We study the application of Bose-Einstein condensates (BECs) to simulations of phenomena across a number of disciplines in physics, using theoretical and computational methods. ¶ Collapsing condensates as created by E. Donley et al. [Nature 415, 39 (2002)] exhibit potentially useful parallels to an inflationary universe. To enable the exploitation of this analogy, we check if current quantum field theories describe collapsing condensates quantitatively, by targeting the discrepancy between experimental and theoretical values for the time to collapse. To this end, we couple the lowest order quantum field correlation functions to the condensate wavefunction, and solve the resulting Hartree-Fock-Bogoliubov equations numerically. Complementarily, we perform stochastic truncated Wigner simulations of the collapse. Both methods also allow us to study finite temperature effects. ¶ We find with neither method that quantum corrections lead to a faster collapse than is predicted by Gross-Pitaevskii theory. We conclude that the discrepancy between the experimental and theoretical values of the collapse time cannot be explained by Gaussian quantum fluctuations or finite temperature effects. Further studies are thus required before the full analogue cosmology potential of collapsing condensates can be utilised. ¶ As the next project, we find experimental parameter regimes in which stable three-dimensional Skyrmions can exist in a condensate. We show that their stability in a harmonic trap depends critically on scattering lengths, atom numbers, trap rotation and trap anisotropy. In particular, for the Rb87 |F=1,m_f=-1>, |F=2,m_f=1> hyperfine states, stability is sensitive to the scattering lengths at the 2% level. We find stable Skyrmions with slightly more than 2*10^6 atoms, which can be stabilised against drifting out of the trap by laser pinning. ¶ As a stepping stone towards Skyrmions, we propose a method for the stabilisation of a stack of parallel vortex rings in a Bose-Einstein condensate. The method makes use of a ``hollow'' laser beam containing an optical vortex, which realises an optical tunnel for the condensate. Using realistic experimental parameters, we demonstrate numerically that our method can stabilise up to 9 vortex rings. ¶ Finally, we focus on analogue gravity, further exploiting the analogy between flowing condensates and general relativistic curved space time. We compare several realistic setups, investigating their suitability for the observation of analogue Hawking radiation. We link our proposal of stable ring flows to analogue gravity, by studying supersonic flows in the optical tunnel. We show that long-living immobile condensate solitons generated in the tunnel exhibit sonic horizons, and discuss whether these could be employed to study extreme cases in analogue gravity. ¶ Beyond these, our survey indicates that for conventional analogue Hawking radiation, simple outflow from a condensate reservoir, in effectively one dimension, has the best properties. We show with three dimensional simulations that stable sonic horizons exist under realistic conditions. However, we highlight that three-body losses impose limitations on the achievable analogue Hawking temperatures. These limitations vary between the atomic species and favour light atoms. ¶ Our results indicate that Bose-Einstein condensates will soon be useful for interdisciplinary studies by analogy, but also show that the experiments will be challenging.

Dans cette thèse nous reprenons, puis développons, le modèle de Davies-Fulling<br />qui présente de nombreuses analogies avec la radiation engendrée par un Trou Noir. <br />Ce modèle consiste en l'étude de la radiation émise par un miroir <br />se déplaçant dans l'espace-temps de Minkowski. <br />Lorsque la trajectoire est non-inertielle, le miroir engendre un flux d'énergie<br />dont les propriétés sont des fonctionnelles de la trajectoire.<br /><br />Dans le but de retrouver les problèmes présents dans l'étude de la radiation de Hawking, <br />nous nous sommes intéressés principalement au cas des miroirs<br />uniformément accélérés, qui possédent un horizon causal.<br />Afin de résoudre ces problèmes présents dans le cadre du modèle de Davies-Fulling,<br />nous avons introduit un nouveau modèle, qui dérive cette fois d'une action.<br />Un choix précis et motivé du Lagrangien permet d'y parvenir.<br />De plus, grâce à deux méthodes complémentaires, nous avons explicité <br />les corrélations quantiques présentes dans le flux émis.

