui simetria axial. Definimos uma variedade produto não-fatorizável entre uma 3-brana e um sóliton de Hamilton resultando em uma solução tipo-corda regular que satisfaz a condição de energia dominante e tem um modo gravitacional não massivo localizado. Outra modificação geométrica proposta foi a Violação da simetria de Lorentz através da introdução de uma estrutura métrica localmente anisotrópica, a chamada geometria de Finsler. Tal abordagem tem sido objeto recente de vários estudos. Escolhemos uma estrutura finsleriana recentemente proposta, chamada bipartite, onde o comprimento dos eventos é calculado não somente com a métrica Lorentziana mas também com uma outra forma bilinear simétrica. O cone de luz desta geometria é deformado para um cone elíptico cujas inclinações das geratrizes dependem dos autovalores do tensor bipartite. Outra propriedade deste espaço-tempo é a de modificar a relação entre o 4-momentum e a 4-velocidade gerando um tensor de inércia. Através de uma ação de Einstein-Hilbert finsleriana em um limite de baixa dependência direcional, encontramos uma analogia entre essa geometria e os modelos bumblebee e aether, que descrevem efetivamente a quebra da simetria de Lorentz em espaços curvos.

Orientadores: Carlos Eduardo Durán Fernandez, Márcio Antônio de Faria Rosa / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-16T14:48:33Z (GMT). No. of bitstreams: 1 ManoOtero_Diego_M.pdf: 1221591 bytes, checksum: 33ae6e3b671523a9602f3398e14d4fb7 (MD5) Previous issue date: 2010 / Resumo: A motivação inicial deste trabalho foi tentar relacionar os conceitos de geometria de Finsler com situações físicas que temos uma certa dependência de direções no nosso espaço. Apresentamos o conceito do cálculo variacional em variedades e sua relação com as geodésicas. Estudamos também o operador laplaciano ?? para espaços de Minkowski, que generaliza o caso Euclideano, e mais especificamente o problema...Observação: O resumo, na íntegra poderá ser visualizado no texto completo da tese digital / Abstract: The initial motivation of this study was to try to relate the concepts of Finsler geometry with physical situations where we have a certain dependence on the directions of our space. We introduce the concept of variational calculus on manifolds and their relationship with the geodesics. We also studied the Laplacian operator ?? in Minkowski space, which generalizes the Euclidean case, and more specifically the problem ...Note: The complete abstract is available with the full electronic digital thesis or dissertations. / Mestrado / Geometria / Mestre em Matemática

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-14T14:51:34Z No. of bitstreams: 2 Dissertação - Carlos Antônio Freitas da Silva - 2015.pdf: 659907 bytes, checksum: c43cf65b3e27833fcd6b4ab11eb79239 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-14T14:53:28Z (GMT) No. of bitstreams: 2 Dissertação - Carlos Antônio Freitas da Silva - 2015.pdf: 659907 bytes, checksum: c43cf65b3e27833fcd6b4ab11eb79239 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-05-14T14:53:28Z (GMT). No. of bitstreams: 2 Dissertação - Carlos Antônio Freitas da Silva - 2015.pdf: 659907 bytes, checksum: c43cf65b3e27833fcd6b4ab11eb79239 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-02-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation we will study Finsler Geometry. In particular, we will study Randers Geometry that which can be viewed as Riemannian Geometry with a pertubation. Furthermore Randers metrics are also obtained as solution to Zermelo’s Navigation Problem. We will also use classification theorems of Randers metrics of constant flag curvature and Einstein Randers metrics in terms of Zermelo’s Navigation Problem. Using Randers metrics we are going to construct a 3-parameter family of Einstein-Finsler metrics with non-constant flag curvature and to get such family we use a Killing vector field and a Riemannian metric which is the Hawking Taub-NUT metric. / Neste trabalho estudaremos a Geometria de Finsler. Em particular, estudaremos a Geometria de Randers que pode ser visto como a mais simples perturbação da Geometria Riemanniana. Além disso, veremos também que métricas de Randers podem ser obtidas como soluções do Problema Navegacional de Zermelo. Utilizaremos também resultados que caracterizam métricas de Randers com curvatura flag constante e métricas de Randers do tipo Einstein em termos do Problema Navegacional de Zermelo. Usando métricas de Randers vamos construir uma família a 3 parâmetros de métricas de Einstein-Finsler com curvatura flag não constante e para obter tal família utilizaremos um campo de Killing e uma métrica Riemanniana que é a métrica de Hawking Taub-NUT.

