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Hermann Minkowski Et La Mathématisation De La Théorie De La Relativité Restreinte 1905-1915

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Hermann Minkowski Et La Mathématisation De La Théorie De La Relativité Restreinte 1905-1915 THÈSE présentée à L’UNIVERSITÉ DE PARIS VII pour obtenir le grade de DOCTEUR Spécialité : Épistémologie et Histoire des Sciences par Scott A. WALTER HERMANN MINKOWSKI ET LA MATHÉMATISATION DE LA THÉORIE DE LA RELATIVITÉ RESTREINTE 1905-1915 Thèse dirigée par M. Christian Houzel et soutenue le 20 décembre 1996 devant la Commission d’Examen composée de : Examinateur : M. Christian Houzel Professeur à l’Université de Paris VII Examinateur : M. Arthur I. Miller Professeur à UniversityCollege London Rapporteur : M. Michel Paty Directeur de recherche, CNRS Rapporteur : M. Jim Ritter Maître de conférences à l’Univ. de Paris VIII S. Walter LA MATHÉMATISATION DE LA RELATIVITÉ RESTREINTE i Résumé Au début du vingtième siècle émergeait l’un des produits les plus remarquables de la physique théorique : la théorie de la relativité. Prise dans son contexte à la fois intellectuel et institution- nel, elle est l’objet central de la dissertation. Toutefois, seul un aspect de cette histoire est abordé de façon continue, à savoir le rôle des mathématiciens dans sa découverte, sa diffusion, sa ré- ception et son développement. Les contributions d’un mathématicien en particulier, Hermann Minkowski, sont étudiées de près, car c’est lui qui trouva la forme mathématique permettant les développements les plus importants, du point de vue des théoriciens de l’époque. Le sujet de la thèse est abordé selon deux axes; l’un se fonde sur l’analyse comparative des documents, l’autre sur l’étude bibliométrique. De cette façon, les conclusions de la première démarche se trouvent encadrées par les résultats de l’analyse globale des données bibliographiques. Les trois premiers chapitres portent chacun sur un aspect différent du travail de Minkow- ski. Le premier chapitre aborde la redescription des origines du principe de relativité annoncée lors de la conférence faite par Minkowski à Cologne en 1908. Cette redescription se comprend comme une expansion des frontières disciplinaires des mathématiques; nous la mettons en rap- port avec l’augmentation fulgurante du nombre d’articles consacrés à la théorie de la relativité à partir de 1909. Une conséquence de la redescription minkowskienne des origines du prin- cipe de relativité a été de fixer l’attention des théoriciens sur le rôle des mathématiques dans la construction des théories physiques. La réussite de sa théorie de l’espace-temps n’était pas due uniquement à la redescription du principe de relativité et aux revendications de priorité par rapport aux théories de Lorentz et d’Einstein ; certains éléments de la théorie minkowskienne plaisaient aussi bien aux physiciens qu’aux mathématiciens. L’acceptation chez les physiciens théoriciens de la théorie de l’espace-temps a été aidée surtout par Arnold Sommerfeld, qui en publia une forme vectorielle. Le deuxième chapitre concerne ce que nous regardons comme une spécialisation mathé- matique dans le domaine des recherches relativistes, à savoir l’application de la géométrie non euclidienne à la relativité minkowskienne. Nous en examinons quelques exemples dans la pé- riode qui va de 1907, lorsque la théorie minkowskienne vit le jour, à 1912, quand Einstein reconnut que la géométrie d’un disque en rotation uniforme n’était pas euclidienne. L’étude des manuscrits de Minkowski jette une lumière nouvelle sur la chronologie de sa découverte de la structure de l’espace-temps et ses rapports avec les lois de l’électrodynamique et de la méca- nique relativiste. Nous montrons, par exemple, que la notion fondamentale de ligne d’Univers a été ignorée de Minkowski jusqu’au mois de novembre 1907. Sa découverte de la géométrie non euclidienne des vecteurs de vitesse est mise en relation avec le développement ultérieur de l’analyse vectorielle à quatre dimensions, qui a gagné la confiance des plus habiles des théori- ciens. Enfin, nous montrons un exemple de la puissance de la démarche minkowskienne chez le mathématicien Theodor Kaluza. Publié en 1910, le travail de Kaluza porta sur le problème du disque tournant dans l’espace-temps à trois dimensions; nous voyons comment il arrivait à quelques-uns des aperçus fondamentaux de la théorie de la relativité générale (cinq ans avant la découverte des équations du champ de gravitation), sans référence aux forces de gravitation ou au principe d’équivalence. À partir de la fréquence à laquelle la géométrie non euclidienne se rencontrait dans les recherches relativistes, nous suggérons que si Einstein basa sa théorie de la relativité générale sur un tenseur métrique riemannien il le fit en réponse–au moins en partie–aux problèmes rencontrés lors de l’élaboration de la notion de mouvement rigide dans l’espace-temps. Dans le troisième chapitre, nous regardons un autre aspect du même sujet, à savoir l’in- fluence de la philosophie de la géométrie de Henri Poincaré sur la réception de la relativité S. Walter LA MATHÉMATISATION DE LA RELATIVITÉ RESTREINTE ii minkowskienne. Nous montrons