Dissertations / Theses on the topic 'Mathematical relativity'
To see the other types of publications on this topic, follow the link: Mathematical relativity.
Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles
Consult the top 50 dissertations / theses for your research on the topic 'Mathematical relativity.'
Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.
You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.
Browse dissertations / theses on a wide variety of disciplines and organise your bibliography correctly.
1
Norgren, Ofelia. "Mathematical Special Relativity." Thesis, Uppsala universitet, Matematiska institutionen, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-435242.
Full text
2
Abdelfattah, Derhham. "General Relativity and penrose process." Master's thesis, University of Cape Town, 2016. http://hdl.handle.net/11427/28961.
Full text
3
Sakovich, Anna. "A study of asymptotically hyperbolic manifolds in mathematical relativity." Doctoral thesis, KTH, Matematik (Avd.), 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-102874.
Full textAbstract:
This thesis consists of ve papers where certain problems arising in mathematical relativity are studied in the context of asymptotically hyperbolic manifolds. In Paper A we deal with constant mean curvature solutions of the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds. Conditions on the scalar field and its potential are given which lead to existence and non-existence results. In Paper B we construct non-constant mean curvature solutions of the constraint equations on asymptotically hyperbolic manifolds. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then letting the exponent tend to its true value. We prove that if a certain limit equation admits no non-trivial solution, then the set of solutions of the constraint equations is non empty and compact. W ealso give conditions ensuring that the limit equation admits no nontrivial solution. This is a joint work with Romain Gicquaud. In this Paper C we obtain Penrose type inequalities for asymptotically hyperbolic graphs. In certain cases we prove that equality is attained only by the anti-de Sitter Schwarzschild metric. This is a joint work with Mattias Dahl and Romain Gicquaud. In Paper D we construct a solution to the Jang equation on an asymptotically hyperbolic manifold with a certain asymptotic behaviour at infinity. In Paper E we study asymptotically hyperbolic manifolds which are also conformally hyperbolic outside a ball of fixed radius, and for which the positive mass theorem holds. For such manifolds we show that when the mass tends to zero the metric converges uniformly tot he hyperbolic metric. This is a joint work with Mattias Dahl and Romain Gicquaud.QC 20120928
4
Sbierski, Jan. "On the initial value problem in general relativity and wave propagation in black-hole spacetimes." Thesis, University of Cambridge, 2014. https://www.repository.cam.ac.uk/handle/1810/248837.
Full textAbstract:
The first part of this thesis is concerned with the question of global uniqueness of solutions to the initial value problem in general relativity. In 1969, Choquet-Bruhat and Geroch proved, that in the class of globally hyperbolic Cauchy developments, there is a unique maximal Cauchy development. The original proof, however, has the peculiar feature that it appeals to Zorn’s lemma in order to guarantee the existence of this maximal development; in particular, the proof is not constructive. In the first part of this thesis we give a proof of the above mentioned theorem that avoids the use of Zorn’s lemma. The second part of this thesis investigates the behaviour of so-called Gaussian beam solutions of the wave equation - highly oscillatory and localised solutions which travel, for some time, along null geodesics. The main result of this part of the thesis is a characterisation of the temporal behaviour of the energy of such Gaussian beams in terms of the underlying null geodesic. We conclude by giving applications of this result to black hole spacetimes. Recalling that the wave equation can be considered a “poor man’s” linearisation of the Einstein equations, these applications are of interest for a better understanding of the black hole stability conjecture, which states that the exterior of our explicit black hole solutions is stable to small perturbations, while the interior is expected to be unstable. The last part of the thesis is concerned with the wave equation in the interior of a black hole. In particular, we show that under certain conditions on the black hole parameters, waves that are compactly supported on the event horizon, have finite energy near the Cauchy horizon. This result is again motivated by the investigation of the conjectured instability of the interior of our explicit black hole solutions.
5
Luo, Xianghui 1983. "Symmetries of Cauchy Horizons and Global Stability of Cosmological Models." Thesis, University of Oregon, 2011. http://hdl.handle.net/1794/11543.
Full textAbstract:
ix, 111 p.This dissertation contains the results obtained from a study of two subjects in mathematical general relativity. The first part of this dissertation is about the existence of Killing symmetries in spacetimes containing a compact Cauchy horizon. We prove the existence of a nontrivial Killing symmetry in a large class of analytic cosmological spacetimes with a compact Cauchy horizon for any spacetime dimension. In doing so, we also remove the restrictive analyticity condition and obtain a generalization to the smooth case. The second part of the dissertation presents our results on the global stability problem for a class of cosmological models. We investigate the power law inflating cosmological models in the presence of electromagnetic fields. A stability result for such cosmological spacetimes is proved. This dissertation includes unpublished co-authored material.
Committee in charge: James Brau, Chair; James Isenberg, Advisor; Paul Csonka, Member; John Toner, Member; Peng Lu, Outside Member
6
Reid, James Andrew. "Conformal holonomy and theoretical gravitational physics." Thesis, University of Aberdeen, 2014. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=215305.
Full textAbstract:
Conformal holonomy theory is the holonomy theory of the tractor connection on a conformal manifold. In this thesis, we present the first application of conformal holonomy theory to theoretical physics and determine the conformal holonomy groups/algebras of physically relevant spaces. After recalling some necessary background on conformal structures, tractor bundles and conformal holonomy theory in chapter 1, we begin in chapter 2 by discussing the role of conformal holonomy in the gauge-theoretic MacDowell-Mansouri formulation of general relativity. We show that the gauge algebra of this formulation is uniquely determined by the conformal structure of spacetime itself, in both Lorentzian and Riemannian metric signatures, through the conformal holonomy algebra. We then show that one may construct a MacDowell-Mansouri action functional for scale-invariant gravity, and we discuss a geometric interpretation for the scalar field therein. In chapter 3 we study a class of spacetimes relevant to Maldacena's AdS5=CFT4 correspondence in quantum gravity. It is well known that a Lie group coincidence lies at the heart of this correspondence: the proper isometry group of the bulk precisely matches the conformal group of the boundary. It has previously been proposed that the AdS5=CFT4 correspondence be extended to so-called Poincar e-Einstein spacetimes, which need not be as symmetric as anti-de Sitter space. We show that the conformal holonomy groups of the boundary and bulk furnish such a Lie group coincidence for 5-dimensional Poincar e-Einstein spacetimes in general. We completely characterise this boundary-bulk conformal holonomy matching for the Riemannian theory and present partial results for the Lorentzian theory. In chapter 4 we use the tools developed in the preceding chapters to further the classiification of the conformal holonomy groups of conformally Einstein spaces. Specifically, we determine the conformal holonomy groups of generic neutral signature conformally Einstein 4-manifolds subject to a condition on the conformal holonomy representation. Lastly, in chapter 5, we investigate the conformal holonomy reduction of the Fefferman conformal structures of residual twistor CR manifolds. A sufficient condition for reducible conformal holonomy is that the (Fefferman conformal structure of a) residual twistor CR manifold admit a parallel tractor. We show that this occurs if and only if the residual twistor CR manifold admits a Sasakian structure.
7
Muir, Stuart. "A relativisitic, 3-dimensional smoothed particle hydrodynamics (SPH) algorithm and its applications." Monash University, School of Mathematical Sciences, 2003. http://arrow.monash.edu.au/hdl/1959.1/9513.
Full text
8
Fama, Christopher J., and -. "Non-smooth differential geometry of pseudo-Riemannian manifolds: Boundary and geodesic structure of gravitational wave space-times in mathematical relativity." The Australian National University. School of Mathematical Sciences, 1998. http://thesis.anu.edu.au./public/adt-ANU20010907.161849.
Full textAbstract:
[No abstract supplied with this thesis - The first page (of three) of the Introduction follows]
¶
This thesis is largely concerned with the changing representations of 'boundary' or 'ideal' points of a
pseudo-Riemannian manifold -- and our primary interest is in the space-times of general relativity. In particular, we are interested in the following question: What assumptions about the 'nature' of 'portions' of a certain 'ideal boundary' construction (essentially the 'abstract boundary' of Scott and Szekeres (1994)) allow us to define precisely the topological type of these 'portions', i.e., to show that different representations of this ideal boundary, corresponding to different embeddings of the manifold into others, have corresponding 'portions' that are homeomorphic?
¶
Certain topological properties of these 'portions' are preserved, even allowing for quite unpleasant properties of the metric (Fama and Scott 1995). These results are given in Appendix D, since they are not used elsewhere and, as well as representing the main portion of work undertaken under the supervision of Scott, which deserves recognition, may serve as an interesting example of the relative ease with which certain simple results about the abstract boundary can be obtained.
¶
An answer to a more precisely formulated version of this question appears very diffcult in general. However, we can give a rather complete answer in certain cases, where we dictate certain 'generalised regularity' requirements for our embeddings, but make no demands on the precise functional form of our metrics apart from these. For example, we get a complete answer to our question for abstract boundary sets which do not 'wiggle about' too much -- i.e., they satisfy a certain Lipschitz condition -- and through which the metric can be extended in a manner which is not required to be differentiable (C[superscript1]), but is continuous and non--degenerate. We allow similar freedoms on the interior of the manifold, thereby bringing gravitational wave space-times within our sphere of discussion. In fact, in the course of developing these results in progressively greater generality, we get, almost 'free', certain abilities to begin looking at geodesic structure on quite general pseudo-Riemannian manifolds.
¶
It is possible to delineate most of this work cleanly into two major parts. Firstly, there are results which use classical geometric constructs and can be given for the original abstract boundary construction, which requires differentiability of both manifolds and metrics, and which we summarise below. The second -- and significantly longer -- part involves extensions of those constructs and results to more general metrics.
