Conformal isometry of the Reissner-Nordström-de Sitter black hole

It was pointed out by Couch and Torrence that the extreme Reissner-Nordström solution possesses a discrete conformal isometry. Using results of Romans, it is shown that such a symmetry also exists when a non-zero cosmological constant is allowed. Introduction In [1], Couch and Torrence found a conformal isometry that interchanges the event horizon and null infinity J of an extremal Reissner-Nordström black hole. Related but distinct ideas have also appeared in a string theory context[2]. Here we will show 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. We will leave global issues aside and refer the reader to [3] for those matters. See also [4] for a discussion of conserved quantities in asymptotically de Sitter spacetimes from a Hamiltonian point of view. The calculation In the Couch-Torrence case the conformal isometry switches the roles of the event horizon and infinity. With a positive cosmological constant, there is another geometrically distinguished object: the cosmological horizon. As Λ → 0, this horizon approaches infinity. Let us therefore provisionally assume that the putative conformal isometry interchanges the black hole horizon and the cosmological horizon since this produces the correct limiting behavior; this assumption will later turn out to be correct. Assume that the BH horizon is at r = a and the cosmological horizon at r = b. It is convenient to introduce the coordinate x defined by x = r − a b− r b a The strategy here is to first re-express the above metric in terms of the x coordinate. In such a coordinate system, the metric will be manifestly conformally invariant under the inversion x 7→ 1/x as we will now show (note that the E-mail address: [email protected]