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

Notre thèse étudie la relation entre les mécaniques classique et relativiste. Il est généralement supposé, à partir de l’hypothèse des petites vitesses, que la mécanique classique correspond à la mécanique relativiste dans les cas où la vitesse des objets est petite par rapport à la vitesse de la lumière. Cette position nous semble inadéquate pour la simple raison que la mécanique classique ne peut être restreinte au seule domaine des petites vitesses. Nous proposons l’hypothèse que les deux mécaniques ont une structure commune et que chacune se distingue sous certaines conditions. Pour appuyer cette hypothèse, nous proposons une axiomatisation de la mécanique suffisamment générale pour servir de structure commune aux mécaniques classique et relativiste. Cette axiomatisation comporte une théorie de la relativité qui précise comment les quantités relatives sont reliées entre elles lorsque déterminées par rapport à différents référentiels, et les lois du mouvement qui précisent comment les forces exercées sur un objet détermine son mouvement. Cette mécanique générale est déterminée à deux constantes près et c’est en déterminant la valeur de ces constantes qu’apparaît le bris de la structure commune qui génère la mécanique classique d’une part et la mécanique relativiste d’autre part. / Our thesis studies the relationship between classical and relativistic mechanics. It is generally assumed, based on the assumption of small velocities, that classical mechanics corresponds to relativistic mechanics in cases where the speed of objects is small compared to the speed of light. This position seems inadequate to us, for the simple reason that classical mechanics cannot be restricted to the realm of small velocities alone. We propose the hypothesis that the two mechanics have a common structure, and that each can be distinguished under certain conditions. To support this hypothesis, we propose an axiomatization of mechanics that is sufficiently general to serve as a common structure for both classical and relativistic mechanics. This axiomatization includes a theory of relativity that specifies how relative quantities are related to each other when determined with respect to different reference frames, and laws of motion that specify how forces exerted on an object determine its motion. This general mechanics is determined to within two constants, and it is by determining the value of these constants that the common structure that generates classical mechanics on the one hand and relativistic mechanics on the other is broken down.

Acompanha: Produto educacional: uma sequência didática para ensinar relatividade restrita no ensino médio com o uso de TIC / FUNPESQ / A Sequência Didática proposta neste trabalho contempla as recomendações dos Parâmetros Curriculares Nacionais (PCN), no sentido de uma progressiva inserção da Física do século XX no ensino médio, tema este que, infelizmente, ainda é muito pouco abordado nas salas de aula do país. O uso das Tecnologias de Informação e Comunicação (TIC) na educação pode proporcionar processos de educação mais participativos, tornando a relação professor‐aluno mais aberta e interativa. Com este intuito, o presente trabalho apresenta a aplicação de uma sequência didática, incluindo suas análises e resultados em sala de aula, para o ensino da Relatividade Restrita aos alunos do ensino médio de um Colégio da rede particular de ensino, no município de Apucarana-Pr. A proposta tem como base teórica a Aprendizagem Significativa de David Ausubel. A sequência em questão faz uso sistemático de recursos computacionais, com destaque para uma planilha original desenvolvida em Excel, com o propósito de melhorar o entendimento de conceitos básicos da Relatividade Restrita, bem como para a dedução de relações matemáticas fundamentais que são decorrência lógica dos postulados de Einstein. Além deste recurso, também foi utilizada a História da Ciência para contextualizar os alunos acerca do momento histórico em que o mundo estava inserido na época que a Teoria da Relatividade Restrita foi desenvolvida. Com isto, os alunos puderam ainda aprender um pouco sobre a vida, a obra e a contribuição deixada por Albert Einstein. Ainda foi utilizado um trecho do filme Interestelar que aborda a viagem no tempo, além de simuladores computacionais. Todos os recursos foram aplicados buscando aproximar a tecnologia (bem conhecida dos jovens) com a sala de aula, com o intuito de melhorar o processo de ensino-aprendizagem. Nossos resultados apontam que a utilização das TIC neste processo propiciam uma aula interativa e dinâmica tanto para o aluno quanto para o professor. Sendo assim, podemos dizer que a proposta aplicada possui um grande potencial pedagógico, pois os alunos, no decorrer da aplicação, mostraram-se mais motivados para as aulas de física. Acreditamos que esse seja o primeiro passo para que a aprendizagem ocorra de forma significativa. / The Didactic sequence proposed in this paper considers the recommendations of the National Curriculum Parameters (PCN), towards a gradual integration of twentiethcentury physics in high school, a topic that unfortunately is still very little explored in the country's classrooms. The use of Information and Communication Technologies (ICT) in education can provide more participatory education process, making it more open and interactive teacher-student relationship. To this end, this work presents the application of a didactic sequence, including its analysis and results in the classroom for teaching relativity to high school students of the College of private schools in the municipality of Apucarana- Pr. The proposal is based on theoretical Meaningful Learning of David Ausubel. The sequence in question is systematic use of computational resources, especially an original spreadsheet developed in Excel, in order to improve understanding of basic concepts of relativity, as well as the deduction of fundamental mathematical relationships that are logical consequence of the postulates Einstein. In addition to this feature, it was also used the History of Science to contextualize students about the historical moment in which the world was inserted at the time that the theory of relativity was developed. With this, students could still learn a little about life, work and contribution left by Albert Einstein. Interstellar a movie part that deals with time travel, and computer simulators also was used. All resources were applied seeking to approach the technology (well-known youth) in the classroom, in order to improve the teachinglearning process. Our results indicate that the use of ICT in this process provide an interactive lesson and dynamic for both the student and the teacher. Thus, we can say that the proposal applied has a great pedagogical potential as students, during the application, were more motivated to physics classes. We believe this is the first step in learning to occur significantly.