9
Muench, Uwe. "Studies in the physical foundations of gravitational theories /." free to MU campus, to others for purchase, 2002. http://wwwlib.umi.com/cr/mo/fullcit?p3060127.
Full text
10
Schlue, Volker. "Linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes." Thesis, University of Cambridge, 2012. https://www.repository.cam.ac.uk/handle/1810/243640.
Full textAbstract:
I study linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes. In the first part of this thesis two decay results are proven for general finite energy solutions to the linear wave equation on higher dimensional Schwarzschild black holes. I establish uniform energy decay and improved interior first order energy decay in all dimensions with rates in accordance with the 3 + 1-dimensional case. The method of proof departs from earlier work on this problem. I apply and extend the new physical space approach to decay of Dafermos and Rodnianski. An integrated local energy decay estimate for the wave equation on higher dimensional Schwarzschild black holes is proven. In the second part of this thesis the global study of solutions to the linear wave equation on expanding de Sitter and Schwarzschild de Sitter spacetimes is initiated. I show that finite energy solutions to the initial value problem are globally bounded and have a limit on the future boundary that can be viewed as a function on the standard cylinder. Both problems are related to the Cauchy problem in General Relativity.
11
Dold, Dominic Nicolas. "Instabilities in asymptotically AdS spacetimes." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/273930.
Full textAbstract:
In recent years, more and more efforts have been expended on the study of $n$-dimensional asymptotically anti-de Sitter spacetimes $(\mathcal{M},g)$ as solutions to the Einstein vacuum equations \begin{align*} \mathrm{Ric}(g)=\frac{2}{n-2}\Lambda\, g \end{align*} with negative cosmological constant $\Lambda$. This has been motivated mainly by the conjectured instability of these solutions. The author of this thesis joins these efforts with two contributions, which are themselves independent of each other. In the first part, we are concerned with a superradiant instability for $n=4$. For any cosmological constant $\Lambda=-3/\ell^2$ and any $\alpha < 9/4$, we find a Kerr-AdS spacetime $(\mathcal{M},g_{\mathrm{KAdS}})$, in which the Klein-Gordon equation \begin{align*} \Box_g\psi+\frac{\alpha}{\ell^2}\psi=0 \end{align*} has an exponentially growing mode solution satisfying a Dirichlet boundary condition at infinity. The spacetime violates the Hawking-Reall bound $r_+^2 > |a|\ell$. We obtain an analogous result for Neumann boundary conditions if $5/4 < \alpha < 9/4$. Moreover, in the Dirichlet case, one can prove that, for any Kerr-AdS spacetime violating the Hawking-Reall bound, there exists an open family of masses $\alpha$ such that the corresponding Klein-Gordon equation permits exponentially growing mode solutions. Our result provides the first rigorous construction of a superradiant instability for a negative cosmological constant. In the second part, we study perturbations of five-dimensional Eguchi-Hanson-AdS spacetimes exhibiting biaxial Bianchi IX symmetry. Within this symmetry class, the Einstein vacuum equations are equivalent to a system of non-linear partial differential equations for the radius $r$ of the spheres, the Hawking mass $m$ and $B$, a quantity measuring the squashing of the spheres, which satisfies a non-linear wave equation. First we prove that the system is well-posed as an initial-boundary value problem around infinity $\mathcal{I}$ with $B$ satisfying a Dirichlet boundary condition. Second, we show that initial data in the biaxial Bianchi IX symmetry class around Eguchi-Hanson-AdS spacetimes cannot form horizons in the dynamical evolution.
12
Garcia, Raphael de Oliveira 1982. "Simulações numericas do colapso gravitacional de um campo escalar sem massa." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306524.
Full textAbstract:
Orientador: Samuel Rocha de OliveiraDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
Made available in DSpace on 2018-08-08T09:44:11Z (GMT). No. of bitstreams: 1 Garcia_RaphaeldeOliveira_M.pdf: 2899892 bytes, checksum: 6befb1d82e7b4effd083097342934f7d (MD5) Previous issue date: 2007
Resumo: Esta dissertação trata de um problema de colapso gravitacional do campo escalar sem massa com simetria esférica, de acordo com as equações de Einstein com acoplamento mínimo. Desenvolvemos algoritmos simples mas com ajuste automático de refinamento para obter soluções numéricas. O refinamento é necessário para descrever os fenômenos do colapso. Utilizamos rotinas comuns para resolver equações diferenciais ordinárias advindas do método das linhas aplicado às equações diferenciais parciais do modelo matemático, a saber, rotinas de Runge-Kutta e de splines cúbicas. Mostramos que é possível obter soluções tão próximas quanto se queria à solução de Buracos Negros e da solução crítica com os algoritmos desenvolvidos e simples computadores de mesa.
Abstract: This dissertation is about a problem of gravitational collapse of the spherically symmetric massless scalar field in accord to the Einstein equations with minimal coupling. We developed simple but with adaptative mesh refinement algorithm to obtain numerical solutions. The mesh refinement is required to describe the collapse phenomena. We used common routines to solve ordinary differential equations that come from the method of lines applied to the partial differential equations of the mathematical model, namely, Runge-Kutta and cubic splines routines. We showed that it is possible to get solutions as close as we want to the Black Holes solution and to the critical solution using our developed algorithms and simple desktop computers
Mestrado
Fisica-Matematica
Mestre em Matemática Aplicada
13
Otani, Yul. "Efeitos cosmológicos induzidos por campos quantizados." Universidade de São Paulo, 2010. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-28032011-095122/.
Full textAbstract:
A presente dissertação revisa um modelo, de autoria de C. Dappiaggi, K. Fredenhagen e N. Pinamonti, de um campo escalar real quântico não-interagente acoplado com a métrica de um espaço-tempo FLRW (Friedmann-Lemaítre-Robertson-Walker). Apresentamos a metodologia de quantização de campos de Klein-Gordon reais em espaçostempos globalmente hiperbólicos e discorremos sobre o procedimento de regularização do tensor de energia-momento via point-splitting. Consideramos os campos em espaços FLRW e estados adiabáticos com flutuação média de campo dado por h2i = m2 +R, com ; constantes provenientes do procedimento de regularização. A retroação do campo quântico gera a equação diferencial para o parâmetro de Hubble H(t) dada por _H (H2H2 c ) = (H2H2+ )(H2H2 ) com Hc uma constante e H pontos críticos estáveis da equação. Esse simples modelo mostra que efeitos quânticos podem, por si só, fornecer fases de de Sitter estáveis sem adição de uma constante cosmológica a priori. Mesmo que de caráter apenas qualitativo, tal resultado indica que análises cautelosas de processos de quantização são importantes para análise de efeitos cosmológicos de teorias quânticas de campos em espaços curvos.The present dissertation reviews the coupling of a scalar non-interacting quantum field with the metric of a FLRW(Friedmann-Lemaítre-Robertson-Walker) spacetime, proposed in a work by C. Dappiaggi, K. Fredenhagen and N. Pinamonti. We present methods for the quantization of a real Klein-Gordon field in globally hyperbolic spacetimes and discuss procedures for the point-splitting regularization of the stress-energy tensor. We consider those fields in FLRWspacetimes and point out adiabatic states with mean field fluctuation given by h2i = m2+R, with ; being constants that emerge from the regularization procedure. The backreaction of the quantum field provides a diferential equation for the Hubble parameter given by _H (H2H2 c ) = (H2H2+)(H2H2) with Hc a constant and H stable critical points of the equation. In this way, this simple model demonstrates that quantum efects may, by themselves, exibit stable de Sitter phases even without an introduction of a cosmological constant by hand. Althoug in a qualitative way, such result shows that, when dealing with the backreaction issue, a careful analysis of the quantization procedures is important for the analysis of cosmological efects of models of quantum field theories in curved spacetimes.
14
Howell, Eric John. "The characterization and temporal distribution of cosmological gravitational wave treatments." University of Western Australia. School of Physics, 2009. http://theses.library.uwa.edu.au/adt-WU2010.0095.
Full textAbstract:
[Truncated abstract] As gravitational wave detectors approach sensitivities that will allow observations to become routine, astrophysics lies on the cusp of an exciting new era. Potential sources will include transients such as merging neutron stars and black holes, supernova explosions or the engines that power gamma-ray bursts. This thesis will be devoted to the astrophysical gravitational wave background signal produced by cosmological populations of such transient signals. Particular attention will be devoted to the observation-time dependence imposed on the individual sources that accumulate to produce a gravitational wave background signal. The ultimate aim is to determine what information is encoded in the temporal evolution of such a signal. To lay the foundations for further investigation, the stochastic gravitational wave background signal from neutron star birth throughout the Universe has been calculated. In view of the uncertainties in both the single-source emissions and source rate histories, several models of each are employed. The results show that that the resulting signals are only weakly dependent on the source-rate evolution model and that prominent features in the single-source spectra can be related to the background spectra. In comparison with previous studies, the use of relativistic single-source gravitational wave waveforms rather than Newtonian models and a more slowly evolving source-rate density results in a 1 { 2 order of magnitude reduction in signal. ... A comparison with the more commonly used brightness distribution of events shows that when applying both methods to a data stream containing a background of Gaussian distributed false alarms, the brightness distribution yielded lower standard errors, but was biased by the false alarms. In comparison, a fitting procedure based on the time evolution of events was less prone to errors resulting from false alarms, but as fewer events contributed to the data, had a lower resolution. In further support of the time dependent signature of transient events, an alternative technique is fiapplied to the same source population. In this case, the local rate density is probed by measuring the statistical compatibility of the filtered data against synthetic time dependent data. Although this method is not as compact as the fitting procedure, the rate estimates are compatible. To further investigate how the observation time dependence of transient populations can be used to constrain global parameters, the method is applied to Swift long gamma-ray burst data. By considering a distribution in peak °ux rather than a gravitational wave amplitude, gamma-ray bursts can be considered as a surrogate for resolved gravitational wave transients. For this application a peak °ux{observation time relation is described that takes the form of a power law that is invariant to the luminosity distribution of the sources. Additionally, the method is enhanced by invoking time reversal invariance and the temporal cosmological principle. Results are presented to show that the peak °ux{observation time relation is in good agreement with recent estimates of source parameters. Additionally, to show that the intrinsic time dependence allows the method to be used as a predictive tool, projections are made to determine the upper limits in peak °ux of future gamma-ray burst detections for Swift.
