Hot spaces with positive cosmological constant in the canonical ensemble: de Sitter solution, Schwarzschild–de Sitter black hole, and Nariai universe

Abstract

In a space with fixed positive cosmological constant Λ, we consider a system with a black hole surrounded by a heat reservoir at radius R and fixed temperature T, i.e., we analyze the Schwarzschild–de Sitter black hole space in a cavity. We use results from the Euclidean path integral approach to quantum gravity to study, in a semiclassical approximation, the corresponding canonical ensemble and its thermodynamics. We give the action for the Schwarzschild–de Sitter black hole space and calculate expressions for the thermodynamic energy, entropy, temperature, and heat capacity. The reservoir radius R gauges the other scales. Thus, the temperature T, the cosmological constant Λ, the black hole horizon radius r+, and the cosmological horizon radius rc, are gauged to RT, ΛR2, r+R, and rcR. The whole extension of ΛR2, 0≤ΛR2≤3, can be split into three ranges. The first range, 0≤ΛR2<1, includes York’s pure Schwarzschild black holes. The other values of ΛR2 within this range also have black holes. The second range, ΛR2=1, opens up a folder containing Nariai universes, rather than black holes. The third range, 1<ΛR2≤3, is unusual. One striking feature here is that it interchanges the cosmological horizon with the black hole horizon. The end of this range, ΛR2=3, only existing for infinite temperature, represents a cavity filled with de Sitter space inside, except for a black hole with zero radius, i.e., a singularity, and with the cosmological horizon coinciding with the reservoir radius. For the three ranges, for sufficiently low temperatures, which for quantum systems involving gravitational fields can be very high when compared to normal scales, there are no black hole solutions and no Nariai universes, and the space inside the reservoir is hot de Sitter. The limiting value RT that divides the nonexistence from existence of black holes or Nariai universes, depends on the value of ΛR2. For each ΛR2 different from one, for sufficiently high temperatures there are two black holes, one small and thermodynamically unstable, and one large and stable. For ΛR2=1, for any sufficiently high temperature there is the small unstable black hole, and the neutrally stable hot Nariai universe. Phase transitions can be analyzed, the dominant phase has the least action. The transitions are between Schwarzschild-de Sitter black hole and hot de Sitter phases and between Nariai and hot de Sitter. For small cosmological constant, the action for the stable black hole equals the pure de Sitter action at a certain black hole radius and temperature, and so the phases coexist equally. For 0<ΛR2<1 the equal action black hole radius is smaller than the Buchdahl radius, the radius for total collapse, and the corresponding Buchdahl temperature is greater than the equal action temperature. So above the Buchdahl temperature, the system collapses and the phase is constituted by a black hole. For ΛR2≥1 a phase analysis is also made. Published by the American Physical Society 2024