The extremal limits of the C-metric: Nariai, Bertotti-Robinson and anti-Nariai C-metrics

Abstract

In two previous papers we have analyzed the C-metric in a background with a cosmological constant, namely the de Sitter (dS) C-metric, and the anti-de Sitter (AdS) C-metric, following the work of Kinnersley and Walker for the flat C-metric. 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. In this paper we analyze the extremal limits of the C-metric in a background with generic cosmological constant. 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 2-dimensional dS and a 2-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 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. One expects that the solutions found in this paper are unstable and decay into a slightly non-extreme 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.