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Lorentz meets Ptolemy

Felix Rott, Zhe-Feng Xu, Matteo Zanardini

TL;DR

The paper develops a Lorentzian analogue of the Ptolemy inequality within Lorentzian pre-length spaces and globally hyperbolic spacetimes, linking it to a global timelike curvature bound $\mathrm{TSec}\le 0$ via hyperbolic inversion and triangle comparison. It proves that a globally hyperbolic spacetime is Ptolemaic if and only if it has global non-positive curvature, with Minkowski space shown to be Ptolemaic; it also establishes that non-positive curvature implies Ptolemy and, conversely, that Ptolemy implies NPC and future one-connectedness, enabling a globalization result. The work further develops the rigidity of the hyperbolic inversion, deriving a Minkowski-space characterization under completeness and simple connectivity through a four-point condition. Overall, it provides a robust Lorentzian counterpart to CAT(0)-type geometry, with concrete implications for spacetime rigidity and the structure of Minkowski space.

Abstract

We consider a Lorentzian analogue of the Ptolemy inequality and we prove that in the setting of globally hyperbolic spacetimes it is equivalent to a global timelike sectional curvature bound from above by zero. We investigate the link between the Ptolemy inequality and the hyperbolic inversion and establish some applications and rigidity properties.

Lorentz meets Ptolemy

TL;DR

The paper develops a Lorentzian analogue of the Ptolemy inequality within Lorentzian pre-length spaces and globally hyperbolic spacetimes, linking it to a global timelike curvature bound via hyperbolic inversion and triangle comparison. It proves that a globally hyperbolic spacetime is Ptolemaic if and only if it has global non-positive curvature, with Minkowski space shown to be Ptolemaic; it also establishes that non-positive curvature implies Ptolemy and, conversely, that Ptolemy implies NPC and future one-connectedness, enabling a globalization result. The work further develops the rigidity of the hyperbolic inversion, deriving a Minkowski-space characterization under completeness and simple connectivity through a four-point condition. Overall, it provides a robust Lorentzian counterpart to CAT(0)-type geometry, with concrete implications for spacetime rigidity and the structure of Minkowski space.

Abstract

We consider a Lorentzian analogue of the Ptolemy inequality and we prove that in the setting of globally hyperbolic spacetimes it is equivalent to a global timelike sectional curvature bound from above by zero. We investigate the link between the Ptolemy inequality and the hyperbolic inversion and establish some applications and rigidity properties.
Paper Structure (10 sections, 17 theorems, 43 equations, 3 figures)

This paper contains 10 sections, 17 theorems, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Ptolemy inequality
  4. First properties and stability
  5. The hyperbolic inversion
  6. The Ptolemy inequality in Minkowski space
  7. Ptolemaic spacetimes
  8. Ptolemaic and non-positive curvature
  9. Geodesic uniqueness and future one-connectedness
  10. Rigidity of the hyperbolic inversion

Key Result

Theorem 1.1

A globally hyperbolic spacetime is Ptolemaic if and only if it is globally non-positively curved.

Figures (3)

  • Figure 1: Hyperbolic inversion in the comparison configuration in the Minkowski plane.
  • Figure 2: The inversion time separation in the Minkowski plane can be calculated via the law of cosines.
  • Figure 3: Two lines pass through the origin and intersect the hyperbola in four points.

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 2.1: Strong causality, global hyperbolicity and future one-connectedness
  • Definition 2.2: Timelike sectional curvature bound
  • Remark 2.3: On the convention of signature
  • Definition 2.4: Lo-rentz-ian pre-length space
  • Definition 2.5: Triangle comparison
  • Definition 3.1: The Ptolemy inequality
  • Remark 3.2: Basic properties of Ptolemaic spaces
  • Remark 3.3: Busemann concavity, stability and Finsler spacetimes
  • Proposition 3.4: Stability of Ptolemaic spaces
  • ...and 36 more