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Maximality of the futures of points in globally hyperbolic maximal conformally flat spacetimes

Rym Smaï

TL;DR

This work identifies and analyzes domains of injectivity for the developing map of simply-connected globally hyperbolic conformally flat spacetimes, proving that IPs/IFs in the universal cover are injective domains and that diamonds serve as the fundamental injectivity domains. By establishing that futures/pasts of points and indecomposable sets are injective, it provides a streamlined proof of Rossi's completeness result and shows that, in the maximal ambient spacetime, IPs/IFs are themselves maximal and conformally equivalent to regular Minkowski domains. The approach hinges on a detailed study of diamonds, Penrose boundary structure, and the conformal compactification via the Einstein universe, including a careful treatment of the intersection properties of diamonds and the shadows on the Penrose boundary. Collectively, the results connect causal completion, maximality, and Minkowski-regular domains, offering a clear geometric characterization of IPs/IFs in non-elliptic GH conformally flat spacetimes and extending the understanding of their maximality and convexity properties.

Abstract

Let M be a globally hyperbolic conformally spacetime. We prove that the indecomposable past/future sets (abbrev. IPs/IFs) -in the sense of Penrose, Kronheimer and Geroch -of the universal cover of M are domains of injectivity of the developing map. This relies on the central observation that diamonds are domains of injectivity of the developing map. Using this, we provide a new proof of a result of completeness by C. Rossi, which notably simplifies the original arguments. Furthermore, we establish that if, in addition, M is maximal, the IPs/IFs are maximal as globally hyperbolic conformally flat spacetimes. More precisely, we show that they are conformally equivalent to regular domains of Minkowski spacetime as defined by F. Bonsante.

Maximality of the futures of points in globally hyperbolic maximal conformally flat spacetimes

TL;DR

This work identifies and analyzes domains of injectivity for the developing map of simply-connected globally hyperbolic conformally flat spacetimes, proving that IPs/IFs in the universal cover are injective domains and that diamonds serve as the fundamental injectivity domains. By establishing that futures/pasts of points and indecomposable sets are injective, it provides a streamlined proof of Rossi's completeness result and shows that, in the maximal ambient spacetime, IPs/IFs are themselves maximal and conformally equivalent to regular Minkowski domains. The approach hinges on a detailed study of diamonds, Penrose boundary structure, and the conformal compactification via the Einstein universe, including a careful treatment of the intersection properties of diamonds and the shadows on the Penrose boundary. Collectively, the results connect causal completion, maximality, and Minkowski-regular domains, offering a clear geometric characterization of IPs/IFs in non-elliptic GH conformally flat spacetimes and extending the understanding of their maximality and convexity properties.

Abstract

Let M be a globally hyperbolic conformally spacetime. We prove that the indecomposable past/future sets (abbrev. IPs/IFs) -in the sense of Penrose, Kronheimer and Geroch -of the universal cover of M are domains of injectivity of the developing map. This relies on the central observation that diamonds are domains of injectivity of the developing map. Using this, we provide a new proof of a result of completeness by C. Rossi, which notably simplifies the original arguments. Furthermore, we establish that if, in addition, M is maximal, the IPs/IFs are maximal as globally hyperbolic conformally flat spacetimes. More precisely, we show that they are conformally equivalent to regular domains of Minkowski spacetime as defined by F. Bonsante.
Paper Structure (37 sections, 36 theorems, 15 equations, 7 figures)

This paper contains 37 sections, 36 theorems, 15 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Conformally flat structures on globally hyperbolic spacetimes
  3. Completeness of GHM conformally flat spacetimes
  4. Domains of injectivity of the developing map
  5. A natural approach that fails: the canonical neighborhoods
  6. Futures/pasts of points in a non-elliptic GHM conformally flat spacetime
  7. Maximality of the IPs/IFs in a non-elliptic GHM conformally flat spacetime
  8. Conformal compactification of Minkowski spacetime
  9. Einstein universe
  10. Klein model
  11. Spatio-temporal decomposition
  12. Lightlike geodesics, lightcones and conformal spheres
  13. Universal Einstein universe
  14. Conformal group
  15. Affine charts
  16. ...and 22 more sections

Key Result

Lemma 1.1

The restriction of the developing map to any diamond $J(p,q)$ of $M$ is injective. Moreover, its image is exactly the diamond $J(D(p), D(q))$ of $\widetilde{Ein}_{1,n-1}$.

Figures (7)

  • Figure 1: Lightcone of a point $\mathrm{x}$ in $Ein_{1,n-1}$ (on the left) and in $\mathsf{Ein}_{1,n-1}$ (on the right).
  • Figure 2: Affine charts defined by $p \in \widetilde{Ein}_{1,n-1}$ for $n = 2$.
  • Figure 3: Intersection of two diamonds in $\widetilde{Ein}_{1,2}$.
  • Figure 4: Intersection of $D$ and $D'$ in the hyperplane $H_1$ in dimension $3$.
  • Figure 5: The diamonds $\gamma_k.D$ and $D'$ in the hyperplane $H_1$ in dimension $3$.
  • ...and 2 more figures

Theorems & Definitions (87)

  • Lemma 1.1: Injectivity on diamonds
  • Theorem 1.1: Salvemini2013Maximal
  • Corollary 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • ...and 77 more