Extremal limits of the C metric:: Nariai, Bertotti-Robinson, and anti-Nariai C metrics -: art. no. 104010

In two previous papers we have analyzed the C metric in a background with a cosmological constant Lambda, namely, the de-Sitter (dS) C metric (Lambda>0), and the anti-de Sitter (AdS) C metric (Lambda<0), extending thus the original work of Kinnersley and Walker for the C metric in flat spacetime (Lambda=0). These exact solutions describe a pair of accelerated black holes in the flat or cosmological constant background, with the acceleration A being provided by a strut in between that pushes away the two black holes or, alternatively, by strings hanging from infinity that pull them in. In this paper we analyze the extremal limits of the C metric in a background with a generic cosmological constant Lambda>0, Lambda=0, and Lambda<0. We follow a procedure first introduced by Ginsparg and Perry in which the Nariai solution, a spacetime which is the direct topological product of the two-dimensional dS and a two-sphere, is generated from the four-dimensional dS-Schwarzschild solution by taking an appropriate limit, where the black hole event horizon approaches the cosmological horizon. Similarly, one can generate the Bertotti-Robinson metric from the Reissner-Nordstrom metric by taking the limit of the Cauchy horizon going into the event horizon of the black hole, as well as the anti-Nariai metric by taking an appropriate solution and limit. Using these methods we generate the C-metric counterparts of the Nariai, Bertotti-Robinson, and anti-Nariai solutions, among others. These C-metric counterparts are conformal to the product of two two-dimensional manifolds of constant curvature, the conformal factor depending on the angular coordinate. In addition, the C-metric extremal solutions have a conical singularity at least at one of the poles of their angular surfaces. We give a physical interpretation to these solutions, e.g., in the Nariai C metric (with topology dS(2)x (S) over bar (2)) to each point in the deformed two-sphere (S) over tilde (2) corresponds a dS(2) spacetime, except for one point which corresponds to a dS(2) spacetime with an infinite straight strut or string. There are other important new features that appear. One expects that the solutions found in this paper are unstable and decay into a slightly nonextreme black hole pair accelerated by a strut or by strings. Moreover, the Euclidean version of these solutions mediate the quantum process of black hole pair creation that accompanies the decay of the dS and AdS spaces.