General Relativity and Gravitation, Vol. 30, No. 2, 1998 Optical Space of the Reissner ±NordstrÈ om Solutions Simon Kristiansson, 1 , 2 Sebastiano Sonego 1 , 3 and Marek A. Abramowicz 1 , 4 Received June 24, 1997 We present an exhaustive discussion of the embedding diagrams for the optical geometry of the Reissner±Nordstr È om solutions. Whereas in the black hole sector there are no qualitative diŒerences with respect to the Schwarzschild case, the diagrams are considerably diŒerent if naked sin- gularities are present. Our treatment is suciently general that it can be applied also to any other static spherically symmetric spacetime. KEY WORDS : Optical geometry 1. INTRODUCTION A static spacetime ( M , g ) with Killing vector ®eld ¶t admits a privileged notion of space in terms of the hypersurfaces t = const. These hyper- surfaces are all diŒeomorphic to some three-dimensional manifold S , and one can thus de®ne space simply as the pair ( S ,s ), where s is a suitable Riemannian metric on S . Usually, one chooses s = h , where h is just the metric induced by g on the hypersurfaces. We shall refer to the Riemann- ian manifold ( S ,h ) as the ordinary space . It is often convenient, however, to imagine that space is endowed not with the metric h , but instead with the so-called optical metric [1±3] Ä h := ( ± g tt ) - 1 h , conformally related to h . The resulting Riemannian ma- 1 Department of Astronomy and Astrophysics, Chalmers University of Technology, S-41296 GÈ oteborg, Sweden 2 E-mail: f95sikr@dd.chalmers.se 3 E-mail: sonego@galileo.sissa.it 4 E-mail: marek@tfa.fy.chalmers.se 275 0001-7701/ 98/ 0200-0275$15.00/ 0 1998 Plenum Publishing Corporation