On ``hyperboloidal'' Cauchy data for vacuum Einstein equations and obstructions to smoothness of ``null infinity''

TL;DR

This paper analyzes the initial data constraints for vacuum Einstein equations near null infinity, revealing generic logarithmic terms in asymptotic expansions and identifying conditions for smoothness of null infinity.

Contribution

It provides a detailed analysis of asymptotically hyperboloidal initial data, showing the generic emergence of log terms and conditions for smooth null infinity in vacuum spacetimes.

Findings

Log terms arise generically in asymptotic expansions of initial data.

Certain non-generic solutions lead to smooth null infinity.

Explicit geometric quantities relate to the presence of log terms.

Abstract

Various works have suggested that the Bondi--Sachs--Penrose decay conditions on the gravitational field at null infinity are not generally representative of asymptotically flat space--times. We have made a detailed analysis of the constraint equations for ``asymptotically hyperboloidal'' initial data and find that log terms arise generically in asymptotic expansions. These terms are absent in the corresponding Bondi--Sachs--Penrose expansions, and can be related to explicit geometric quantities. We have nevertheless shown that there exists a large class of ``non--generic'' solutions of the constraint equations, the evolution of which leads to space--times satisfying the Bondi--Sachs--Penrose smoothness conditions.