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

TL;DR

This paper compares Friedrich's and Ashtekar's frameworks for understanding spatial infinity in general relativity, establishing their relation and providing methods to connect their structures through conformal factors.

Contribution

It demonstrates the link between Friedrich's cylinder at spatial infinity and Ashtekar's asymptote, including existence proofs for the conformal factor in general spacetimes.

Findings

The conformal geodesic equations link the two frameworks in Minkowski spacetime.

Existence of the conformal factor is proven for general asymptotically Minkowskian spacetimes.

A conformal Gaussian system can be constructed near spatial infinity using these results.

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 $\mathcal{H}$.