que cette doctrine philosophique n’a pas convaincu quelques- uns des géomètre les plus éminents au début du XXe siècle. Les sources sociales possibles de ce rejet sont examinées brièvement. En ce qui concerne la théorie de l’espace-temps, le fait que certains savants y voyaient une confirmation des vues de Helmholtz a pu favoriser sa réception. La doctrine conventionnaliste a pu la gêner, comme on le voit dans deux cas. D’une façon gé- nérale, les découvertes des théories de la relativité d’Einstein (1905) et de Minkowski (1908) n’ont pu donner qu’une confirmation supplémentaire aux opinions sur le fondement physique de la géométrie tenues depuis longtemps par plusieurs géomètres. Le quatrième chapitre présente une étude bibliométrique des publications sur la théorie de la relativité restreinte sorties entre 1905 et 1915. Les références de sept cents articles et plus de cent livres sont classées selon la discipline de l’auteur et son lieu de travail. Sont également pris en compte le type de journal et le pays de publication. L’étude concerne l’ensemble des revues publiées dans les langues de l’Europe occidentale. L’analyse des données montre qu’en général, le nombre de contributions par discipline était comparable entre les mathématiques, la physique théorique, et la physique non théorique. Déjà en 1911, l’engagement des mathé- maticiens universitaires dans les recherches relativistes en Allemagne égalait celui de leurs collègues physiciens non théoriciens, alors que l’engagement des physiciens théoriciens dé- passait largement celui de ces deux groupes. Deux ans plus tard, davantage de mathématiciens écrivaient des articles sur la relativité restreinte que de physiciens théoriciens ou de physiciens non théoriciens. A la suite du quatrième chapitre se trouvent quelques observations sur le rôle de Minkowski et des mathématiciens de son époque dans l’histoire de la théorie de la relativité. Abstract The first decade of the twentieth century saw the emergence of what is generally viewed to be one of its crowning theoretical artifacts: the special theory of relativity. This theory, embedded in a web of intellectual and institutional history, constitutes the locus of my research. One aspect of this history forms the central theme of the thesis: the role of mathematicians in the discovery, scientific reception and development of the theory of relativity. The work of one mathematician in particular, Hermann Minkowski, is examined in detail, since it was the form provided by his space-time theory that led to the most important developments of the theory of relativity, as viewed by contemporary theorists. The approach to the subject is bilateral, combining textual analysis with a bibliometric study, such that the conclusions drawn from case studies are placed in the context of a global analysis of publication data. In the first three chapters of the thesis, separate topics relating to Minkowski’s work on the theory of relativity are addressed. In the first chapter, the vociferous disciplinary advo- cacy of Hermann Minkowski’s famous lecture on space and time is discussed, and correlated to the rapid increase in publications on the theory of relativity. We show in particular how Minkowski’s lecture effectively expanded the disciplinary frontier of mathematics to include the principle of relativity, thereby paving the way for mathematical engagement with the theory of relativity. An important consequence of this view was the heightened sense among physicists and mathematicians of the role of mathematics in the construction of physical theories. The transdisciplinary success of the space-time theory, it is argued, can not be attributed solely to Minkowski’s redescription of the origins of the principle of relativity and his claim for the tech- nical superiority of his view over those of Lorentz and Einstein. Some features of Minkowskian relativity, for example, the “absolute” nature of space-time, appealed to physicists as well as to S. Walter LA MATHÉMATISATION DE LA RELATIVITÉ RESTREINTE iii mathematicians. Acceptance by physicists of formal elements of Minkowskian relativity was further enhanced by their recasting in vectorial form by Arnold Sommerfeld and others. Interpretations of the theory of relativity are found to be particularly well correlated with discipline in applications invoking non-Euclidean geometry, as I show in the second chapter. This field is studied in detail for the period from 1907, when Minkowskian relativity appeared, until 1912, when Einstein recognized that the geometry of a disk in uniform rotation was not Euclidean. A close study of Minkowski’s manuscripts and published work throws new light on the chronology of his discovery of the structure of space-time and its relation to the laws of electrodynamics and mechanics. One surprising result of this analysis is that the basic notion of a worldline in spacetime was unknown to Minkowski before November, 1907. The con- sequences of Minkowski’s recognition of the non-Euclidean geometry of velocity vectors are investigated, and related to the subsequent developments of four-dimensional vector analysis, which won the favor of leading theorists.
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