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We provide a definitive treatment, including sharp decay and the preciselate-time asymptotic profile, for generic solutions of linear wave equationswith a (singular) inverse-square potential in (3+1)-dimensional Minkowskispacetime. Such equations are scale-invariant and we show their solutions decayin time at a rate determined by the coefficient in the inverse-squarepotential.  We present a novel, geometric, physical-space approach for determininglate-time asymptotics, based around embedding Minkowski spacetime conformallyinto the spacetime $AdS_2 \times \mathbb{S}^2$ (with $AdS_2$ thetwo-dimensional anti de-Sitter spacetime) to turn a global late-timeasymptotics 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 thenear-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 presenceof a static electric field, which can be viewed as a simplification of thecharged Maxwell-Klein-Gordon equations on a black hole spacetime.

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