15
Fluder, Martin Felix. "Aspects of the class S superconformal index, and gauge/gravity duality in five/six dimensions." Thesis, University of Oxford, 2015. http://ora.ox.ac.uk/objects/uuid:3362dc52-f87c-4d58-bf82-eeb9a46f8fb3.
Full textAbstract:
In the first part of this thesis, we discuss some aspects of the four-dimensional N = 2 superconformal index of theories of class S. We first consider a generalized index on S1 × S3/Zr, and prove S-duality in a particular fugacity slice. We then go on to study the (round) superconformal index in the presence of surface defects. We develop a systematic prescription to compute surface defects labeled by arbitrary irreducible representations of the gauge group and subject those defects to various tests in several different limits. Each of these limits is interesting in its own right, and we go on to explore them in some depth. In the second part of this thesis, we construct the gravity duals of large N supersymmetric gauge theories defined on squashed five-spheres with SU(3) × U(1) symmetry. The gravity duals are constructed in Euclidean Romans F(4) gauged supergravity in six- dimensions, and uplift to massive type IIA supergravity. We compute the partition function and Wilson loop in the large N limit of the gauge theory and compare them to their corresponding supergravity dual quantities. As expected from AdS/CFT, both sides agree perfectly. Based on these results, we conjecture a general formula for the partition function and Wilson loop on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector. We then go on to construct rigid supersymmetric gauge theories on more general Riemannian five-manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of the six-dimensional bulk supergravity solution. This leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals.
16
Lopes, Pedro Tavares Paes. "Uma estratégia euclidiana para o estudo do efeito Unruh." Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-29022008-125203/.
Full textAbstract:
Neste trabalho nós propomos uma estratégia Euclidiana para entender o efeito Unruh. Com este objetivo, nós inicialmente o estudamos para campos livres escalares sem massa, numa forma que é normalmente apresentada aos físicos e que é mais próxima ao trabalho original de Unruh I321| . Logo em seguida, deduzimos o efeito de um ponto de vista algébrico. Com este objetivo, estudamos as propriedades e as definições de estados KMS para compreender como um estado de equilíbrio é descrito na abordagem algébrica. Apresentamos os axiomas de Wightman para campos escalares assim como os de Osterwalder-Schrader. Usamos, então, o Teorema de Bisognano-Wichmann para estes campos e concluímos, baseados no trabalho de Sewell [27], que um observador uniformemente acelerado vê o estado de vácuo dos observadores inerciais como um estado KMS, e portanto, como um estado de equilíbrio. Novamente, concluímos a existência do efeito Unruh. Finalmente estudamos algumas relações entre probabilidade e análise funcional. Este estudo é fundamental para o entendimento do trabalho de Klein e Landau [15] e de Gérard e Jakel [7]. Estes trabalhos afirmam que existe uma relação biunívoca entre certos estados KMS e certos processos estocásticos (Klein e Landau) e uma relação entre certos processos estocásticos e espaços de trajetórias generalizados (Gérard e Jakel). Usando estes trabalhos e as funções de Schwinger para campos escalares, deduzimos o efeito Unruh de uma nova maneira. Acreditamos que este trabalho mostra um ponto de vista interessante do efeito Unruh e ilustra o uso do formalismo Euclidiano em teorias quânticas dos campos. Mesmo que algumas demonstrações para uma prova completa do efeito, usando técnicas Euclidianas, não são obtidas, devido às dificuldades técnicas encontradas, acreditamos que o material apresentado neste trabalho fornece, no mínimo, uma boa estratégia para a compreensão completa deste fenômeno físico. Além disto, as técnicas que são mostradas podem ser usadas em diversos problemas, como a construção de campos interagentes a uma temperatura finita, que permanecem atuais e promissores.This paper proposes a Euclidean strategy to understand the Unruh effect. On that ground we first study it for free massless scalar fields the way it is usually presented to pliysicists, which is closer to Unruh\'s original work [32]. Then we infer the effect from an algebraic perspective. We study the proprieties and definitions of KklS states in order to understand the description of an equilibrium state in the algebraic approach. We present the Wightman\'s as well as Osterwalder-Schrader\'s axioms for scalar fields. Then we use the Bisognano-Wichmann theorem for these fields and conclude, based on Sewell work 1271, that a uniformly accelerated observer will observe tlie vacuum state of inertial observers as a KMS state and thus as an equilibrium state. Once again we infer the existence of the Cnruh effect. Finally we study some relations between probability and functional analysis. This study is crucial for understanding the work of Klein and Landau 1151 as well as of Gérard and Jakel (71. They state there is a biunivocal relation between certain KMS states and certain stochastic processes (Klein and Landau) and a relation between certain stochastic processes and generalized path spaces (Gérard and Jakel). Lsing these works and Schwinger functions for scalar fields, we deduce tlie Unruh effect in a new way. LVe believe this work shows an interesting aspect of the Unruh effect and represents the use of Euclidean formalism in quantum field theory. Although some demonstrations for a complete proof of the Unruh effect using Euclidean techniques were not obtained due to technical difficulties we faced, we believe the material presented in this paper provides at least a good strategy for the complete understanding of this physical phenomenon. Furthermore the techniques shown, which remain current and promising, can be used in different problems, sudy as the construction of interacting fields at a finite temperature.
17
Cattoën, Céline. "Applied mathematics of space-time & space+time : problems in general relativity and cosmology : a thesis submitted to the Victoria University of Wellington in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics /." ResearchArchive@Victoria e-thesis, 2009. http://hdl.handle.net/10063/972.
Full text
18
Valcu, Caterina. "Autour des équations de contrainte en relativité générale." Thesis, Lyon, 2019. http://www.theses.fr/2019LYSE1180/document.
Full textAbstract:
Le but à long terme de mon travail de recherche est de trouver une alternative viable à la méthode conforme, qui nous permettrait de mieux comprendre la structure géométrique de l'espace des solutions des équations de contrainte. L'avantage du modèle de Maxwell (the drift model) par rapport aux modèles plus classiques est la présence des paramètres supplémentaires. Le prix à payer, par contre, sera que la complexité analytique du système correspondant. Ma thèse a été structuré en deux parties : a. Existence sous la condition de petitesse des données initiales. Nous avons montré que le système de Maxwell est raisonnable dans le sens où nous pouvons le résoudre, malgré sa forte nonliniarité, sous des conditions de petitesse sur ses coefficients, en dimension 3, 4 et 5. Par conséquent, l'ensemble des solutions est non-vide. b. Stabilité Nous montrons la stabilité des solutions du système: ce résultat est obtenu en dimension 3,4 et 5, dans le cas où la métrique est conformément plate, et le drift et petitThe long-term goal of my work is to find a viable alternative to the conformal method, which would allow us to better understand the geometry of the space of solutions of the constraint equations. The advantage of Maxwell's model (the drift model) is the presence of additional parameters. Its downside, however, is that it proves to be much more difficult from an analytic standpoint. My thesis is structued in two parts: a. Existence under suitable smallness conditions. We show that Maxwell's system is sufficiently reasonable: it can be solved even given the presence of focusing non linearities. We prove this under smallness conditions of its coefficients, and in dimensions 3,4 and 5. An immediate consequence is that the set of solutions is non-empty. b. Stability. We verify that the solutions of the system are stable: this result holds in dimensions 3,4 and 5, when the metric is conformally flat and the drift is small
19
Kragh, Helge. "Mathematics, Relativity, and Quantum Wave Equations." Wissenschaftlicher Verlag Harri Deutsch, 2011. https://slub.qucosa.de/id/qucosa%3A16265.
Full text
20
Smerlak, Matteo. "Divergence des mousses de spins : comptage de puissance et resommation dans le modèle plat." Phd thesis, Université de la Méditerranée - Aix-Marseille II, 2011. http://tel.archives-ouvertes.fr/tel-00662170.
Full textAbstract:
L'objet de cette thèse est l'étude du modèle plat, l'ingrédient principal du programme de quantification de la gravité par les mousses de spins, avec un accent particulier sur ses divergences. Outre une introduction personnelle au problème de la gravité quantique, le manuscrit se compose de deux parties. Dans la première, nous obtenons une formule exacte pour le comptage de puissances des divergences de bulles dans le modèle plat, notamment grâce à des outils de théorie de jauge discrète et de cohomologie tordue. Dans la seconde partie, nous considérons le problème de la limite continue des mousses de spins, tant du point de vue des théorie de jauge sur réseau que du point de vue de la "group field theory". Nous avançons en particulier une nouvelle preuve de la sommabilité de Borel du modèle de Boulatov-Freidel-Louapre, permettant un contrôle accru du comportement d'échelle dans la limite de grands spins. Nous concluons par une discussion prospective du programme de renormalisation pour les mousses de spins.
