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

TL;DR

This paper explores the extremal limits of the C-metric with a cosmological constant, deriving new solutions like Nariai, Bertotti-Robinson, and anti-Nariai C-metrics, and discusses their potential instabilities and quantum implications.

Contribution

It introduces the C-metric counterparts of extremal solutions such as Nariai and Bertotti-Robinson, extending previous analyses to generic cosmological constants.

Findings

Generated C-metric Nariai, Bertotti-Robinson, and anti-Nariai solutions.

Indicated these solutions are likely unstable and decay into black hole pairs.

Suggested these solutions mediate quantum black hole pair creation.

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.