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A comparison of Ashtekar's and Friedrich's formalisms of spatial infinity

Mariem Magdy, Juan A. Valiente Kroon

TL;DR

The paper establishes a rigorous link between Ashtekar–Hansen’s asymptote at spatial infinity and Friedrich’s cylinder by using conformal-geodesic methods and stability arguments. It first clarifies the Minkowski spacetime case, showing that the cylinder and the asymptote are related by a conformal factor and a time-compactifying transformation, then extends these ideas to asymptotically Minkowskian spacetimes at spatial infinity (AMSI) by deriving the conformal constraint equations on the asymptote and proving the existence of a conformal Gaussian system near ${\mathcal H}$. A two-step program relates a general AMSI asymptote to Friedrich’s cylinder (via a local $1+2$-Minkowski link) and demonstrates that a small neighborhood of ${\mathcal H}$ can be covered by conformal-geodesic congruences, enabling a stable, regular evolution of conformal fields. The results provide a robust bridge between the two spatial infinity formalisms, facilitating transfer of asymptotic data and shedding light on the interplay between spatial and null infinity in the conformal framework.

Abstract

Penrose's idea of asymptotic flatness provides a framework for understanding the asymptotic structure of gravitational fields of isolated systems at null infinity. However, the studies of the asymptotic behaviour of fields near spatial infinity are more challenging due to the singular nature of spatial infinity in a regular point compactification for spacetimes with non-vanishing ADM mass. Two different frameworks that address this challenge are Friedrich's cylinder at spatial infinity and Ashtekar's definition of asymptotically Minkowskian spacetimes at spatial infinity that give rise to the 3-dimensional asymptote at spatial infinity $\mathcal{H}$. Both frameworks address the singularity at spatial infinity although the link between the two approaches had not been investigated in the literature. This article aims to show the relation between Friedrich's cylinder and the asymptote as spatial infinity. To do so, we initially consider this relation for Minkowski spacetime. It can be shown that the solution to the conformal geodesic equations provides a conformal factor that links the cylinder and the asymptote. For general spacetimes satisfying Ashtekar's definition, the conformal factor cannot be determined explicitly. However, proof of the existence of this conformal factor is provided in this article. Additionally, the conditions satisfied by physical fields on the asymptote $\mathcal{H}$ are derived systematically using the conformal constraint equations. Finally, it is shown that a solution to the conformal geodesic equations on the asymptote can be extended to a small neighbourhood of spatial infinity by making use of the stability theorem for ordinary differential equations. This solution can be used to construct a conformal Gaussian system in a neighbourhood of $\mathcal{H}$.

A comparison of Ashtekar's and Friedrich's formalisms of spatial infinity

TL;DR

The paper establishes a rigorous link between Ashtekar–Hansen’s asymptote at spatial infinity and Friedrich’s cylinder by using conformal-geodesic methods and stability arguments. It first clarifies the Minkowski spacetime case, showing that the cylinder and the asymptote are related by a conformal factor and a time-compactifying transformation, then extends these ideas to asymptotically Minkowskian spacetimes at spatial infinity (AMSI) by deriving the conformal constraint equations on the asymptote and proving the existence of a conformal Gaussian system near . A two-step program relates a general AMSI asymptote to Friedrich’s cylinder (via a local -Minkowski link) and demonstrates that a small neighborhood of can be covered by conformal-geodesic congruences, enabling a stable, regular evolution of conformal fields. The results provide a robust bridge between the two spatial infinity formalisms, facilitating transfer of asymptotic data and shedding light on the interplay between spatial and null infinity in the conformal framework.

Abstract

Penrose's idea of asymptotic flatness provides a framework for understanding the asymptotic structure of gravitational fields of isolated systems at null infinity. However, the studies of the asymptotic behaviour of fields near spatial infinity are more challenging due to the singular nature of spatial infinity in a regular point compactification for spacetimes with non-vanishing ADM mass. Two different frameworks that address this challenge are Friedrich's cylinder at spatial infinity and Ashtekar's definition of asymptotically Minkowskian spacetimes at spatial infinity that give rise to the 3-dimensional asymptote at spatial infinity . Both frameworks address the singularity at spatial infinity although the link between the two approaches had not been investigated in the literature. This article aims to show the relation between Friedrich's cylinder and the asymptote as spatial infinity. To do so, we initially consider this relation for Minkowski spacetime. It can be shown that the solution to the conformal geodesic equations provides a conformal factor that links the cylinder and the asymptote. For general spacetimes satisfying Ashtekar's definition, the conformal factor cannot be determined explicitly. However, proof of the existence of this conformal factor is provided in this article. Additionally, the conditions satisfied by physical fields on the asymptote are derived systematically using the conformal constraint equations. Finally, it is shown that a solution to the conformal geodesic equations on the asymptote can be extended to a small neighbourhood of spatial infinity by making use of the stability theorem for ordinary differential equations. This solution can be used to construct a conformal Gaussian system in a neighbourhood of .

Paper Structure

This paper contains 41 sections, 11 theorems, 198 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Tools of conformal methods
  3. Conformal geodesics
  4. Conformal field equations
  5. Representations of spatial infinity in Minkowski spacetime
  6. Spatial infinity as point
  7. The cylinder at spatial infinity
  8. Conformal geodesics
  9. Gauge freedom
  10. The hyperboloid at spatial infinity
  11. Standard representation of spatial infinity
  12. The hyperboloid at spatial infinity
  13. Relating the Ashtekar and F-gauge construction in the Minkowski spacetime
  14. From the hyperboloid at infinity to the cylinder at infinity
  15. Relating the neighbourhoods of spatial infinity
  16. ...and 26 more sections

Key Result

Theorem 1

Given an AMSI spacetime that can be conformally extended to null infinity, there exists a sufficiently small neighbourhood of the asymptote at spatial infinity in which is possible to construct conformal Gaussian coordinates based on curves which extend beyond null infinity.

Figures (4)

  • Figure 1: (a) Diagrammatic depiction of the domain $\tilde{\mathcal{D}}$ containing spatial infinity, (b) The domain $\tilde{\mathcal{D}}$ on the conformal diagram of Minkowski spacetime.
  • Figure 2: (a) The timelike hyperboloids in the domain $\tilde{\mathcal{D}}$ of the Minkowski spacetime used in the construction of Ashtekar's hyperboloid at spatial infinity. (b) The timelike hyperboloids shown on the conformal diagram of Minkowski spacetime.
  • Figure 3: Example of one of the timelike hyperboloids used in Ashtekar's representation of spatial infinity for the Minkowski as seen from the point of view of the F-gauge. Observe that the hyperboloid asymptotes the sets described by the conditions $\tau=\pm 1$. Notice, however, that in this description the hyperboloid at spatial infinity is compact and corresponds to the portion of the vertical axis between the values $\tau=-1$ and $\tau=1$.
  • Figure 4: Schematic summary of the argument showing that the an asymptote $\mathcal{H}$ satisfying Definition \ref{['Definition:AMSI']} and the $1+2$ Einstein static Universe are conformally related.

Theorems & Definitions (44)

  • Definition 1
  • Theorem
  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • Remark 6
  • ...and 34 more