Late-time tails for scale-invariant wave equations with a potential and the near-horizon geometry of null infinity

TL;DR

This paper analyzes the late-time decay of solutions to scale-invariant wave equations with inverse-square potentials in Minkowski spacetime, introducing a geometric approach based on conformal embedding into $AdS_2 \times \mathbb{S}^2$ to determine precise asymptotics.

Contribution

It develops a novel geometric method using conformal embedding to derive sharp decay rates and asymptotics for wave equations with inverse-square potentials, inspired by near-horizon black hole geometry.

Findings

Solutions decay at rates determined by the potential coefficient.

The method applies to charged wave equations with static electric fields.

Provides a local existence framework for late-time asymptotics.

Abstract

We provide a definitive treatment, including sharp decay and the precise late-time asymptotic profile, for generic solutions of linear wave equations with a (singular) inverse-square potential in (3+1)-dimensional Minkowski spacetime. Such equations are scale-invariant and we show their solutions decay in time at a rate determined by the coefficient in the inverse-square potential. We present a novel, geometric, physical-space approach for determining late-time asymptotics, based around embedding Minkowski spacetime conformally into the spacetime $AdS_2 \times \mathbb{S}^2$ (with $AdS_2$ the two-dimensional anti de-Sitter spacetime) to turn a global late-time asymptotics problem into a local existence problem for the wave equation in $AdS_2 \times \mathbb{S}^2$. Our approach is inspired by the treatment of the near-horizon geometry of extremal black holes in the physics literature. We moreover apply our method to another scale-invariant model: the (complex-valued) charged wave equation on Minkowski spacetime in the presence of a static electric field, which can be viewed as a simplification of the charged Maxwell-Klein-Gordon equations on a black hole spacetime.