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Conformal transformations of spacetimes without observer horizons

Leonardo García-Heveling, Abdelghani Zeghib

TL;DR

The paper proves that for causal spacetimes without observer horizons, conformal transformations split into future-escaping, past-escaping, or non-escaping types, with escaping maps acting freely, properly, and cocompactly. It applies this dichotomy to the Einstein static universe, providing a detailed Jordan-type decomposition in the central extension $\widehat{\mathrm{O}}(2,n+1)$ and a complete classification of conformal transformations of $\widehat{\mathsf{Eins}}^{1,n}$, including a precise description of elliptic, hyperbolic, and parabolic elements and their combinations. The work then investigates essentiality, connecting Lorentzian Lichnerowicz-type conjectures to conformal flatness and to the Einstein static universe, proving that escaping elements are inessential and offering partial classifications of conformally flat NOH spacetimes with finite fundamental group. Overall, the results clarify the structure of Lorentzian conformal groups, reveal how causal boundary conditions shape that structure, and provide a coherent framework linking causal geometry, group actions, and conformal flatness in the Lorentzian setting.

Abstract

We prove that for a certain class of Lorentzian manifolds, namely causal spacetimes without observer horizons, conformal transformations can be classified into two types: escaping and non-escaping. This means that successive powers of a given conformal transformation will either send all points to infinity, or none. As an application, we classify the conformal transformations of Einstein's static universe. We also study the question of essentiality in this context, i.e. which conformal transformations are isometric for some metric in the conformal class.

Conformal transformations of spacetimes without observer horizons

TL;DR

The paper proves that for causal spacetimes without observer horizons, conformal transformations split into future-escaping, past-escaping, or non-escaping types, with escaping maps acting freely, properly, and cocompactly. It applies this dichotomy to the Einstein static universe, providing a detailed Jordan-type decomposition in the central extension and a complete classification of conformal transformations of , including a precise description of elliptic, hyperbolic, and parabolic elements and their combinations. The work then investigates essentiality, connecting Lorentzian Lichnerowicz-type conjectures to conformal flatness and to the Einstein static universe, proving that escaping elements are inessential and offering partial classifications of conformally flat NOH spacetimes with finite fundamental group. Overall, the results clarify the structure of Lorentzian conformal groups, reveal how causal boundary conditions shape that structure, and provide a coherent framework linking causal geometry, group actions, and conformal flatness in the Lorentzian setting.

Abstract

We prove that for a certain class of Lorentzian manifolds, namely causal spacetimes without observer horizons, conformal transformations can be classified into two types: escaping and non-escaping. This means that successive powers of a given conformal transformation will either send all points to infinity, or none. As an application, we classify the conformal transformations of Einstein's static universe. We also study the question of essentiality in this context, i.e. which conformal transformations are isometric for some metric in the conformal class.
Paper Structure (17 sections, 27 theorems, 40 equations, 2 figures)

This paper contains 17 sections, 27 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lorentzian geometry
  4. Group actions
  5. Escaping vs non-escaping conformal transformations
  6. The Einstein static universe
  7. Description of the Einstein universes
  8. Structure of $\mathrm{Conf}(\mathsf{Eins}^{1,n})$ and $\mathrm{Conf}(\widehat{\mathsf{Eins}}{}^{1,n})$
  9. Elliptic elements
  10. Hyperbolic and parabolic elements
  11. Mixed elements
  12. Conclusion
  13. Conformal transformations of $\widehat{\mathsf{Eins}}{}^{1,n}$ with a fixed point
  14. Essentiality
  15. The Lichnerowicz conjecture
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a causal spacetime satisfying the no observer horizons condition, and let $\phi \colon M \to M$ be a time-orientation preserving conformal transformation. Then, there are three mutually exclusive possibilities: In cases (i$^+$) and (i$^-$), the action of $\phi$ is free, proper, and cocompact.

Figures (2)

  • Figure 1: The points $p_0,p_1,p_2 \in \widehat{\mathsf{Eins}}{}^{1,1}$ as described in Proposition \ref{['prop:fixed']}. The spiral-shaped curves are null geodesics going through these points, and the shaded region is the diamond $I(p_0,p_2)$.
  • Figure 2: The flowlines of a time-translation of Minkowski $\mathbb{R}^{1,1}$ mapped onto a diamond $I(p_0,p_2)$ in $\widehat{\mathsf{Eins}}{}^{1,1}$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Theorem 2.2: GerBeSa1
  • Definition 2.3
  • Definition 2.4: Paeng
  • Lemma 2.5
  • Theorem 2.6
  • Proposition 2.7: Lee
  • ...and 50 more