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

Dias, Oscar J.C. and Lemos, Jose P.S. (2003) The extremal limits of the C metric: Nariai, Bertotti-Robinson, and anti-Nariai C metrics. Phys.Rev.D, 68 (10), [104010].

Record type: Article

## Abstract

In two previous papers we have analyzed the C metric in a background with a cosmological constant Λ, namely, the de–Sitter (dS) C metric (Λ>0), and the anti–de Sitter (AdS) C metric (Λ<0), extending thus the original work of Kinnersley and Walker for the C metric in flat spacetime (Λ=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 Λ>0,Λ=0, and Λ<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-Nordström 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 dS2×˜S2) to each point in the deformed two-sphere ˜˜S2 corresponds a dS2 spacetime, except for one point which corresponds to a dS2 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.

e-pub ahead of print date: 11 November 2003
Published date: 15 November 2003

## Identifiers

Local EPrints ID: 468165
URI: http://eprints.soton.ac.uk/id/eprint/468165
PURE UUID: 4d0d654b-1828-4c0a-8f75-2d9df50c7eba
ORCID for Oscar J.C. Dias: orcid.org/0000-0003-4855-4750

## Catalogue record

Date deposited: 04 Aug 2022 16:38

## Contributors

Author: Oscar J.C. Dias
Author: Jose P.S. Lemos