21
Lopez-Monsalvo, Cesar Simon. "Covariant thermodynamics & relativity." Thesis, University of Southampton, 2011. https://eprints.soton.ac.uk/197299/.
Full textAbstract:
This thesis deals with the dynamics of irreversible processes within the context of the general theory of relativity. In particular, we address the problem of the `infinite' speed of propagation of thermal disturbances in a dissipative fluid. Although this problem is not new, its best known solution - the Israel and Stewart second order expansion - has an effective, rather than fundamental, character. The present work builds on the multi-fluid variational approach to relativistic dissipation, pioneered by Carter, and provides a dynamical theory of heat conduction. The novel property of such approach is the thermodynamic interpretation associated with a two-fluid system whose constituents are matter and entropy. The dynamics of this model leads to a relativistic generalisation of the Cattaneo equation; the constitutive relation for causal heat transport. A comparison with the Israel and Stewart model is presented and its equivalence is shown. This discussion provides new insights into the not-well understood definition of a non-equilibrium temperature. A crucial feature of the multi-fluid approach is the interaction between its constituents. It is a well known fact that when two, or more, fluids interact, instabilities may occur. Within this work, the two-stream instability analysis is extended to the relativistic domain. As far as the author is aware, such extension has not been discussed in the literature. The analysis allows to assess the stability and causality of relativistic models of matter and their linear deviations from thermodynamic equilibrium directly from their equations of state or, equivalently, their Lagrangian densities. For completeness, a brief digression on a consistent (stable and causal) `first-order' model is also included. Finally, the road to follow is laid by posing some physical applications together with some future perspectives and closing remarks. To sum up, the variational approach to heat conduction presented in this thesis constitutes a mathematically promising formalism to explore the relativistic evolution towards equilibrium of dissipative fluids in a dynamical manner and to get a deeper conceptual understanding of non-equilibrium thermodynamic quantities. Moreover, it might also be useful to explore the more fundamental issues of the irreversible dynamics of relativity and its connections with the time asymmetry of nature
22
Nascimento, Danilo Borim do. "Natação em espaços curvos via teoria de calibre." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306425.
Full textAbstract:
Orientador: Ricardo Antonio MosnaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
Made available in DSpace on 2018-08-16T04:10:22Z (GMT). No. of bitstreams: 1 Nascimento_DaniloBorimdo_M.pdf: 2195612 bytes, checksum: 9db76fa3dc3fca1c9128da411fe240b5 (MD5) Previous issue date: 2010
Resumo: No espaço euclidiano, deformações de corpos quase-rígidos podem gerar rotações globais líquidas que obedecem, em cada instante, a lei de conservação do momento angular (o problema do gato caindo é um exemplo). Em espaços curvos, um ciclo de deformações de um corpo pode gerar não só rotações, mas também translações globais. Este fenômeno é conhecido como efeito swimming, ou natação. Avron e Kenneth apresentaram recentemente um modelo físico para descrever este fenômeno [Avron JE, Kenneth O, New J. Phys. 8, 68 (2006)]. Os autores tratam de corpos compostos por um conjunto de massas puntiformes em variedades estáticas (no contexto não-relativístico) e calculam o deslocamento obtido por um ciclo de deformações infinitesimais. Tal deslocamento é então relacionado, no caso de corpos pequenos, à curvatura do espaço ambiente. Nesta dissertação, propomos uma nova formulação para o efeito swimming utilizando formalismo de fibrados e conexões. O espaço de configurações do sistema é descrito como o espaço total de um fibrado principal, cujo espaço base é dado pelo espaço dos formatos do sistema e o grupo estrutural é (essencialmente) dado pelas isometrias da variedade ambiente. Dotando o fibrado de uma conexão que carrega consigo a informação sobre as leis físicas de conservação, expressamos o ciclo de deformações como uma curva fechada no espaço base, o movimento do corpo como o levantamento horizontal desta curva e o deslocamento resultante como a holonomia da mesma. Por meio deste formalismo, sistematizamos o cálculo do deslocamento gerado por ciclos de deformações arbitrárias, além de obter, em cada instante e analiticamente, a evolução temporal do sistema em questão
Abstract: In Euclidean space, cyclic deformations of quasi-rigid bodies can lead to net global rotations even though they satisfy, at each moment, the angular momentum conservation law (the falling cat problem is an example). In curved spaces, cyclic changes in the body shape can also lead to rotations, but also to global translations. This phenomenon is known as the swimming effect. In a recent work, Avron and Kenneth developed a formalism to describe this phenomenon in the non-relativistic context [Avron JE, Kenneth O, New J. Phys. 8, 68 (2006)], which may be used to calculate the net displacement caused by an infinitesimal cycle of deformations of a given body. This displacement is then related, for small swimmers, to the curvature of the ambient space. In the present work, we propose a new formulation for the swimming effect in terms of principal bundles and connections. The configuration space of the system is described by the total space of a principal bundle, whose base space is given by the space of shapes of the body and whose structural group is (essentially) given by the isometries of the ambient manifold. A given deformation cycleof the body then corresponds to a loop in the base space. By defining a connection in this bundle which conveys the physical conservation laws of the system, the corresponding physical motion of the body is then given by the horizontal lift of this curve in the base space, while the net displacement of the body is given by the holonomy associated with this loop. As a result we obtain, in a systematical way, the displacement generated by arbitrary deformation cycles and we get, for each instant of time, the time evolution of the system analytically
Mestrado
Geometria e Topologia
Mestre em Matemática
23
Mongwane, Bishop. "Problems in cosmology and numerical relativity." Doctoral thesis, University of Cape Town, 2015. http://hdl.handle.net/11427/15689.
Full textAbstract:
Includes bibliographical references.A generic feature of most inflationary scenarios is the generation of primordial perturbations. Ordinarily, such perturbations can interact with a weak magnetic field in a plasma, resulting in a wide range of phenomena, such as the parametric excitation of plasma waves by gravitational waves. This mechanism has been studied in different contexts in the literature, such as the possibility of indirect detection of gravitational waves through electromagnetic signatures of the interaction. In this work, we consider this concept in the particular case of magnetic field amplification. Specifically, we use non-linear gauge-in variant perturbation theory to study the interaction of a primordial seed magnetic field with density and gravitational wave perturbations in an almost Friedmann-Lemaıtre-Robertson- Walker (FLRW) spacetime with zero spatial curvature. We compare the effects of this coupling under the assumptions of poor conductivity, perfect conductivity and the case where the electric field is sourced via the coupling of velocity perturbations to the seed field in the ideal magnetohydrodynamic (MHD) regime, thus generalizing, improving on and correcting previous results. We solve our equations for long wavelength limits and numerically integrate the resulting equations to generate power spectra for the electromagnetic field variables, showing where the modes cross the horizon. We find that the interaction can seed Electric fields with non-zero curl and that the curl of the electric field dominates the power spectrum on small scales, in agreement with previous arguments. The second focus area of the thesis is the development a stable high order mesh refinement scheme for the solution of hyperbolic partial differential equations. It has now become customary in the field of numerical relativity to couple high order finite difference schemes to mesh refinement algorithms. This approach combines the efficiency of local mesh refinement with the robustness and accuracy of higher order methods. To this end, different modifications of the standard Berger-Oliger adaptive mesh refinement a logarithm have been proposed. In this work we present a new fourth order convergent mesh refinement scheme with sub- cycling in time for numerical relativity applications. One of the distinctive features of our algorithm is that we do not use buffer zones to deal with refinement boundaries, as is currently done in the literature, but explicitly specify boundary data for refined grids instead. We argue that the incompatibility of the standard mesh refinement algorithm with higher order Runge Kutta methods is a manifestation of order reduction phenomena which is caused by inconsistent application of boundary data in the refined grids. Indeed, a peculiar feature of high order explicit Runge Kutta schemes is that they behave like low order schemes when applied to hyperbolic problems with time dependent Dirichlet boundary conditions. We present a new algorithm to deal with this phenomenon and through a series of examples demonstrate fourth order convergence. Our scheme also addresses the problem of spurious reflections that are generated when propagating waves cross mesh refinement boundaries. We introduce a transition zone on refined levels within which the phase velocity of propagating modes is allowed to decelerate in order to smoothly match the phase velocity of coarser grids. We apply the method to test problems involving propagating waves and show a significant reduction in spurious reflections.
24
Delbem, Nilton Flávio [UNESP]. "Introdução matemática aos modelos cosmológicos." Universidade Estadual Paulista (UNESP), 2010. http://hdl.handle.net/11449/94340.
Full textAbstract:
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
25
Assirati, João Luis Meloni. "Quantização covariante de sistemas mecânicos." Universidade de São Paulo, 2010. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-01072010-130800/.
Full textAbstract:
Estudamos as restrições impostas pelo princípio da covariância sobre o procedimento de quantização em espaços planos e curvos. Mostramos que o conjunto de todas as quantizações covariantes em espaços planos em coordenadas retangulares é composto de ordenamentos de operadores de posição e momento e exibimos uma parametrização funcional deste conjunto. Deduzimos regras para a quantização covariante em espaços planos em coordenadas gerais. Generalizamos estas quantizações para espaços curvos e mostramos que nestes espaços, além da ambiguidade de ordenamento, surge uma nova ambiguidade relacionada à curvatura. Este novo tipo de ambiguidade explica o surgimento de uma classe grande de potenciais quânticos no problema da quantização de uma partícula não relativística em um espaço curvo.We study the restrictions imposed by the covariance principle on the quantization procedure in flat and curved spaces. We show that the set of all covariant quantizations in flat spaces in rectangular coordinates is composed of position and momentum operator orderings and exhibit a functional parametrization of this set. We deduce rules for the covariant quantization in flat spaces in general coordinates. We generalize these quantizations for curved spaces and show that in such spaces, besides the ordering ambiguity, it appears a new ambiguity related to the curvature. This new kind of ambiguity explains the appearence of a wide class of quantum potentials in the problem of quantization of a non-relativistic particle in curved space.
