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Geometries Presented by Chinese Scholars for Possible Space-time Kinematics
Mon, 11/26/2012 - 12:58pm

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 [1] .

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 [2], and all of those kinematical algebras are subalgebras of a 4-dimensional "inertial motion algebra".

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.

More importantly, a kinematic action should be established on a suitable geometry so that the geometry is invariant under the transformations generated by the kinematical algebra. In Ref. [1], the geometries for all 22 kinematical algebras were presented using the contraction technique. As arranged in figure 2, there are 45 different 4-dimensional geometries in total. Each geometry is defined on a portion of a 4-dimensional real projective manifold. Among these geometries, some are non-degenerate like the de Sitter and Minkowski geometries, but most are degenerate as for example for the Galilei and Carroll geometries. In these geometries, including the degenerate ones, some have Lorentzian signature that may serve as space-time geometries, and some have Euclidean signature that may serve as Euclidean versions of space-time. Moreover, there are geometries which have a (+, +, -, -)-signature and are interpreted as double-time geometries. Many of the geometries are related by the transformation t « 1/(n2t). When this transformation is regarded as a coordinate transformation, these geometries are not independent from the view of differential geometry.

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.

Therefore, the requirements suitable in identifying genuine possible kinematics have been revised to the following: (1) space is isotropic with respect to any point in the manifold; (2) space-reflection and time-reversal are automorphisms of the kinematical group; and (3) the geometry has Lorentz-like signature. According to those requirements, the genuine possible kinematics from the viewpoint of differential geometry correspond to three relativistic geometries, three absolute-time geometries and three absolute-space geometries