Late-time tails for linear waves on radially symmetric stationary spacetimes of two space dimensions

TL;DR

This paper analyzes the late-time behavior of linear waves on radially symmetric stationary spacetimes in two dimensions, showing the dominant decay rate and extending energy estimate techniques.

Contribution

It extends $r^p$-weighted energy estimates to two spatial dimensions and characterizes the leading asymptotic decay of solutions.

Findings

Leading late-time decay term proportional to u^{-1/2}v^{-1/2}

Extension of $r^p$-weighted energy estimates to 2D

Solution behavior similar to Minkowski space in late-time regime

Abstract

We show that the leading-order term in the late-time asymptotics of solutions to the linear wave equation on radially symmetric stationary perturbations of $(2 + 1)$-dimensional Minkowski space is proportional to $u^{-1/2}v^{-1/2}$ (which solves the wave equation on Minkowski space), where $u$ and $v$ are double null coordinates. Our proof adapts the physical space techniques in the work of Gajic (arXiv:2203.15838) on the wave equation with an inverse-square potential on the Schwarzschild spacetime. In particular, we extend the $r^p$-weighted energy estimates of Dafermos--Rodnianski (arXiv:0910.4957) to two space dimensions.