26
Delbem, Nilton Flávio. "Introdução matemática aos modelos cosmológicos /." Rio Claro : [s.n.], 2010. http://hdl.handle.net/11449/94340.
Full textAbstract:
Orientador: Wladimir SeixasBanca: 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
27
Sjödin, Robert. "Cosmic strings in general relativity." Thesis, University of Southampton, 2001. https://eprints.soton.ac.uk/50619/.
Full textAbstract:
In this thesis we examine the properties of Cosmic Strings in the theory of General Relativity. We begin by considering static Cosmic Strings in flat space-time. We derive the field equations for the Cosmic String and show that the solution depends upon a single scaling parameter a which is constructed from the physical constants. Using this result we construct 1-parameter families of solutions which depend on an auxiliary parameter e and which describe the thin-string limit of a Cosmic String. By interpreting these solutions as elements of the simplified Colombeau algebra we may interpret the relativistic energy density Too of the thin string as an element of the Colombeau algebra with delta-function mass-per-unit-length. Furthermore, for a critically coupled Cosmic String the energymomentum tensor in the thin-string limit may be given a distributional interpretation. We also solve the string equations numerically for various values of a. This is done by compactifying the space-time to include infinity as part of the numerical grid and then using a relaxation method to suppress exponentially growing un-physical solutions. In curved space-time we derive the equations for the scalar and vector fields which are now coupled to the geometric variables through Einstein's equations. We again examine the thin-string limit in the Colombeau algebra by considering a 1-parameter family of solutions. W'e derive an expression for the deficit angle in terms of the distributional energy-momentum tensor of the thin string. We use this result to investigate the gravitational lensing properties of the string and relate this to the deficit angle. In the special case of a cone we find the scattering angle is equal to the deficit angle. We also solve the coupled equations numerically using techniques similar to those used in flat space-time. The second part of the thesis involves the dynamics of Cosmic Strings. Einstein's equations then lead to wave equations for both the matter and metric variables. However, the space-time is not asymptotically flat and this leads to problems in applying the appropriate boundary conditions. By using a Geroch transformation it is possible to reformulate the equations in terms of geometrical variables defined on an asymptotically flat (2+l)-dimensional space-time. Three exact vacuum solutions describing gravitational radiation due to Weber-Wheeler, Xanthopoulos and Piran et al. are used to excite the string which is found to oscillate with frequencies which are proportional to the masses of the scalar and vector fields of the string. This is in agreement with the exact results obtained using the linearised equations of the thin dynamic string. The behaviour of the dynamic string is studied by solving the equations numerically using an implicit fully characteristic scheme. The use of the Geroch transformation allows us to compactify the space-time and include null infinity as part of the numerical grid. This enables us to use the correct boundary conditions at infinity and hence suppress un-physical divergent solutions. The code is tested by comparing the results with exact solutions, by checking that it agrees with the static code and by undertaking a time dependent convergence test. The code is found to be accurate, stable and exhibit clear second order convergence.
28
Masetlwa, Nkosinathi. "Relativistic neutron stars in general relativity and fourth order gravity." Master's thesis, Faculty of Science, 2021. http://hdl.handle.net/11427/33791.
Full textAbstract:
This thesis investigates numerical instabilities arising from stiffness in the models of nonrotating, spherically symmetric single neutron star systems. The work deals with two distinct problems, each of which involves a stiff system of differential equations. In each case, we deal with stiffness by employing an IMEX Runge-Kutta scheme as opposed to the more computationally intensive fully implicit schemes or other adaptive Runge Kutta methods that may be impractical for partial differential equations. The first problem is focused on the mass-radius relation of a neutron star under a quadratic f(R) = R+αR2 theory for various realistic equations of state. This results in a coupled system of ODEs with stiff source terms which we discretize using an IMEX scheme. The observed maximum masses for different values of α, were consistent with the current neutron star maximum mass limit for some equations of state in both GR and beyond. In the second problem, we compute the frequencies of radial oscillations of neutron stars in the context of general relativity. This is achieved by linearly perturbing the ADM equations coupled to a matter source term. We discretize the resulting coupled system of PDEs with a third order WENO scheme in space and an IMEX scheme in time. We obtained 18 frequencies from the Fast Fourier Transform (FFT) of the evolved perturbation equations, which were consistent with the frequencies of the neutron star's Sturm-Liouville problem. The efficiency of the IMEX scheme as compared to other methods such as fully implicit schemes or adaptive methods makes it ideal for implementation in fully 3D numerical relativity codes for modified gravity.
29
Piper, Matthew Sebastian. "Power series and computer algebra in general relativity." Thesis, Queen Mary, University of London, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.264988.
Full text
30
Rowlingson, Robert R. "A class of perfect fluids in general relativity." Thesis, Aston University, 1990. http://publications.aston.ac.uk/12060/.
Full textAbstract:
This thesis is concerned with exact solutions of Einstein's field equations of general relativity, in particular, when the source of the gravitational field is a perfect fluid with a purely electric Weyl tensor. General relativity, cosmology and computer algebra are discussed briefly. A mathematical introduction to Riemannian geometry and the tetrad formalism is then given. This is followed by a review of some previous results and known solutions concerning purely electric perfect fluids. In addition, some orthonormal and null tetrad equations of the Ricci and Bianchi identities are displayed in a form suitable for investigating these space-times. Conformally flat perfect fluids are characterised by the vanishing of the Weyl tensor and form a sub-class of the purely electric fields in which all solutions are known (Stephani 1967). The number of Killing vectors in these space-times is investigated and results presented for the non-expanding space-times. The existence of stationary fields that may also admit 0, 1 or 3 spacelike Killing vectors is demonstrated. Shear-free fluids in the class under consideration are shown to be either non-expanding or irrotational (Collins 1984) using both orthonormal and null tetrads. A discrepancy between Collins (1984) and Wolf (1986) is resolved by explicitly solving the field equations to prove that the only purely electric, shear-free, geodesic but rotating perfect fluid is the Godel (1949) solution. The irrotational fluids with shear are then studied and solutions due to Szafron (1977) and Allnutt (1982) are characterised. The metric is simplified in several cases where new solutions may be found. The geodesic space-times in this class and all Bianchi type 1 perfect fluid metrics are shown to have a metric expressible in a diagonal form. The position of spherically symmetric and Bianchi type 1 space-times in relation to the general case is also illustrated.
31
Durkee, Mark N. "New approaches to higher-dimensional general relativity." Thesis, University of Cambridge, 2011. https://www.repository.cam.ac.uk/handle/1810/240580.
Full textAbstract:
This thesis considers various aspects of general relativity in more than four spacetime dimensions. Firstly, I review the generalization to higher dimensions of the algebraic classification of the Weyl tensor and the Newman-Penrose formalism. In four dimensions, these techniques have proved useful for studying many aspects of general relativity, and it is hoped that their higher dimensional generalizations will prove equally useful in the future. Unfortunately, many calculations using the Newman-Penrose formalism can be unnecessarily complicated. To address this, I describe new work introducing a higher-dimensional generalization of the so-called Geroch-Held-Penrose formalism, which allows for a partially covariant reformulation of general relativity. This approach provides great simplifications for many calculations involving spacetimes which admit one or two preferred null directions. The next chapter describes the proof of an important result regarding algebraic classification in higher dimensions. The classification is based upon the existence of a particular null direction that is aligned with the Weyl tensor of the geometry in some appropriate sense. In four dimensions, it is known that a null vector field is such a multiple Weyl aligned null direction (WAND) if and only if it is tangent to a shearfree null geodesic congruence. This is not the case in higher dimensions. However, I have formulated and proved a partial generalization of the result to arbitrary dimension, namely that a spacetime admits a multiple WAND if and only if it admits a geodesic multiple WAND.Moving onto more physical applications, I describe how the formalism that we have developed can be applied to study certain aspects of the stability of extremal black holes in arbitrary dimension. The final chapter of the thesis has a rather different flavour. I give a detailed analysis of the properties of a particular solution to the Einstein equations in five dimensions: the Pomeransky-Sen'kov doubly spinning black ring. I study geodesic motion around this black ring and demonstrate the separability of the Hamilton-Jacobi equation for null, zero energy geodesics. I show that this unexpected separability can be understood in terms of a symmetry described by a conformal Killing tensor on a four dimensional spacetime obtained by a Kaluza-Klein reduction of the original black ring spacetime.
32
Mokdad, Mokdad. "Champs de Maxwell en espace-temps de Reissner - Nordstr∫m- De Sitter : décroissance et scattering conforme." Thesis, Brest, 2016. http://www.theses.fr/2016BRES0060/document.
