Abstract
Simply-connected homogeneous spacetimes for kinematical and aristotelian Lie algebras (with space isotropy) have recently been classified in all dimensions. In this paper, we continue the study of these “maximally symmetric” spacetimes by investigating their local geometry. For each such spacetime and relative to exponential coordinates, we calculate the (infinitesimal) action of the kinematical symmetries, paying particular attention to the action of the boosts, showing in almost all cases that they act with generic non-compact orbits. We also calculate the soldering form, the associated vielbein and any invariant aristotelian, galilean or carrollian structures. The (conformal) symmetries of the galilean and carrollian structures we determine are typically infinite-dimensional and reminiscent of BMS Lie algebras. We also determine the space of invariant affine connections on each homogeneous spacetime and work out their torsion and curvature.
Highlights
Half a century ago, Bacry and Levy-Leblond [1] asked what were the possible kinematics
They provided an answer to this question by classifying kinematical Lie algebras in 3 + 1 dimensions subject to the assumptions of invariance under parity and time-reversal. They showed that the kinematical Lie algebras in their classification could be related by contractions. They observed that each such Lie algebra acts transitively on some (3 + 1)-dimensional spatially isotropic homogeneous spacetime and that the contractions could be interpreted as geometric limits of the corresponding spacetimes
The de Sitter spacetime is important for cosmology, the anti de Sitter spacetime currently drives much of our understanding of quantum gravity due to the AdS/CFT correspondence [2], and, in the limit where the cosmological constant goes to zero, Minkowski spacetime is fundamental in particle physics
Summary
Bacry and Levy-Leblond [1] asked what were the possible kinematics. Based on a recent deformation-theoretic classification of kinematical Lie algebras [19,20,21], we revisited this problem and in [6] classified and showed the existence of -connected spatially isotropic homogeneous spacetimes in arbitrary dimension, making en passant a small correction to the (3 + 1)-dimensional classification in [18] Another novel aspect of [6] was the classification of aristotelian spacetimes, which lack boost symmetry. We will prove that the boosts do act with (generic) non-compact orbits in all spacetimes with the unsurprising exceptions of the aristotelian spacetimes (which have no boosts) and the riemannian symmetric spaces, where the “boosts” are rotations.3 To those ends we introduce exponential coordinates for each of the spacetimes in [6], relative to which we write down the fundamental vector fields which generate the action of the transitive Lie algebra. The paper contains two appendices: in appendix A we discuss the carrollian and galilean spacetimes in terms of modified exponential coordinates, which are the most convenient coordinates in order to discuss their symmetries, and in appendix B we record for convenience the Lie algebras of conformal Killing vectors on low-dimensional maximally symmetric riemannian manifolds
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