Dans cette thèse je me suis intéressé au caractère relativiste de matériaux topologiques tridimensionnels : les semi-métaux de Weyl et les isolants topologiques. Après une introduction aux états de surfaces et aux matériaux topologiques, je discute leurs propriétés de covariance sous les rotations trigonométriques et hyperboliques. Ces transformations me permettent de traiter les équations du mouvement d'un électron dans un champ magnétique ou à la surface, sous l'influence d'un champ électrique ou d'une inclinaison de la relation de dispersion. En première partie, je l'illustre dans le cas de la réponse magnéto-optique des semi-métaux de Weyl, en présence d'une inclinaison. Ces calculs sont en lien avec ma collaboration avec les expérimentateurs du LNCMI à Grenoble pour la caractérisation de la structure de bande de Cd₃As₂ où l'on montre que ce matériau est un semi-métal de Kane et non un semi-métal de Dirac dans la gamme de potentiels chimiques expérimentalement accessible. L'autre partie de cette thèse porte sur les états de surface des isolants topologiques où l'on montre qu'il existe des états de surface massifs au-delà de l'état de surface chiral. Ces états semblent avoir été observés par des études en ARPES d'échantillons de Bi₂Se₃ et Bi₂Te₃ oxydés et par des mesures de transport sur HgTe déformé. J'ai ainsi eu l'occasion de travailler avec les expérimentateurs du LPA à Paris sur le comportement des états de surface de HgTe sous forts effets de champ. Je termine par une discussion des états à l'interface entre un semi-métal de Weyl et un isolant dans le cas où le gap de ce dernier est suffisamment petit pour observer l'effet d'un champ magnétique et d'une inclinaison de la relation de dispersion sur les états de surface. / During my PhD studies I focused on the relativistic properties of threedimensional topological materials, namely Weyl semimetals and topological insulators. After introducing surface states and topological materials I discuss their covariance in trigonometric and hyperbolic rotations. These transformations help to solve the equations of motion of an electron in a magnetic field or at the surface with an applied electric field or with a tilt in the band dispersion. In a first place, I illustrate these transformations for the magneto-optical response of tilted Weyl semimetals. This work is related to my collaboration with experimentalists at LNCMI, Grenoble for characterizing the band structure of Cd₃As₂ where we show that this material is a Kane semi-metal instead of a Dirac semi-metal in the experimentally accessible range of chemical doping. The other part of this thesis is concerned with the surface states of topological insulators. I show that massive surface states can also exist in addition to the chiral surface state due to band inversion. Such states may have already been observed in ARPES measurement of oxidized Bi₂Se₃ and Bi₂Te₃ and in transport measurement of strained bulk HgTe. I show the work we performed with experimentalists at LPA, Paris on the behavior of HgTe surface states for strong field effects. Finally, I discuss the states at the interface of a Weyl semimetal and a small gap insulator. In this situation, an applied magnetic field or the tilt of the band dispersion can strongly affect the observed surface states.

Nous nous intéressons dans cette thèse à un aspect assez peu documenté de l'histoire de la théorie de la relativité restreinte, la recherche d'une vérification expérimentale de ses prédictions sur la variation de l'inertie en fonction de leur vitesse. Nous complétons les études historiques antérieures sur les expériences de Kaufmann (1906) et de Bucherer (1908), et montrons que la vérification de la formule de Lorentz-Einstein constitue encore un enjeu expérimental après 1911. Nous étudions plus particulièrement les recherches dirigées par Charles-Eugène Guye en collaboration avec ses étudiants Simon Ratnowsky (1907-1910) et Charles Lavanchy (1913-1915). Nous montrons que la seconde phase de ce travail est très largement considérée dans les années 1920 comme la vérifiation expérimentale la plus précise de la formule de Lorentz-Einstein. Nous utilisons la méthode de réplication, appliquée à l'expérience de Guye et Lavanchy. La très grande maîtrise de l'émission cathodique, associée à une méthode d'investigation spécifique, leur permet de surmonter toutes les difficultés identifiées alors comme préjudiciables au succès de ce type d'expérience. / We focus on an aspect of the history of special relativity theory that has not received much attention yet, namely the search for an experimental verification of the relativistic velocity dependency of inertia. Former historical studies on Kaufmann's 1906 and Bucherer's 1908 experiments are pursued. It is shown that verifying Lorentz-Einstein's formula is still a challenge after 1911. We concentrate here on Charles-Eugène Guye's works with students Simon Ratnowsky (1907-1910) and Charles Lavanchy (1913-1915). The second experiment is shown to be considered as the most precise verification of Lorentz-Einstein's formula by a large number of 1920s physicists. The replication method is used to probe Guye and Lavanchy's experiment. They are able to solve every then known difficulty in successfully performing such an experiment, through a great mastership of the crucial issue of cathode ray emission, together with a specific investigation method.