Dutailly

Actually in Relativity there are 2 different theories : one addresses the geometry of the universe, the other one gravitation. The General relativity Geometry is based on simple facts. i) To locate an event you need 4 coordinates, in a chart (whatever it is, it is conventional) : 2 for space location, 1 for time location. So the right representation of the universe is with a 4 dimensional manifold. For any observer the "space" corresponds to the event to which it has assigned a given time. This is a folliation of the universe in 3 dimensional hypersurfaces. ii) There is a breakdown of symmetry : time is not measured with the same units and protocols than space. There should be some universal constant relating both : x=ct. iii) The Principle is causality is the same for all observers. This implies the existence of a bilinear form - the metric - which is not euclidean, the Lorentz metric, which is the physical part of the geometry. It changes from a point to another. iv) The main property of material bodies - including the observers - is that they occupy a single point at any time. So they travel along a world line in the universe. The tangent to their world line is a quantity which has an absolute meaning : it does not depend on an observer. There is a unique (up to an origin) parameter t such that the world line x(t) has a given tangent. This is the proper time, which is also the biological time of any observer. The issue is then "at what rate clocks of different observers run ?" The basic assumption is that the 4 dimensional velocity of an observer (and any material body) is the same. A simple computation shows that the Lorentz length of the velocity is null, the spatial speed is such that v=ct. And from there we get the usual formulas, which hold for any observer using an orthonormal basis, without involving the speed of light, train or whatever. The assumption about the "expanding universe" involves only the metric : the container does not change, only the metric. This is an assumption about cosmology. And of course a material body, be it a particle or a star, does not change, only the geometric measures. The Einstein's theory of gravitation is distinct. It is based on 2 assumptions. i) Material bodies follow geodesics. So we need to define a geodesic, and this is done through the metric. ii) The metric is defined, through a classic implementation of the Principle of Least Action using the scalar curvature. So only the metric is involved. This is a beautiful, consistent, theory, but it does not account for the motion of stars in the Galaxy. The usual answer of lazzy physicists is to say that it requires a new mystery (good for the media) : dark matter. Actually the Einsitein's theory of gravitation denies the existence of a gravitational field. Einstein, and many others, tried to adjust the theory, but it needs one way or another to give up the central role given to the metric.