Comment on "Late-time tails of a self-gravitating massless scalar field revisited" by Bizon et al: The leading order asymptotics

TL;DR

This paper revisits the late-time decay behavior of a self-gravitating massless scalar field, revealing that the decay rate results from specific nonlinear cancellations, and proposes a more direct calculation method.

Contribution

It introduces a new approach to compute asymptotics by identifying structural cancellations in the wave equation, improving upon previous series expansion methods.

Findings

Decay rate is influenced by nonlinear cancellations.

Direct calculation method simplifies asymptotic analysis.

Highlights the importance of wave equation structure in decay behavior.

Abstract

In Class. Quantum Grav. 26 (2009) 175006 (arXiv:0812.4333v3) Bizon et al discuss the power-law tail in the long-time evolution of a spherically symmetric self-gravitating massless scalar field in odd spatial dimensions. They derive explicit expressions for the leading order asymptotics for solutions with small initial data by using formal series expansions. Unfortunately, this approach misses an interesting observation that the actual decay rate is a product of asymptotic cancellations occurring due to a special structure of the nonlinear terms. Here, we show that one can calculate the leading asymptotics more directly by recognizing the special structure and cancellations already on the level of the wave equation.