Les trous noirs primordiaux sont une sonde exceptionnelle pour rechercher des effets de nouvelle physique, à l'intersection de la relativité générale, de la mécanique quantique, de la physique des particules et de la cosmologie. Ce mémoire présente quelques pistes d'études relatives à ces objets astrophysiques fascinants. D'abord, autour de leur recherche via l'étude des rayons cosmiques qui seraient émis par évaporation de Hawking. Des liens entre les limites obtenues et les modèles d'inflation sont ensuite proposés afin d'obtenir une borne supérieure très contraignante - et totalement inaccessible aux observables usuelles que sont le fond diffus et les grandes structures - sur la puissance aux petites échelles dans l'Univers primordial. La fin de l'évaporation des trous noirs est etudiée en gravité de corde et leur statut de candidat à la matière noire froide revisité dans le cadre des modèles à brisure d'invariance d'échelle. Enfin, dans le cadre des modèles à basse échelle de Planck (c'est-à-dire présentant de larges dimensions supplémentaires), la formation de trous noirs auprès des collisionneurs est envisagée. Nous montrons que des effets de gravité quantique (couplage de Gauss-Bonnet) pourraient être sondés au LHC. Quelques voies d'investigations futures, liées à la présence d'une constante cosmologique ou au rayonnement cosmique d'énergie extrême sont esquissées.

L'expérience AMS sera mise en orbite à partir de 2006 pour une durée de 3 ans afin d'étudier le rayonnement cosmique et d'ouvrir de nouvelles perspectives pour la recherche d'antimatière et de matière noire. Cette thèse présente d'abord le travail mené pour les développements du détecteur Tcherenkov (RICH) d'AMS qui conduira à une mesure précise de la vitesse et de la charge des particules le traversant. Nous exposons le choix des photodétecteurs, les tests de l'électronique, les caractéristiques générales du compteur ainsi que la mise en oeuvre de l'analyse des données obtenues avec deux prototypes (incluant des mesures au CERN). La seconde partie du travail est consacrée à l'étude théorique du rayonnement cosmique et d'un signal exotique potentiel pour AMS : les trous noirs en évaporation. Les conséquences astrophysiques, cosmologiques et gravitationnelles de l'existence de ces objets sont considérées en détails.

<p>In this thesis we calculate the Hawking mass numerically for surfaces in Kerr spacetime. The Hawking mass is a useful tool for proving the Penrose inequality and the result does not contradict the inequality. It also does not contradict the assumption that the Hawking mass should be monotonic for surfaces in Kerr spacetime. The Hawking mass is quasi-local and defined by the spin coefficents of Newman and Penrose, so first we give a discussion about quasi-local quantities and then a short description of the Newman-Penrose formalism.</p>

In this thesis we calculate the Hawking mass numerically for surfaces in Kerr spacetime. The Hawking mass is a useful tool for proving the Penrose inequality and the result does not contradict the inequality. It also does not contradict the assumption that the Hawking mass should be monotonic for surfaces in Kerr spacetime. The Hawking mass is quasi-local and defined by the spin coefficents of Newman and Penrose, so first we give a discussion about quasi-local quantities and then a short description of the Newman-Penrose formalism.

The semi-classical black hole tunnelling method is a useful technique to calculate black hole temperature and understand black hole thermodynamics. I will investigate the black hole tunnelling method in detail. I will compare two different approaches used to calculate black hole tunnelling. The tunnelling method can be applied to a broad range of spacetimes and I will show this explicitly in order to demonstrate the robustness of the tunnelling technique. In particular, I will apply the tunnelling method to spacetimes including: Rindler (the method can recover the Unruh temperature), and more general spacetimes (such as Kerr-Newman and Taub-NUT). I will also discuss the 5d Kerr-Gödel spacetimes in detail (while showing a previous unobserved property of these spaces). Once the parameter space of Kerr-Gödel is understood in detail, I will show how the tunnelling method can also be successfully applied to the Kerr-Gödel black hole. Finally, the key result of my thesis involves extending the tunnelling method to model fermion emission. The previous tunnelling calculations all involved the emission of scalar particles. I will model the emission of spin-1/2 fermions from various spacetimes including the Rindler spacetime and general non-rotating black holes. I will also model the emission of charged spin-1/2 fermions from the Kerr-Newman spacetime to show that the method is also applicable to rotating spacetimes. In all these cases I show that the correct Hawking temperature (Unruh temperature in the case of Rindler) is recovered for spin-1/2 fermion emission. Although this final result is not surprising, it is an important result because it confirms that Dirac particles will radiate from the black hole at the same temperature as scalar particles. It has always been assumed that this is the case but there is very little literature involving fermion radiation of black holes. So the results of my calculations are twofold, I demonstrate that Dirac particles are emitted at the same temperature as scalar particles from a black hole and it shows how robust the semi-classical tunnelling technique is.

