Generalized Couch--Torrence inversions

The Laplace equation on Euclidean flat space admits a discrete radial inversion symmetry. In 1983, Couch and Torrence (CT) found -- surprisingly -- that the massless wave equation continues to display this symmetry on the background of an extremal (and asymptotically flat) black hole, where the inversion interchanges horizon and infinity while preserving the singularity structure of the separated radial mode equation. We revisit this CT inversion symmetry and investigate its possible extensions beyond the extremal (Reissner-Nordström or Kerr) setting in which it was originally identified. Using the example of the static lukewarm de Sitter black hole, we show that neither the exchange of horizon with infinity, nor the preservation of radial singularities, are essential features needed for a CT inversion to exist. Instead, we interpret CT transformations through their action on photon spheres, providing a unified viewpoint that extends to the (phase-space-dependent) CT inversions of the extremal Kerr-Newman geometry. For scalar fields on that spacetime, we find a simple relation between the fixed point of a CT inversion and the coefficient of superradiant scattering. Finally, we exhibit a hidden CT inversion symmetry that arises in the static limit of the Kerr Laplacian for all spins. Together, these results suggest that CT symmetry may admit a broader generalization than previously understood.