Nariai

Nariai spacetime, or the Nariai solution, is an exact solution to Einstein’s field equations with a positive cosmological constant Λ. It describes a four-dimensional spacetime that is the direct product of a two-dimensional de Sitter space with a two-sphere, denoted dS2 × S2. This geometry arises as a limiting case in the family of Schwarzschild–de Sitter black holes.

A commonly used form of the metric is ds^2 = (1/Λ)[ -dτ^2 + cosh^2 τ dχ^2 + dΩ^2_2 ], where dΩ^2_2 = dθ^2 + sin^2 θ dφ^2. In this representation the two-dimensional part has constant positive curvature and the S2 factor has radius 1/√Λ. The spacetime satisfies the vacuum Einstein equations with a cosmological constant, Rμν = Λ gμν.

History and origin: The solution was identified by Hideki Nariai in the 1950s as a limiting case of Schwarzschild–de Sitter black holes in which the event horizon and the cosmological horizon coincide. The name reflects this limiting process and the resulting distinct geometric structure.

Properties and interpretation: As a product space, Nariai spacetime is homogeneous along the dS2 directions and isotropic on the S2 factor, though it is not a maximally symmetric four-dimensional space. Its curvature scales are set by the cosmological constant, with both factors carrying curvature 1/√Λ. In the Schwarzschild–de Sitter interpretation, it corresponds to an extremal horizon configuration.

Stability and significance: The Nariai solution exhibits perturbative instabilities in certain contexts, known as the Nariai instability, and plays a role in studies of horizon thermodynamics, extremal black holes in de Sitter space, and the behavior of quantum fields in curved backgrounds.