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Spatially isotropic homogeneous spacetimes

José Figueroa-O'Farrill, Stefan Prohazka

TL;DR

The paper delivers a comprehensive geometrical realisation of all simply-connected homogeneous spacetimes arising from kinematical and aristotelian Lie groups across arbitrary spacetime dimensions. By separating algebraic classifications (Lie pairs) from their geometric realizations, it shows that the same Lie algebra can act on multiple spacetimes, and that some spacetimes lack invariant metrics yet retain maximal symmetry in other invariant structures. A key achievement is the construction of extensive tables and contraction diagrams linking Minkowski, (A)dS, Galilean, Carrollian, and novel torsional/aristotelian geometries, including several low-dimensional spacetimes unique to D≤2. The results illuminate how spacetimes interrelate through contractions and limits, with implications for holography beyond AdS/CFT and for effective field theories in Newton–Cartan and related geometries. Overall, the work provides a robust framework for exploring the landscape of homogeneous spacetimes beyond traditional maximally symmetric geometries and sets the stage for further geometric and dynamical investigations.

Abstract

We classify simply-connected homogeneous ($D+1$)-dimensional spacetimes for kinematical and aristotelian Lie groups with $D$-dimensional space isotropy for all $D\geq 0$. Besides well-known spacetimes like Minkowski and (anti) de Sitter we find several new classes of geometries, some of which exist only for $D=1,2$. These geometries share the same amount of symmetry (spatial rotations, boosts and spatio-temporal translations) as the maximally symmetric spacetimes, but unlike them they do not necessarily admit an invariant metric. We determine the possible limits between the spacetimes and interpret them in terms of contractions of the corresponding transitive Lie algebras. We investigate geometrical properties of the spacetimes such as whether they are reductive or symmetric as well as the existence of invariant structures (riemannian, lorentzian, galilean, carrollian, aristotelian) and, when appropriate, discuss the torsion and curvature of the canonical invariant connection as a means of characterising the different spacetimes.

Spatially isotropic homogeneous spacetimes

TL;DR

The paper delivers a comprehensive geometrical realisation of all simply-connected homogeneous spacetimes arising from kinematical and aristotelian Lie groups across arbitrary spacetime dimensions. By separating algebraic classifications (Lie pairs) from their geometric realizations, it shows that the same Lie algebra can act on multiple spacetimes, and that some spacetimes lack invariant metrics yet retain maximal symmetry in other invariant structures. A key achievement is the construction of extensive tables and contraction diagrams linking Minkowski, (A)dS, Galilean, Carrollian, and novel torsional/aristotelian geometries, including several low-dimensional spacetimes unique to D≤2. The results illuminate how spacetimes interrelate through contractions and limits, with implications for holography beyond AdS/CFT and for effective field theories in Newton–Cartan and related geometries. Overall, the work provides a robust framework for exploring the landscape of homogeneous spacetimes beyond traditional maximally symmetric geometries and sets the stage for further geometric and dynamical investigations.

Abstract

We classify simply-connected homogeneous ()-dimensional spacetimes for kinematical and aristotelian Lie groups with -dimensional space isotropy for all . Besides well-known spacetimes like Minkowski and (anti) de Sitter we find several new classes of geometries, some of which exist only for . These geometries share the same amount of symmetry (spatial rotations, boosts and spatio-temporal translations) as the maximally symmetric spacetimes, but unlike them they do not necessarily admit an invariant metric. We determine the possible limits between the spacetimes and interpret them in terms of contractions of the corresponding transitive Lie algebras. We investigate geometrical properties of the spacetimes such as whether they are reductive or symmetric as well as the existence of invariant structures (riemannian, lorentzian, galilean, carrollian, aristotelian) and, when appropriate, discuss the torsion and curvature of the canonical invariant connection as a means of characterising the different spacetimes.

Paper Structure

This paper contains 107 sections, 5 theorems, 194 equations, 6 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Motivation and contextualisation
  3. Overview of results
  4. Organisation of the paper
  5. Reader's guide
  6. Kinematical spacetimes
  7. Basic definitions
  8. Summary of main results
  9. Classification of kinematical Lie pairs
  10. Lie pairs for $D\geq 3$
  11. Lie pairs associated to Lie algebra #1
  12. Lie pairs associated to Lie algebra #2
  13. Lie pairs associated to Lie algebra #3$_\gamma$
  14. Lie pairs associated to Lie algebra #4
  15. Lie pairs associated to Lie algebra #5
  16. ...and 92 more sections

Key Result

Lemma 1

Let $M$ and $M'$ be homogeneous spaces of $\mathcal{G}$ and $\mathcal{G}'$, respectively. If $M$ and $M'$ are isomorphic, then so are any Lie pairs associated to $M$ and $M'$.

Figures (6)

  • Figure 1: Effect on light cone of ultra- (left) and non-relativistic (right) limits.
  • Figure 2: Maximally symmetric spaces and their limits: non-relativistic (vertical), ultra-relativistic (horizontal) and flat (diagonal).
  • Figure 3: Homogeneous spacetimes in dimension $D+1 \geq 4$ and their limits.
  • Figure 4: Three-dimensional homogeneous spacetimes and their limits.
  • Figure 5: Two-dimensional homogeneous spacetimes and their limits
  • ...and 1 more figures

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 1
  • proof
  • Definition 7
  • Lemma 2
  • ...and 7 more