Space and Spacetime

This lecture comprises two parts. The first one presents a short analysis of the evolution of ideas about space and spacetime. In particular, it suggests an analogy between the introduction of homogeneous space by Newton, and the introduction of spacetime in special relativity. The second part provides, in the frame of general relativity, a prescription to define space, at a given moment, for an arbitrary observer in an arbitrary (sufficiently regular) curved spacetime. Based on synchronicity arguments, this prescription defines a foliation of spacetime, which provides a natural global reference frame (with space and time coordinates) for the observer, in spacetime, which remains Minkowskian along his world‐line. This definition remains valid in curved spacetime, and/or for non inertial observers. Application to Minkowski spacetime illustrates clearly the fact that different observers see different spaces. It allows, for instance, to define space everywhere without ambiguity, for the Langevin observer (involved in the Langevin pseudoparadox of twins). Applied to the Rindler observer (with uniform acceleration) it leads to the Rindler coordinates, whose choice is so justified with a physical basis. This allows to interpret the Unruh effect as due to the observer dependence of the space‐time splitting. Finally an application if given for a rotating observer in circular motion. We also apply this prescription in cosmology, to inertial observers in the Friedmann‐Lemaître models: space constructed in this manner differs from the hypersurfaces of homogeneity, which do not obey the simultaneity requirement. I work out two examples: the Einstein ‐ de Sitter model, in which space, for an inertial observer, is not flat nor homogeneous, and the de Sitter case.