En 1949, C. Loewner a demontré dans un travail non publié l'inégalité systolique optimale du tore T reliant l'aire au carré de la systole. Par la systole on désigne la longueur du plus court lacet non contractile de T. De plus, l' égalité est atteinte si et seulement si le tore est plat hexagonal. Ce résultat a donné naissance à la géométrie systolique. Dans cette thèse, nous étudions des inégalités de type systolique portant sur les longueurs minimales de différentes courbes et pas seulement la systole.Dans un premier temps, nous démontrons trois inégalités géométriques optimales conformes sur la bouteille de Klein reliant l'aire au produit des longueurs des plus courts lacets noncontractiles dans des classes d'homotopie libres différentes. Pour chaque classe conforme, nous décrivons la métrique extrémale réalisant le cas d'égalité.Nous établissons ensuite des inégalités géométriques optimales sur le ruban deMobius muni d'une métrique de Finsler. Ces inégalités géométriques relient la systole et la hauteur du ruban de Mobius à son volume de Holmes-Thompson. Nous en déduisons une inégalité systolique optimale sur la bouteille de Klein munie d'une métrique de Finsler avec des symétries. Nous décrivons également une famille de métriques extrémales dans les deux cas.Dans le troisième travail, nous démontrons une inégalité systolique critique sur la surface de genre deux. Plus précisément, il est connu que la surface de genre deux admet une métrique Riemannienne plate à singularités coniques qui est extrémale parmi les métriques à courbure nonpositive pour l' inégalité systolique. Nous montrons que cette métrique est en fait critique pour des variations lentes de métriques, cette fois-ci sans hypothèse de courbure, pour un autre problème systolique portant sur les longueurs des plus courts lacets non contractiles dans certaines classes d'homotopie libres données. Ces classes d'homotopie correspondent aux lacets systoliques et deux-systoliques de la surface extrémale / In 1949, C. Loewner proved in an unpublished work that the two-torus T satisfies an optimal systolic inequality relating the area of the torus to the square of its systole. By a systole here we mean the smallest length of a noncontractible loop in T. Furthermore, the equality is attained if and only if the torus is flat hexagonal. This result led to whatwas called later systolic geometry. In this thesis, we study several systolic-like inequalities. These inequalities involve the minimal length of various curves and not merely the systole.First we obtain three optimal conformal geometric inequalities on Riemannian Klein bottles relating the area to the product of the lengths of the shortest noncontractible loops in different free homotopy classes. We describe the extremal metrics in each conformal class.Then we prove optimal systolic inequalities on Finsler Mobius bands relating the systoleand the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimalsystolic inequality for Finsler Klein bottles with symmetries. We describe extremal metric families in both cases.Finally, we prove a critical systolic inequality on genus two surface. More precisely, it is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved Riemannian metrics. We show that this piecewise flat metric is also critical for slow metric variations, this time without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes. The free homotopy classes considered correspond to those of the systolic loops and the second-systolic loops of the extremal surface

Which tensor fields G on a smooth manifold M can serve as a spacetime structure? In the first part of this thesis, it is found that only a severely restricted class of tensor fields can provide classical spacetime geometries, namely those that can carry predictive, interpretable and quantizable matter dynamics. The obvious dependence of this characterization of admissible tensorial spacetime geometries on specific matter is not a weakness, but rather presents an insight: it was Maxwell theory that justified Einstein to promote Lorentzian manifolds to the status of a spacetime geometry. Any matter that does not mimick the structure of Maxwell theory, will force us to choose another geometry on which the matter dynamics of interest are predictive, interpretable and quantizable. These three physical conditions on matter impose three corresponding algebraic conditions on the totally symmetric contravariant coefficient tensor field P that determines the principal symbol of the matter field equations in terms of the geometric tensor G: the tensor field P must be hyperbolic, time-orientable and energy-distinguishing. Remarkably, these physically necessary conditions on the geometry are mathematically already sufficient to realize all kinematical constructions familiar from Lorentzian geometry, for precisely the same structural reasons. This we were able to show employing a subtle interplay of convex analysis, the theory of partial differential equations and real algebraic geometry. In the second part of this thesis, we then explore general properties of any hyperbolic, time-orientable and energy-distinguishing tensorial geometry. Physically most important are the construction of freely falling non-rotating laboratories, the appearance of admissible modified dispersion relations to particular observers, and the identification of a mechanism that explains why massive particles that are faster than some massless particles can radiate off energy until they are slower than all massless particles in any hyperbolic, time-orientable and energy-distinguishing geometry. In the third part of the thesis, we explore how tensorial spacetime geometries fare when one wants to quantize particles and fields on