Full textAbstract:
Nous étudions les champs de Maxwell à l'extérieur de trous noirs de Reissner-Nordstrom-de Sitter. Nous commençons par étudier la géométrie de ces espaces-temps : nous donnons une condition sous laquelle la métrique admet trois horizons puis dans ce cadre nous construisons l'extension analytique maximale d'un trou noir de Reissner-Nordstrom-de Sitter. Nous donnons ensuite une description générale des champs de Maxwell en espace-temps courbe, de leur décomposition en composantes spinorielle ainsi que de leur énergie. La première étude analytique établit la décroissance ponctuelle de champs de Maxwell à l'extérieur d'un trou noir de Reissner-Nordstrom-de Sitter ainsi que la décroissance uniforme de l'énergie sur un hyperboloïde qui s'éloigne dans le futur. Ce chapitre utilise des méthodes de champs de vecteurs (estimations d'énergie géométriques) dans l'esprit des travaux de Pieter Blue. Enfin nous construisons une théorie du scattering conforme pour les champs de Maxwell à l'extérieur du trou noir. Ceci consiste en la résolution du problème de Goursat pour les champs de Maxwell à la frontière isotrope de l'extérieur du trou noir, constituée des horizons du trou noir et horizons cosmologiques futurs et passés. Les estimations de décroissance uniforme de l'énergie sont cruciales dans cette partieWe study Maxwell fields on the exterior of Reissner-Nordstrom-de Sitter black holes. We start by studying the geometry of these spacetimes: we give the condition under which the metric admits three horizons and in this case we construct the maximal analytic extension of the Reissner-Nordstrom-de Sitter black hole. We then give a general description of Maxwell fields on curves spacetimes, their decomposition into spin components, and their energies. The first result establishes the pointwise decay of the Maxwell field in the exterior of a Reissner-Nordstrom-de Sitter black hole, as well as the uniform decay of the energy flux across a hyperboloid that recedes in the future. This chapter uses the vector fields methods (geometric energy estimates) in the spirit of the work of Pieter Blue. Finally, we construct a conformal scattering theory for Maxwell fields in the exterior of the black hole. This amounts to solving the Goursat problem for Maxwell fields on the null boundary of the exterior region, consisting of the future and past black hole and cosmological horizons. The uniform decay estimates of the energy are crucial to the construction of the conformal scattering theory
33
Taliotis, Anastasios S. "Evolving Geometries in General Relativity." The Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1274838401.
Full text
34
Van, der Walt Petrus Johannes. "Numerical relativity on cosmological past null cones." Thesis, Rhodes University, 2013. http://hdl.handle.net/10962/d1002959.
Full textAbstract:
The observational approach to cosmology is the endeavour to reconstruct the geometry of the Universe using only data that is theoretically verifiable within the causal boundaries of a cosmological observer. Using this approach, it was shown in [36] that given ideal cosmological observations, the only essential assumption necessary to determine the geometry of the Universe is a theory of gravity. Assuming General Relativity, the full set of Einstein field equations (EFEs) can be used to reconstruct the geometry of the Universe using direct observations on the past null cone (PNC) as initial conditions. Observationally and theoretically this is a very ambitious task and therefore, current developments have been restricted to spherically symmetric dust models while only relaxing the usual assumption of homogeneity in the radial direction. These restricted models are important for the development of theoretical foundations and also useful as verification models since they avoid the circularity of verifying what has already been assumed. The work presented in this thesis is the development of such a model where numerical relativity (NR) is used to simulate the observable universe. Similar to the work of Ellis and co-workers [36], a reference frame based on the PNC is used. The reference frame used here, however, is based on that of the characteristic formalism of NR, which has developed for calculating the propagation of gravitational waves. This provides a formalism that is well established in NR, making the use of existing algorithms possible. The Bondi-Sachs coordinates of the characteristic formalism is, however, not suitable for calculations beyond the observer apparent horizon (AH) since the diameter distance used as a radial coordinate becomes multi-valued when the cosmological PNC reconverges in the history of a universe, smaller in the past. With this taken into consideration, the Bondi-Sachs characteristic formalism is implemented for cosmology and the problem approaching the AH is investigated. Further developments address the limitations approaching the AH by introducing a metric based on the Bondi-Sachs metric where the radial coordinate is replaced with an affine parameter. The model is derived with a cosmological constant Λ incorporated into the EFEs where Λ is taken as a parameter of the theory of gravity rather than as a matter source term. Similar to the conventional characteristic formalism, this model consists of a system of differential equations for numerically evolving the EFEs as a characteristic initial value problem (CIVP). A numerical code implemented for the method has been found to be second order convergent. This code enables simulations of different models given identical data on the initial null cone and provides a method to investigate their physical consistency within the causally connected region of our current PNC. These developments closely follow existing 3D schemes developed for gravitational wave simulations, which should make it natural to extend the affine CIVP beyond spherical symmetric simulations. The developments presented in this thesis is an extended version of two papers published earlier.
35
Lim, Hyun. "Dynamical Compact Objects in Numerical Relativity." BYU ScholarsArchive, 2019. https://scholarsarchive.byu.edu/etd/7729.
Full textAbstract:
The work of this dissertation will study various aspects of the dynamics of compact objects using numerical simulations.We consider BH dynamics within two modified or alternative theories of gravity. Within a family of Einstein-Maxwell-Dilaton-Axion theories, we find that the GW waveforms from binary black hole (BBH) mergers differ from the standard GW waveform prediction of GR for especially large axion values. For more astrophysically realistic (i.e. smaller) values, the differences become negligible and undetectable. Weestablish the existence of a well-posed initial value problem for a second alternative theory fo gravity (quadratic gravity) and demonstrate in spherical symmetry that a linear instability is effectively removed on consideration of the full nonlinear theory.We describe the key components and development of a code for studying BBH mergers for which the mass ratio of the binaries is not close to one. Such intermediate mass ratio inspirals (IMRIs) are much more difficult to simulate and present greater demands on resolution, distributed computing, accuracy and efficiency. To this end, we present a highly-scalable framework that combines a parallel octree-refined adaptive mesh with a wavelet adaptive multiresolution approach. We give results for IMRIs with mass ratios up to 100:1. We study the ejecta from BNS in Newtonian gravity. Using smoothed particle hydrodynamics we develop and present the highly scalable FleCSPH code to simulate such mergers. As part of the ejecta analysis, we consider these mergers and their aftermath as prime candidates for heavy element creation and calculate r-process nucleosynthesis within the post-merger ejecta. Lastly we consider a non-standard, yet increasingly explored, interaction between a BH and a NS that serves as a toy model for primordial black holes (PBH) and their possible role as dark matter candidates. We present results from a study of such systems in which a small BH forms at the center of a NS. Evolving the spherically symmetric system in full GR, we follow the complete dynamics as the small BH consumes the NS from within. Using numerical simulations, we examine the time scale for the NS to collapse into the PBH and show that essentially nothing remains behind. As a result, and in contradiction to other claims in the literature, we conclude that thisis an unlikely site for ejecta and nucleosynthesis, at least in spherical symmetry.
36
Gasperin, Garcia. "Applications of conformal methods to the analysis of global properties of solutions to the Einstein field equations." Thesis, Queen Mary, University of London, 2017. http://qmro.qmul.ac.uk/xmlui/handle/123456789/25820.
Full textAbstract:
Although the study of the initial value problem in General Relativity started in the decade of 1950 with the work of Foures-Bruhat, addressing the problem of global non-linear stability of solutions to the Einstein field equations is in general a hard problem. The first non-linear global stability result in General Relativity was obtained for the de-Sitter spacetime by means of the so-called conformal Einstein field equations introduced by H. Friedrich in the decade of 1980. The latter constitutes the main conceptual and technical tool for the results discussed in this thesis. In Chapter 1 the physical and geometrical motivation for these equations is discussed. In Chapter 2 the conformal Einstein equations are presented and first order hyperbolic reduction strategies are discussed. Chapter 3 contains the first result of this work; a second order hyperbolic reduction of the spinorial formulation of the conformal Einstein field equations. Chapter 4 makes use of the latter equations to give a discussion of the non-linear stability of the Milne universe. Chapter 5 is devoted to the analysis of perturbations of the Schwarzschild-de Sitter spacetime via suitably posed asymptotic initial value problems. Chapter 6 gives a partial generalisation of the results of Chapter 5. Finally a result relating the Newman-Penrose constants at future and past null infinity for spin-1 and spin-2 fields propagating on Minkowski spacetime close to spatial infinity is discussed in Chapter 7 exploiting the framework of the cylinder at spatial in nity. Collectively, these results show how the conformal Einstein field equations and more generally conformal methods can be employed for analysing perturbations of spacetimes of interest and extract information about their conformal structure.
37
Kolenick, Joseph F. "On exponentially perfect numbers relatively prime to 15 /." Connect to resource online, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1196698780.
Full text
38
Le, Tiec Alexandre. "Coalescence de trous noirs en relativité générale & Le problème de la matière noire en astrophysique." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2010. http://tel.archives-ouvertes.fr/tel-00521645.
Full textAbstract:
La première partie de cette thèse s'inscrit dans le cadre de la modélisation des ondes gravitationnelles en provenance des systèmes binaires coalescents de trous noirs, dans la perspective de leur détection par les antennes gravitationnelles terrestres LIGO/VIRGO et spatiale LISA. Nous étudions la dynamique relativiste de tels systèmes binaires d'objets compacts à l'aide de deux méthodes d'approximation en relativité générale : les développements post-newtoniens, et le formalisme de la force propre, une extension naturelle de la théorie des perturbations d'un trou noir ; nous démontrons la cohérence des résultats ainsi obtenus dans leur domaine de validité commun. Dans un second temps, nous combinons ces deux méthodes perturbatives afin d'estimer l'effet de recul gravitationnel lors de la coalescence de deux trous noirs de Schwarzschild ; nos résultats sont en très bon accord avec ceux obtenus par des simulations en relativité numérique. La seconde partie de cette thèse traite du problème de la matière noire en astrophysique. L'hypothèse de la matière noire rend compte de nombreuses observations indépendantes de l'échelle des amas de galaxies jusqu'aux échelles cosmologiques. Les observations à l'échelle galactique sont toutefois en bien meilleur accord avec la phénoménologie de la dynamique newtonienne modifiée (MOND), qui postule une modification des lois de la gravité en l'absence de matière noire. Nous proposons une troisième alternative : conserver la théorie de la gravitation standard, mais doter la matière noire d'une propriété de polarisabilité dans un champ gravitationnel, de façon à rendre compte de la phénoménologie de MOND à l'échelle des galaxies.
