La gravitation dans le cadre de la géométrie non-commutative
Institution:
Paris 11Disciplines:
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Abstract EN:
In this work several aspects of gravitation theory are examined at the light of non-commutative geometry. It is argued that quantum corrections of gravity might be calculated using this framework. As a first step the quantum plane equipped with a differential calculus is examined. By implementing a set of hermiticity conditions in conjunction with metric compatible constraints, the admissible linear connections and the related differential structure are determined and classified in several families according to the parameters involved in the solutions. Thereafter, we consider the case of classical solutions to general relativity with emphasis on kasner's space-time. For each one of the space-times discussed, a frame adapted to the metric is given and a set of duality conditions is deduced. A non-commutative algebra intended to reproduce the field equations of general relativity is proposed for each case. The commutation relations among the coordinates are found and the consistency of these expressions with the algebra is discussed in detail. We conclude with some general remarks on these results.
Abstract FR:
Dans ce travail divers aspects de la theorie de la gravitation sont examines dans le cadre de la geometrie non-commutative. Nous partons du principe que certaines corrections d'origine quantique a la gravite pourraient etre determinees au moyen de cet outil mathematique. Tout d'abord nous etudions le plan quantique muni d'un calcul differentiel. A l'aide des conditions d'hermiticite et de compatibilite entre la metrique et ce calcul differentiel, les connections lineaires possibles ainsi que leur structures differentielles associees sont decrites et classees par familles selon les parametres apparaissant dans les solutions. Nous reconsiderons ensuite des solutions classiques de la relativite generale, plus particulierment pour l'espace-temps de kasner. Pour chacun des espace-temps consideres, un repere adapte a la metrique est donne et un ensemble de relations de dualite en est deduit. Une algebre non-commutative visant a decrire les equations de champ de la relativite generale est proposee pour chaque cas considere. Les relations de commutation entre les coordonnees sont calculees et la coherence entre ces solutions et l'algebra est examinee en detail. Nous concluons avec une discussion generale de ces resultats.