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 three-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 .

彭罗斯的渐近平坦性思想为理解零无穷远孤立系统引力场的渐近结构提供了一个框架。然而，由于空间无穷大在具有非零 ADM 质量的时空的规则点压缩中的奇异性质，因此对空间无穷大附近场的渐近行为的研究更具挑战性。解决这一挑战的两个不同框架是 Friedrich 的空间无穷大圆柱体和 Ashtekar 对空间无穷远渐近 Minkowski 时空的定义，它们产生了空间无穷大的三维渐近线. 尽管文献中尚未研究这两种方法之间的联系，但这两种框架都解决了空间无穷大的奇点问题。本文旨在展示弗里德里希圆柱体和渐近线作为空间无穷大的关系。为此，我们首先考虑 Minkowski 时空的这种关系。可以证明，共形测地线方程的解提供了连接圆柱和渐近线的共形因子。对于满足 Ashtekar 定义的一般时空，不能明确确定共形因子。但是，本文提供了此适形因子存在的证明。此外，渐近线上的物理场满足的条件使用保形约束方程系统地导出。最后，证明了利用常微分方程的稳定性定理，渐近线上的共形测地线方程的解可以推广到空间无穷小的小邻域。该解决方案可用于在 邻域内构造共形高斯系统。