Can one detect a non-smooth null infinity?

TL;DR

This work investigates whether the smoothness of null infinity in asymptotically flat spacetimes can be inferred from physical observables, focusing on polyhomogeneous spacetimes where asymptotic expansions include $\ln r$ terms. By modeling gyroscope precession within the Bondi framework, it shows that peeling (smooth) spacetimes produce leading precession terms of order $O(r^{-2})$ (or $O(r^{-3})$ if the mass aspect is isotropic), while polyhomogeneous (non-smooth) spacetimes generate a logarithmically enhanced $O(r^{-2}\ln r)$ contribution. This distinct $r$-scaling of precession constitutes a practical fingerprint distinguishing the two classes of null infinity. The analysis also connects logarithmic terms to incoming radiation and wave tails, providing a physical interpretation for these asymptotic features and a potential diagnostic tool for gravitational radiation studies.

Abstract

It is shown that the precession of a gyroscope can be used to elucidate the nature of the smoothness of the null infinity of an asymptotically flat spacetime (describing an isolated body). A model for which the effects of precession in the non-smooth null infinity case are of order $r^{-2}\ln r$ is proposed. By contrast, in the smooth version the effects are of order $r^{-3}$. This difference should provide an effective criterion to decide on the nature of the smoothness of null infinity.