On Special Relativity with Cosmological Constant

Abstract

Based on the principle of relativity and the postulate of invariant speed and length, we propose the theory of special relativity with cosmological constant ${\mathcal{S}\mathcal{R}}_{c,R}$ if the invariant length whose square is the inverse of the one-third cosmological constant of the universe. It is on the Beltrami-de Sitter spacetime ${\mathcal{B}}_R$ with de Sitter invariance. We define the observables of free particles and generalize famous Einstein's formula. We also define two kinds of simultaneity. The first is for local experiments and inertial motions. The second is for cosmological observations. Thus there is a relation between the relativity principle and the cosmological principle. We predict that the 3-d cosmic space is then of positive spatial curvature of order cosmological constant. The relation between ${\mathcal{S}\mathcal{R}}_{c,R}$ and the doubly special relativity is briefly disucssed.

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