Non-genericity of the Nariai solutions: I. Asymptotics and spatially homogeneous perturbations

Abstract

This is the first of two papers in which we study the asymptotics of the generalized Nariai solutions and its relation to the cosmic no-hair conjecture. According to the cosmic no-hair conjecture, generic expanding solutions of Einstein's field equations in vacuum with a positive cosmological constant isotropize and approach the de-Sitter solution asymptotically. The family of solutions which we introduce as ‘generalized Nariai solutions’, however, shows quite unusual asymptotics and hence should be non-generic in some sense. In this paper, we list basic facts for the Nariai solutions and characterize their asymptotic behavior geometrically. One particular result is a rigorous proof of the fact that the Nariai solutions do not possess smooth conformal boundaries. We proceed by explaining the non-genericity within the class of spatially homogeneous solutions. It turns out that perturbations of the three isometry classes of generalized Nariai solutions are related to different mass regimes of Schwarzschild–de-Sitter solutions. A motivation for our work here is to prepare the second paper devoted to the study of the instability of the Nariai solutions for Gowdy symmetry. We will be particularly interested in the construction of new and in principle arbitrarily complicated cosmological black hole solutions.