Letter: Conformal isometry of the Reissner-Nordström-de Sitter black hole (Q1431737)

In \textit{W.~E.~Couch} and \textit{R.~J.~Torrence} [Gen. Relativ. Gravitation 16, 789 (1984; Zbl 0539.53024)], the authors found a conformal isometry that interchanges the event horizon and null infinity \(\mathcal J\) of an extremal Reissner-Nordström black hole. One may ask the question whether there are other black-hole spacetimes for which a conformal isometry of the Couch-Torrence type exists. This seems to be tricky to answer; since the isometry is discrete, and infinitesimal methods using Killing vectors are not likely to be very useful. Using results of \textit{L. J. Romans} [Nucl. Phys. B 383, 395 (1992)], it is shown that such a conformal isometry exists also for the case of a positive cosmological constant, provided that the surface gravities of the two horizons are equal.