The possible kinematics and their corresponding geometries were once regarded as an already-solved problem. The de Sitter relativity research group formed by researchers from Chinese Academy of Sciences, Tsinghua University, and Beijing Normal University, restudied the problem and showed that additional, previously unknown realizations exist of possible kinematical algebras, each of which has so(3) isotropy and a ten-generators symmetry group. They presented these geometries corresponding to all these realizations and provided a classification in an article, entitled "Geometries for Possible Kinematics", published in the 2012 10th issue of SCIENCE CHINA.
In the 1960s Bacry and Lévy-Leblond established connections among eleven kinematical algebras of eight types. Each kinematical algebra was supposed to possess (i) an so(3) isotropy, (ii) parity and time-reversal automorphisms, and (iii) a non-compact one-dimensional subgroup generated by each boost. For a long time, it was widely accepted that the accounting for all kinematical algebras satisfying those three conditions had been exhausted. Two years ago, the de Sitter relativity research group showed by using linear combinations of generators that there are 24 kinematical algebras if the third condition is relaxed, and all of those kinematical algebras are subalgebras of a 4-dimensional "inertial motion algebra".
This shows the contraction scheme for the possible kinematics in a more symmetric way. (Photo Credit: ©Science China Press)
In Ref. [1], it was shown that, with the exception of two static algebras, the 22 possible kinematical algebras with so(3) isotropy can be obtained by the Inönü-Wigner contraction from the Riemann, Lobachevsky, de Sitter, and anti-de Sitter algebras (r, l, d±), respectively. The existence of more possible kinematical algebras than obtained by Bacry and Lévy-Leblond arises from taking different realizations of the generators and then performing the contraction under two opposite limits. For example, it is well known that the de Sitter and anti-de Sitter algebras contract to the Poincaré algebra (p) when a certain length parameter tends to infinity. What was overlooked was that when the length parameter tends to zero, [3] these algebras contract to other realizations of the Poincaré algebra, called the second Poincaré algebras and denoted p2±, for brevity. Although these are isomorphic, p and p2± have very different geometrical significance. The 22 possible kinematical algebras are related, as depicted in figure 1.This shows the contraction scheme for the geometries for he possible kinematics. (Photo Credit: ©Science China Press)
The explicit geometric structures show that the requirement that transformations generated by boosts in any given direction form a noncompact subgroup does not guarantee a geometry having Lorentz-like signature. Some geometries satisfying that requirement possess Euclidean signature, whereas others violating that requirement possess Lorentz-like signature. In addition, before the geometry is presented, the isotropy (rotational invariance) of the space is expressed by an so(3) subalgebra. However, although many geometries are invariant under transformations generated by so(3), they might not have spatial isotropy with respect to each point in the manifold.
This shows the geometries for the genuine possible kinematics. (Photo Credit: ©Science China Press)
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