Late time tail of waves on dynamic asymptotically flat spacetimes of odd space dimensions
Jonathan Luk, Sung-Jin Oh
TL;DR
This work introduces a general, geometry-driven framework to analyze late-time tails of wave equations in odd spatial dimensions on asymptotically flat spacetimes, including dynamical and nonlinear backgrounds. By expanding solutions into a hierarchy of higher radiation fields at null infinity and establishing recurrence relations, the authors derive sharp upper bounds and precise tail asymptotics that depend only on these asymptotic charges. The method recovers Price's law in stationary linear settings and reveals how nonlinearities and background dynamics can alter decay, sometimes accelerating or decelerating tails depending on angular modes and region (wave/med/near). The results are supported by a robust vector-field toolkit, Minkowski-model comparisons, and a detailed exterior Cauchy-data theory, with applications to classical Minkowski, obstacle problems, and black-hole spacetimes in various dimensions. Overall, the paper provides a versatile, principled approach to tail behavior with broad implications for stability analyses and cosmic censorship in general relativity.
Abstract
We introduce a general method for understanding the late time tail for solutions to wave equations on asymptotically flat spacetimes with odd space dimensions. In particular, for a large class of equations, we prove that the precise late time tail is determined by the limits of higher radiation field at future null infinity. In the setting of stationary linear equations, we recover and generalize the Price law decay rates. In particular, in addition to reproving known results on $(3+1)$-dimensional black holes, this allows one to obtain the sharp decay rate for the wave equation on higher dimensional black hole spacetimes, which exhibits an anomalous rate due to subtle cancellations. More interesting, our method goes beyond the stationary linear case and applies to both equations on dynamical background and nonlinear equations. In this case, our results can be used to show that in general there is a correction to the Price law rates.