them. This study is motivated, in part, in order to provide the tools to calculate the rate at which superluminal particles radiate off energy to become infraluminal, as explained above. Remarkably, it is again the three geometric conditions of hyperbolicity, time-orientability and energy-distinguishability that allow the quantization of general linear electrodynamics on an area metric spacetime and the quantization of massive point particles obeying any admissible dispersion relation. We explore the issue of field equations of all possible derivative order in rather systematic fashion, and prove a practically most useful theorem that determines Dirac algebras allowing the reduction of derivative orders. The final part of the thesis presents the sketch of a truly remarkable result that was obtained building on the work of the present thesis. Particularly based on the subtle duality maps between momenta and velocities in general tensorial spacetimes, it could be shown that gravitational dynamics for hyperbolic, time-orientable and energy distinguishable geometries need not be postulated, but the formidable physical problem of their construction can be reduced to a mere mathematical task: the solution of a system of homogeneous linear partial differential equations. This far-reaching physical result on modified gravity theories is a direct, but difficult to derive, outcome of the findings in the present thesis. Throughout the thesis, the abstract theory is illustrated through instructive examples. / Welche Tensorfelder G auf einer glatten Mannigfaltigkeit M können eine Raumzeit-Geometrie beschreiben? Im ersten Teil dieser Dissertation wird es gezeigt, dass nur stark eingeschränkte Klassen von Tensorfeldern eine Raumzeit-Geometrie darstellen können, nämlich Tensorfelder, die eine prädiktive, interpretierbare und quantisierbare Dynamik für Materiefelder ermöglichen. Die offensichtliche Abhängigkeit dieser Charakterisierung erlaubter tensorieller Raumzeiten von einer spezifischen Materiefelder-Dynamik ist keine Schwäche der Theorie, sondern ist letztlich genau das Prinzip, das die üblicherweise betrachteten Lorentzschen Mannigfaltigkeiten auszeichnet: diese stellen die metrische Geometrie dar, welche die Maxwellsche Elektrodynamik prädiktiv, interpretierbar und quantisierbar macht. Materiefeld-Dynamiken, welche die kausale Struktur von Maxwell-Elektrodynamik nicht respektieren, zwingen uns, eine andere Geometrie auszuwählen, auf der die Materiefelder-Dynamik aber immer noch prädiktiv, interpretierbar und quantisierbar sein muss. Diesen drei Voraussetzungen an die Materie entsprechen drei algebraische Voraussetzungen an das total symmetrische kontravariante Tensorfeld P, welches das Prinzipalpolynom der Materiefeldgleichungen (ausgedrückt durch das grundlegende Tensorfeld G) bestimmt: das Tensorfeld P muss hyperbolisch, zeitorientierbar und energie-differenzierend sein. Diese drei notwendigen Bedingungen an die Geometrie genügen, um alle aus der Lorentzschen Geometrie bekannten kinematischen Konstruktionen zu realisieren. Dies zeigen wir im ersten Teil der vorliegenden Arbeit unter Verwendung eines teilweise recht subtilen Wechselspiels zwischen konvexer Analysis, der Theorie partieller Differentialgleichungen und reeller algebraischer Geometrie. Im zweiten Teil dieser Dissertation erforschen wir allgemeine Eigenschaften aller solcher hyperbolischen, zeit-orientierbaren und energie-differenzierenden Geometrien. Physikalisch wichtig sind der Aufbau von frei fallenden und nicht rotierenden Laboratorien, das Auftreten modifizierter Energie-Impuls-Beziehungen und die Identifizierung eines Mechanismus, der erklärt, warum massive Teilchen, die sich schneller als einige masselosse Teilchen bewegen, Energie abstrahlen können, aber nur bis sie sich langsamer als alle masselossen Teilchen bewegen. Im dritten Teil der Dissertation ergründen wir die Quantisierung von Teilchen und Feldern auf tensoriellen Raumzeit-Geometrien, die die obigen physikalischen Bedingungen erfüllen. Eine wichtige Motivation dieser Untersuchung ist es, Techniken zur Berechnung der Zerfallsrate von Teilchen zu berechnen, die sich schneller als langsame masselose Teilchen bewegen. Wir finden, dass es wiederum die drei zuvor im klassischen Kontext identifizierten Voraussetzungen (der Hyperbolizität, Zeit-Orientierbarkeit und Energie-Differenzierbarkeit) sind, welche die Quantisierung allgemeiner linearer Elektrodynamik auf einer flächenmetrischen Raumzeit und die Quantizierung massiver Teilchen, die eine physikalische Energie-Impuls-Beziehung respektieren, erlauben. Wir erkunden auch systematisch, wie man Feldgleichungen aller Ableitungsordnungen generieren kann und beweisen einen Satz, der verallgemeinerte Dirac-Algebren bestimmt und die damit Reduzierung des Ableitungsgrades einer physikalischen Materiefeldgleichung ermöglicht. Der letzte Teil der vorliegenden Schrift skizziert ein bemerkenswertes Ergebnis, das mit den in dieser Dissertation dargestellten Techniken erzielt wurde. Insbesondere aufgrund der hier identifizierten dualen Abbildungen zwischen Teilchenimpulsen und -geschwindigkeiten auf allgemeinen tensoriellen Raumzeiten war es möglich zu zeigen, dass man die Gravitationsdynamik für hyperbolische, zeit-orientierbare und energie-differenzierende Geometrien nicht postulieren muss, sondern dass sich das Problem ihrer Konstruktion auf eine rein mathematische Aufgabe reduziert: die Lösung eines homogenen linearen Differentialgleichungssystems. Dieses weitreichende Ergebnis über modifizierte Gravitationstheorien ist eine direkte (aber schwer herzuleitende) Folgerung der Forschungsergebnisse dieser Dissertation. Die abstrakte Theorie dieser Doktorarbeit wird durch mehrere instruktive Beispiele illustriert.