# Couch-Torrence inversion isometry The **Couch-Torrence inversion isometry** is a discrete conformal isometry that maps between (future) horizon and null infinity in the extreme RN black hole. It helps to map [[0340 Aretakis instability|Aretakis charges]] to [[0456 Newman-Penrose charges|Newman-Penrose charges]]. It is possible essentially because the extreme RN black hole has Ricci scalar $R=0$, which implies that the massless scalar wave equation in conformally covariant. Hence conformal isometry can be used to generate a new solution from an old one. (nice explanation in [[2012#Lucietti, Murata, Reall, Tanahashi]]) ## Examples - for extreme RN with a probe Maxwell field: - [[FernandesGhoshVirmani2020]][](https://arxiv.org/pdf/2008.04365.pdf)