39
Sabree, Benjamin David. "A Pedagogical Investigation of the Development of General Relativity Using Differential Forms." Case Western Reserve University School of Graduate Studies / OhioLINK, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=case1212173046.
Full text
40
Radermacher, Katharina Maria. "Strong Cosmic Censorship and Cosmic No-Hair in spacetimes with symmetries." Doctoral thesis, KTH, Matematik (Avd.), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-220400.
Full textAbstract:
This thesis consists of three articles investigating the asymptotic behaviour of cosmological spacetimes with symmetries arising in Mathematical General Relativity. In Paper A and B, we consider spacetimes with Bianchi symmetry and where the matter model is that of a perfect fluid. We investigate the behaviour of such spacetimes close to the initial singularity ('Big Bang'). In Paper A, we prove that the Strong Cosmic Censorship conjecture holds in non-exceptional Bianchi class B spacetimes. Using expansion-normalised variables, we further show detailed asymptotic estimates. In Paper B, we prove similar estimates in the case of stiff fluids. In Paper C, we consider T2-symmetric spacetimes satisfying the Einstein equations for a non-linear scalar field. To given initial data, we show global existence and uniqueness of solutions to the corresponding differential equations for all future times. In the special case of a constant potential, a setting which is equivalent to a linear scalar field on a background with a positive cosmological constant, we investigate in detail the asymptotic behaviour towards the future. We prove that the Cosmic No-Hair conjecture holds for solutions satisfying an additional a priori estimate, an estimate which we show to hold in T3-Gowdy symmetry.Denna avhandling består av tre artiklar som undersöker det asymptotiska beteendet hos kosmologiska rumstider med symmetrier som uppstår i Matematisk Allmän Relativitetsteori. I Artikel A och B studerar vi rumstider med Bianchi symmetri och där materiemodellen är en ideal fluid. Vi undersöker beteendet av sådana rumstider nära ursprungssingulariteten ('Big Bang'). I Artikel A bevisar vi att den Starka Kosmiska Censur-förmodan håller för icke-exceptionella Bianchi klass B-rumstider. Med hjälp av expansions-normaliserade variabler visar vi detaljerade asymptotiska uppskattningar. I Artikel B visar vi liknande uppskattningar för stela fluider. I Artikel C betraktar vi T2-symmetriska rumstider som uppfyller Einsteins ekvationer för ett icke-linjärt skalärfält. För givna begynnelsedata visar vi global existens och entydighet av lösningar till motsvarande differentialekvationer för all framtid. I det speciella fallet med en konstant potential, en situation som motsvarar ett linjärt skalärfält på en bakgrund med en positiv kosmologisk konstant, undersöker vi i detalj det asymptotiska beteendet mot framtiden. Vi visar att den Kosmiska Inget-Hår-förmodan håller för lösningar som uppfyller en ytterligare a priori uppskattning, en uppskattning som vi visar gäller i T3-Gowdy-symmetri.
QC 20171220
41
Boonserm, Petarpa. "Some exact solutions in general relativity : a thesis submitted to the Victoria University of Wellington in fulfilment of the requirements for the degree of Master of Science in Mathematics /." ResearchArchive@Victoria e-Thesis, 2005. http://hdl.handle.net/10063/16.
Full text
42
Angst, Jürgen. "Etude de diffusions à valeurs dans des variétés lorentziennes." Phd thesis, Université de Strasbourg, 2009. http://tel.archives-ouvertes.fr/tel-00418842.
Full textAbstract:
L'objet de ce mémoire est l'étude de processus stochastiques à valeurs dans des variétés lorentziennes. En particulier, on s'intéresse au comportement asymptotique en temps long de ces processus et on souhaite voir en quoi celui-ci reflète la géométrie des variétés sous-jacentes. Nous limitons notre étude à celle de diffusions, c'est-à-dire de processus markoviens continus, à valeurs dans le fibré tangent unitaire de variétés lorentziennes fortement symétriques. L'introduction et l'étude de tels processus ont des motivations purement mathématiques mais aussi physiques. Ce mémoire est composé de deux parties. La première est consacrée à la preuve d'un théorème limite central pour une classe de diffusions minkowskiennes. Elle est motivée par des questions ouvertes de la littérature physique. La seconde partie du manuscrit est consacrée à l'étude détaillée d'une diffusion relativiste à valeurs dans les espaces de Robertson-Walker. En fonction de la courbure et de la vitesse d'expansion de ces espaces, nous déterminons précisément le comportement asymptotique de la diffusion relativiste et montrons que ses trajectoires approchent asymptotiquement des géodésiques de lumière aléatoires. Pour une classe d'espaces de Robertson-Walker, nous explicitons en outre la frontière de Poisson de la diffusion relativiste.
43
Bigdely, Hadi. "Subgroup theorems in relatively hyperbolic groups and small- cancellation theory." Thesis, McGill University, 2013. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=119606.
Full textAbstract:
In the first part, we study amalgams of relatively hyperbolic groups and also therelatively quasiconvex subgroups of such amalgams. We prove relative hyperbolicityfor a group that splits as a finite graph of relatively hyperbolic groups with parabolicedge groups; this generalizes a result proved independently by Dahmani, Osin andAlibegovic. More generally, we prove a combination theorem for a group that splitsas a finite graph of relatively hyperbolic groups with total, almost malnormal andrelative quasiconvex edge groups. Moreover, we provide a criterion for detectingquasiconvexity of subgroups in relatively hyperbolic groups that split as above. As anapplication, we show local relative quasiconvexity of any f.g. group that is hyperbolicrelative to Noetherian subgroups and has a small-hierarchy. Studying free subgroupsof relatively hyperbolic groups, we reprove the existence of a malnormal, relativelyquasiconvex, rank 2 free subgroup F in a non-elementary relatively hyperbolic groupG. Using this result and with the aid of a variation on a result of Arzhantseva, weshow that if G is also torsion-free then "generically" any subgroup of F is aparabolic,malnormal in G and quasiconvex relative to P and therefore hyperbolically embedded.As an application, generalizing a result of I. Kapovich, we prove that for any f.g.,non-elementary, torsion-free group G that is hyperbolic relative to P, there exists agroup G∗ containing G such that G∗ is hyperbolic relative to P and G is not relativelyquasiconvex in G∗ .In the second part, we investigate the existence of F2 × F2 in the non-metric small-cancellation groups. We show that a C(6)-T(3) small-cancellation group cannotcontain a subgroup isomorphic to F2 × F2 . The analogous result is also proven in theC(3)-T(6) case.Dans la premiere partie, nous etudions les amalgames de groupes relativement hyperboliques et egalement les sous-groupes relativement quasiconvexes de ces amalgames. Nous prouvons l'hyperbolicie relative pour un groupe qui se separe comme un graphe fini de groupes relativement hyperboliques avec des groupes d'aretes paraboliques, ce qui generalise un resultat prouve independamment par Dahmani,Osin et Alibegovic. Nous l'etendons au cas ou les groupes d'aretes sont totalaux, malnormal et relativement quasiconvexes. En outre, nous fournissons un critere de detection de quasiconvexite relative des sous-groupes dans les groupes hyperboliques qui divisent. Comme application, nous montrons la quasiconvexite locale relative d'un groupe qui est relativement hyperbolique a certains sous-groupes noetheriens et qui a une petite hierarchie. Nous etudions egalement les sous-groupes libres de groupes relativement hyperboliques, et reprouvons l'existence d'un sous-groupe libre, malnormal, relativement quasiconvexe F2 dans un groupe non- elementaire relativement hyperbolique G. En combinant ce resultat avec une variation sur un theoremede Arzhantseva, nous montrons que si G est aussi sans-torsion, "generiquement" tout sous-groupe de F2 est aparabolique, malnormal dans G et quasiconvexe par rapport a P. Comme application, nous montrons que pour tout groupe G non-elementaire, sans-torsion, qui est hyperbolique par rapport a P, il existe un groupe G∗ contenant G tel que G∗ est hyperbolique par rapport a P et G n'est pas quasiconvexe dans G∗. Dans la deuxieme partie, nous etudions l'existence de sous-groupe F2 × F2 dans desgroupes a petite simplification. Nous montrons que les groupes C(6) ne peuvent pas contenir un sous-groupe isomorphe a F2 × F2 . Le resultat analogue est egalement prouve dans le dossier C(3)-T(6) affaire.
44
Adamo, Timothy M. "Twistor actions for gauge theory and gravity." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:9749662e-cbb3-4f6e-b81c-4ee17ba752fa.
