The principle of relativity and the special relativity triple

Abstract

Based on the principle of relativity and the postulate on universal invariant constants (c, l) as well as Einstein's isotropy conditions, three kinds of special relativity form a triple with a common Lorentz group as isotropy group under full Umov-WeylFock-Lorentz transformations among inertial motions.

Geometries with the Second Poincaré Symmetry

“…However, the second Poincaré algebra is the different realization of iso (1,3) from the ordinary realization. The second Poincaré algebra generates the second Poincaré transformations 6) where b µ are dimensionless parameters, which can be expressed again in terms of 5 × 5 matrix…”

“…3 To be distinguished from the ordinary time and space translation generators H and P , H ′ and P ′ are called the pseudo-time-and pseudo-space-translation generators because they cannot generate time or space translation in Minkowski space-time.…”

“…Theorem 1 There is no tensor field g = g µν dx µ ⊗ dx ν with the following three conditions satisfied simultaneously: (1) g is smooth; (2) g is non-degenerate everywhere; (3) g is invariant under the p 2 -translations.…”

“…The third purpose of the paper is to show that there exists the geometrical structure which satisfies all three assumptions in [1] and the P oR c,l in [3,4] even after re-construction of the algebra in terms of new space and time coordinates.…”

“…Recently, it is pointed out that the triality of special relativity with Poincaré, de Sitter, anti-de Sitter invariance, respectively, can be set up based on the P oR and the postulate on two universal invariant constants c of speed dimension and l of length dimension denoted as the P oR c,l [3,4]. It is also found in [3,4] that there is another realization of Poincaré group being called the second Poincaré group and denoted as P 2 , with the corresponding realization of algebra being denoted as p 2 . 1 Unlike the ordinary Poincaré transformation under which the metric of Minkowski space-time is invariant, the second Poincaré transformations do not generate the automorphism of the Minkowski space-time.…”