  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.
