Relativité Générale (2026) – Séance 2a

This is the second session of a master’s level course on general relativity, taught by Etienne Parizot at Université Paris Cité. The session focuses on the conceptual foundations of spacetime geometry. The professor begins by revisiting the idea that a reference frame is just a choice of coordinates in spacetime, and that the law of inertia corresponds to straight lines in spacetime. He then poses the question: what is a straight line? He discusses two possible definitions: the shortest path (requiring a notion of distance) and the path that keeps a constant direction (requiring a notion of direction). In Euclidean geometry, these coincide, but in general relativity they may not. He emphasizes that in a general curved spacetime, there is no global affine structure, so vectors cannot be defined between distant points. He distinguishes between ‘space vectors’ (connecting two points) and ‘vectors at a point’ (like velocity and force), which belong to different vector spaces. He explains that in a curved space, the concept of a vector connecting two points is not well-defined, and that physics must be reformulated without relying on such vectors. He introduces the idea that differentiability is a key assumption in physics, and that it will be preserved, but the notion of a vector field will need to be constructed locally. The session sets the stage for introducing differential geometry and the concept of geodesics.