The semi-classical black hole tunnelling method is a useful technique to calculate black hole temperature and understand black hole thermodynamics. I will investigate the black hole tunnelling method in detail. I will compare two different approaches used to calculate black hole tunnelling. The tunnelling method can be applied to a broad range of spacetimes and I will show this explicitly in order to demonstrate the robustness of the tunnelling technique. In particular, I will apply the tunnelling method to spacetimes including: Rindler (the method can recover the Unruh temperature), and more general spacetimes (such as Kerr-Newman and Taub-NUT). I will also discuss the 5d Kerr-Gödel spacetimes in detail (while showing a previous unobserved property of these spaces). Once the parameter space of Kerr-Gödel is understood in detail, I will show how the tunnelling method can also be successfully applied to the Kerr-Gödel black hole. Finally, the key result of my thesis involves extending the tunnelling method to model fermion emission. The previous tunnelling calculations all involved the emission of scalar particles. I will model the emission of spin-1/2 fermions from various spacetimes including the Rindler spacetime and general non-rotating black holes. I will also model the emission of charged spin-1/2 fermions from the Kerr-Newman spacetime to show that the method is also applicable to rotating spacetimes. In all these cases I show that the correct Hawking temperature (Unruh temperature in the case of Rindler) is recovered for spin-1/2 fermion emission. Although this final result is not surprising, it is an important result because it confirms that Dirac particles will radiate from the black hole at the same temperature as scalar particles. It has always been assumed that this is the case but there is very little literature involving fermion radiation of black holes. So the results of my calculations are twofold, I demonstrate that Dirac particles are emitted at the same temperature as scalar particles from a black hole and it shows how robust the semi-classical tunnelling technique is.

<p>The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist infinitely many spacetimes, not necessarily spherically symmetric or static, that admit smooth global solutions to inverse mean curvature vector flow. Prior to our work, such solutions were only known in spherically symmetric and static spacetimes. The technique used in this thesis might be important to prove the Spacetime Penrose Conjecture, which remains open today. </p><p>Given a spacetime $(N^{4}, \gbar)$ and a spacelike hypersurface $M$. For any closed surface $\Sigma$ embedded in $M$ satisfying some natural conditions, one can ``steer'' the spacetime metric $\gbar$ such that the mean curvature vector field of $\Sigma$ becomes tangential to $M$ while keeping the induced metric on $M$. This can be used to construct more examples of smooth solutions to inverse mean curvature vector flow from smooth solutions to inverse mean curvature flow in a spacelike hypersurface.</p> / Dissertation

An investigation of information loss in black hole spacetimes is performed. We demon- strate that the definition of particles as energy levels of the Harmonic oscillator will not have physical significance in general and is thus not a good instrument to study the ra- diation of black holes. This is due to the ambiguity of the choice of coordinates on the phase space of the quantum field. We demonstrate how to identify quantum states in the functional Schr ̈dinger picture. o We demonstrate that information is truly lost in the case of a Vaidya black hole (a black hole formed from null dust) if we neglect back reaction. This is done by quantizing the constrained classical system of a Klein-Gordon field in a Vaidya background. The interaction picture of quantum mechanics can be applied to this system. We find a physically well motivated vacuum state for a spherically symmetric space- time with an extra conformal Killing vector. We also demonstrate how to calculate the response of a particle detector in the a LeMaitre-Tolman-Bondi spacetime with a self- similarity. Finally, some of the claims and confusion surrounding Unruh radiation, Hawking radiation and the equivalence principle are investigated and shown to be false.