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

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.