Full textAbstract:
We first consider four-dimensional gauge theory on twistor space, taking as a case study maximally supersymmetric Yang-Mills theory. Using a twistor action functional, we show that gauge theory scattering amplitudes are naturally computed on twistor space in a manner that is much more efficient than traditional space-time Lagrangian techniques at tree-level and beyond. In particular, by rigorously studying the Feynman rules of a gauge-fixed version of the twistor action, we arrive at the MHV formalism. This provides evidence for the naturality of computing scattering amplitudes in twistor space as well as an alternative proof of the MHV formalism itself. Next, we study other gauge theory observables in twistor space including gauge invariant local operators and Wilson loops, and discuss how to compute their expectation values with the twistor action. This enables us to provide proofs for the supersymmetric correlation function / Wilson loop correspondence as well as conjectures on mixed Wilson loop - local operator correlators at the level of the loop integrand. Furthermore, the twistorial formulation of such observables is naturally algebro-geometric; this leads to novel recursion relations for computing mixed correlators by performing BCFW-like deformations of the observables in twistor space. Finally, we apply these twistor actions to gravity. Using the on-shell equivalence between Einstein and conformal gravity in de Sitter space, we argue that the twistor action for conformal gravity should encode the tree-level graviton scattering amplitudes of Einstein's theory. We prove this in terms of generating functionals, and derive the flat space MHV amplitude as well as a recursive version of the MHV amplitude with cosmological constant. We also include some discussion of super-connections and Coulomb branch regularization on twistor space.
45
Hicks, Jesse W. "Classification of Spacetimes with Symmetry." DigitalCommons@USU, 2016. https://digitalcommons.usu.edu/etd/5054.
Full textAbstract:
Spacetimes with symmetry play a critical role in Einstein's Theory of General Relativity. Missing from the literature is a correct, usable, and computer accessible classification of such spacetimes. This dissertation fills this gap; specifically, we
i) give a new and different approach to the classification of spacetimes with symmetry using modern methods and tools such as the Schmidt method and computer algebra systems, resulting in ninety-two spacetimes;
ii) create digital databases of the classification for easy access and use for researchers;
iii) create software to classify any spacetime metric with symmetry against the new database;
iv) compare results of our classification with those of Petrov and find that Petrov missed six cases and incorrectly normalized a significant number of metrics;
v) classify spacetimes with symmetry in the book Exact Solutions to Einstein’s Field Equations Second Edition by Stephani, Kramer, Macallum, Hoenselaers, and Herlt and in Komrakov’s paper Einstein-Maxwell equation on four-dimensional homogeneous spaces using the new software.
46
Kolenick, Joseph F. Jr. "On Exponentially Perfect Numbers Relatively Prime to 15." Youngstown State University / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1196698780.
Full text
47
Eltzner, Benjamin. "Local Thermal Equilibrium on Curved Spacetimes and Linear Cosmological Perturbation Theory." Doctoral thesis, Universitätsbibliothek Leipzig, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-117472.
Full textAbstract:
In this work the extension of the criterion for local thermal equilibrium by Buchholz, Ojima and Roos to curved spacetime as introduced by Schlemmer is investigated. Several problems are identified and especially the instability under time evolution which was already observed by Schlemmer is inspected. An alternative approach to local thermal equilibrium in quantum field theories on curved spacetimes is presented and discussed. In the following the dynamic system of the linear field and matter perturbations in the generic model of inflation is studied in the view of ambiguity of quantisation. In the last part the compatibility of the temperature fluctuations of the cosmic microwave background radiation with local thermal equilibrium is investigatedIn dieser Arbeit wird die von Schlemmer eingeführte Erweiterung des Kriteriums für lokales thermisches Gleichgewicht in Quantenfeldtheorien von Buchholz, Ojima und Roos auf gekrümmte Raumzeiten untersucht. Dabei werden verschiedene Probleme identifiziert und insbesondere die bereits von Schlemmer gezeigte Instabilität unter Zeitentwicklung untersucht. Es wird eine alternative Herangehensweise an lokales thermisches Gleichgewicht in Quantenfeldtheorien auf gekrümmten Raumzeiten vorgestellt und deren Probleme diskutiert. Es wird dann eine Untersuchung des dynamischen Systems der linearen Feld- und Metrikstörungen im üblichen Inflationsmodell mit Blick auf Uneindeutigkeit der Quantisierung durchgeführt. Zuletzt werden die Temperaturfluktuationen der kosmischen Hintergrundstrahlung auf Kompatibilität mit lokalem thermalem Gleichgewicht überprüft
48
Barbaroux, Jean-Marie. "Modèles mathématiques de la chimie quantique atomique & dynamique quantique et spectre multifractal." Habilitation à diriger des recherches, Université de Nantes, 2005. http://tel.archives-ouvertes.fr/tel-00010385.
Full textAbstract:
Les électrons dans les atomes lourds, en particulier ceux qui sont proches du noyau, sont soumis à des effets relativistes importants. Il est nécessaire de prendre en compte ces effets si l'on veut, par exemple, décrire précisément les niveaux d'énergies des atomes. L'étude des modèles atomiques quantiques relativistes remonte aux travaux fondateurs de P.A.M. Dirac, dès 1928. Ses travaux ont permis d'anticiper la découverte des antiparticules. En effet, le hamiltonien quantique qu'il obtient pour l'atome d'hydrogène n'a de sens physique que si l'on peut interpréter ses énergies négatives comme celles d'une mer infinie de particules virtuelles. Un « trou » dans le spectre des énergies négatives est alors interprété comme l'apparition d'une anti-particule : le positron. Peu après, en 1938, pour étudier les atomes à plusieurs électrons Swirles propose un modèle d'approximation qui donnera lieu aux fameuses équations de Dirac-Fock. Cette approche qui est auto-consistante, et pour laquelle les équations obtenues sont non linéaires, permet une étude numérique dont les résultats sont en très bon accord avec les mesures expérimentales. Pour autant, la motivation physique de cette approche reste incomplète. Elle s'appuie essentiellement sur l'analogue non relativiste des modèles atomiques quantiques, mais ne tient pas compte de l'interprétation de Dirac. De plus, le lien des équations de Dirac-Fock avec l'approche théorique donnée par l'électrodynamique quantique (QED) reste à établir clairement. En particulier, en QED, la question de la définition d'un espace qui décrit les états électroniques reste posée. Le travail présenté ici est une tentative d'apporter quelques réponses mathématiques rigoureuses sur ces problèmes. Nous commencerons par construire une famille de fonctionnelles à partir du hamiltonien formel de la QED qui dépendra du choix de l'espace à un électron. On se placera dans l'approximation de Hartree-Fock. On étudiera alors le problème de la stabilité, celui de l'existence de minima pour ces fonctionnelles (avec ou sans condition de charge totale fixée). On se consacrera ensuite à l'exposé des résultats obtenus qui permettent de comparer les deux approches : « Equations de Dirac-Fock » et « QED dans l'approximation de Hartree-Fock ». On distinguera en particulier le cas des couches pleines qui conduit aux mêmes résultats dans les deux cas, tout au moins pour des constantes de couplages faibles.
49
Miani, Lorenzo. "Highlighting interdisciplinarity between physics and mathematics in historical papers on special relativity: design of blended activities for pre-service teacher education." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23544/.
Full textAbstract:
This thesis is framed within the Erasmus+ project titled IDENTITIES, whose aim is to develop
interdisciplinary teaching materials for preservice teacher education. In collaboration with the research
group in STEM education of Crete, a blended module on special relativity has been developed. The
module is based on an analysis of the original texts by Lorentz, Poincaré, Einstein and Minkowski
(written between 1904 and 1908), aimed to recognise the interplay between mathematics and physics
implemented in the four papers. The analysis has been carried out by applying the 'Boundary Crossing
and Boundary Object' research framework developed in 2011 by Akkermann and Bakker. The results
of the analysis show that Lorentz Transformations can be read as a Boundary Object and this lens allows
for different nuances of the interplay between mathematics and physics to be recognised in the four
papers. A series of activities to be conducted in blended mode in a pre-service teacher education course
have been designed with the goal of exploiting special relativity as a context to develop interdisciplinary
skills.
50
Poujade, Olivier. "Iteration post-Newtonienne du champ de gravitation d'un systeme isole en Relativite Generale." Phd thesis, Université Paris-Diderot - Paris VII, 2002. http://tel.archives-ouvertes.fr/tel-00002179.
Full textAbstract:
Pour extraire l'empreinte laissee par le passage d'une onde gravitationnelle dans le signal bruite d'un des futurs detecteurs (VIRGO/LIGO), il est necessaire de connaitre avec une grande precision la forme de cette onde. Or, les equations de la Relativite Generale (RG), qui regissent cette forme, sont d'une grande complexite et ne sont pas solubles analytiquement pour des systemes dynamiques tels que les binaires d'etoiles a neutrons. Celles ci representent pourtant les sources les plus prometteuses de rayonnement gravitationnel. On a donc recours a des methodes d'approximation. Dans cette these nous avons etudie l'approximation post-Newtonienne (developpement perturbatif en puissance de 1/c) dans le cadre de la RG. Dans un premier temps nous avons etudie cette approximation d'un point de vue general afin de demontrer un theoreme qui permet d'affirmer que, sous certaines hypotheses, le developpement post-Newtonien du champ de gravitation d'une source isolee auto-gravitante peut etre itere a tout les ordres en 1/c, ce qui n'avait jamais ete fait jusqu'a present. Nous avons montre que ce developpement se raccorde a tous les ordres a une solution exterieur, valable notamment a l'infini, et satisfaisant aux conditions d'absence de radiation rentrante. Dans un deuxieme temps, sous l'impulsion de Luc Blanchet et en collaboration avec Gilles Esposito-Farese, je me suis interesse au probleme de l'energie post-Newtonienne a l'ordre 1/c^6 d'un systeme binaire de corps compacts etendus. Nous nous interessons notamment au probleme du calcul d'un coefficient indetermine a l'ordre 3PN du a une incompletude des methodes de regularisation. Pour lever cette indetermination, nous nous sommes interesses au calcul de cette meme energie pour des corps etendus lorsque l'on tient compte de la structure interne. Nous presentons dans cette these les raisons possibles de l'indetermination, la methode et les problemes poses par l'extension finie des corps.