Late-time tails of a self-gravitating massless scalar field, revisited

TL;DR

This paper investigates the nonlinear origins of power-law tails in the evolution of self-gravitating scalar fields, deriving explicit decay expressions and showing differences between four and higher dimensions through analytical and numerical methods.

Contribution

It provides explicit third-order perturbation formulas for tail decay in higher dimensions and clarifies the dimensional dependence of tail behavior.

Findings

Decay rates differ between four and higher dimensions.

Explicit formulas for tail amplitude and decay rate are derived.

Numerical verification confirms analytical predictions.

Abstract

We discuss the nonlinear origin of the power-law tail in the long-time evolution of a spherically symmetric self-gravitating massless scalar field in even-dimensional spacetimes. Using third-order perturbation method, we derive explicit expressions for the tail (the decay rate and the amplitude) for solutions starting from small initial data and we verify this prediction via numerical integration of the Einstein-scalar field equations in four and six dimensions. Our results show that the coincidence of decay rates of linear and nonlinear tails in four dimensions (which has misguided some tail hunters in the past) is in a sense accidental and does not hold in higher dimensions.