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

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 $\mathcal{H}$.

33 pages, 4 figures

References in corpus (2)

• Polyhomogeneous expansions from time symmetric initial data
• Asymptotic properties of the development of conformally flat data near spatial infinity

Cited by in corpus (11)

• Charge and Antipodal Matching across Spatial Infinity
• Unified Treatment of Null and Spatial Infinity III: Asymptotically Minkowski Space-times
• The Maxwell-scalar field system near spatial infinity
• Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime
• Asymptotic symmetries of projectively compact order one Einstein manifolds
• Calculation of asymptotic charges at the critical sets of null infinity
• Projective and Carrollian geometry at time/space-like infinity on projectively compact Ricci flat Einstein manifolds
• Asymptotics of spin-0 fields and conserved charges on n-dimensional Minkowski spaces
• Ti and Spi, Carrollian extended boundaries at timelike and spatial infinity
• Variationality of conformal geodesics in dimension 3
• Conformal Scattering of Maxwell Potentials