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A special relativity based on the de Sitter group is introduced, which is the theory that might hold up in the presence of a non-vanishing cosmological constant. Like ordinary special relativity, it retains the quotient character of spacetime, and a notion of homogeneity. As a consequence, the underlying spacetime will be a de Sitter spacetime, whose associated kinematics will differ from that of ordinary special relativity. The corresponding modified notions of energy and momentum are obtained, and the exact relationship between them, which is invariant under a re-scaling of the involved quantities, explicitly exhibited. Since the de Sitter group can be considered a particular deformation of the Poincaré group, this theory turns out to be a specific kind of deformed (or doubly) special relativity. Some experimental consequences, as well as the causal structure of spacetime-modified by the presence of the de Sitter horizon-are briefly discussed.

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Recent data on supernovae favor high values of the cosmological constant. Spacetimes with a cosmological constant have non-relativistic kinematics quite different from Galilean kinematics. De Sitter spacetimes, vacuum solutions of Einstein's equations with a cosmological constant, reduce in the non-relativistic limit to Newton-Hooke spacetimes, which are non-metric homogeneous spacetimes with non-vanishing curvature. The whole nonrelativistic kinematics would then be modified, with possible consequences to cosmology, and in particular to the missing-mass problem.(flat, vacuum) solution of Einstein's equations, Minkowski spacetime M is a true, even paradigmatic spacetime. It is taken as the local, "kinematical" spacetime and can also be identified with the space tangent to (real, curved) spacetime at each point. Above all, it has a consistent kinematics, in the sense that its metric is invariant under the appropriate kinematical (Poincaré) group P . This group contains the Lorentz group L = SO(3, 1) and includes the translation subgroup T , which acts transitively on M and is, in a sense, its "double". Indeed, Minkowski spacetime appears as a homogeneous space under P , actually as the quotient M = T = P/L. If we prefer, the manifold of P is a principal bundle (P/L, L) with T = M as base space and L as the typical fiber.The invariance of M under the transformations of P reflects its uniformity. Also in this "Copernican" aspect, Minkowski spacetime establishes a paradigm. P has the maximum possible number of Killing vectors, which is ten for a 4-dimensional flat spacetime. The Lorentz subgroup provides an isotropy around a given point of M, and the translation invariance enforces this isotropy around any other point. This is the meaning of "uniformity": all the points of spacetime are ultimately equivalent.The reduction of relativistic to Galilean kinematics in the non-relativistic limit is the standard example of Inönü-Wigner contraction, 3 by which the Poincaré group is contracted to the Galilei group. However, if we insist on the central role of the metric, there is no such a thing as a real "Galilean spacetime". The original metric is somehow "lost" in the process of contraction, and no metric exists which is invariant under the Galilei group. Minkowski spacetime tends, in the non-relativistic limit, to something that is not a spacetime. Nevertheless, there exists a meaningful connection which survives, even though the metric becomes undefined. This is not easily visible in the Minkowski-Galilei case, because both the initial and the final connections are flat.Actually, in all local, or tangential physics, what happens is that the laws of Physics are invariant under transformations related to an uniformity as that described above. It includes homogeneity of space and of time, isotropy of space and the equivalence of inertial frames. This holds for Galilean and for special-relativistic physics, their difference being grounded only in their different "kinematical groups". However, as was clearly shown by Bacry an...