Adventures in de Sitter space

TL;DR

The article surveys the semiclassical thermodynamics of de Sitter space, focusing on horizon entropy and temperature and their role in entropy bounds (Bekenstein, D-bound) and covariant formulations. It analyzes how different asymptotic structures (dS+, dS±, all(Λ)) constrain observable entropy and the challenges of defining quantum gravity in spacetimes with Λ>0, including finite Hilbert-space proposals and the Λ–N correspondence. A central theme is the Nariai instability of maximal Schwarzschild–de Sitter black holes, which can drive fragmentation into multiple de Sitter regions and reveal rich global structures relevant to holography and observables. The discussion situates these issues within the broader effort to formulate quantum gravity in de Sitter space, including potential string-theoretic approaches and the role of asymptotic boundaries. Overall, the work highlights how entropy bounds, horizon thermodynamics, and nontrivial instabilities shape our understanding of quantum gravity in cosmological spacetimes with positive cosmological constant.

Abstract

This is my contribution to the Festschrift honoring Stephen Hawking on his 60th birthday. Twenty-five years ago, Gibbons and Hawking laid out the semi-classical properties of de Sitter space. After a summary of their main results, I discuss some further quantum aspects that have since been understood. The largest de Sitter black hole displays an intriguing pattern of instabilities, which can render the boundary structure arbitrarily complicated. I review entropy bounds specific to de Sitter space and outline a few of the strategies and problems in the search for a full quantum theory of the spacetime.

Figures (5)

• Figure 1: de Sitter space as a hyperboloid. Time goes up.---Right: Penrose diagram. Horizontal lines represent three-spheres.
• Figure 2: Past and future event horizon (diagonal lines). The static slicing covers the interior of the cosmological horizon (shaded).
• Figure 3: Causal diamond $C(p,q)$, bounded by top (T) and bottom (B) cone, which intersect on the edge (E). A spatial region that fails to fit into any causal diamond cannot be probed in any experiment.
• Figure 4: Penrose diagram of a Schwarzschild-de Sitter spacetime. The curved line is a slice of equal time in the static coordinates; its geometry is a warped product of $S^1$ and $S^2$. The $S^2$ directions are suppressed in this diagram; the $S^1$ arises because the left and right ends are identified. The black hole (b) and cosmological (c) horizons are indicated. The static coordinates, Eq. (\ref{['eq-schds']}), cover one of the diamond-shaped regions. The black hole singularity and the de Sitter infinity are shown as dashed and bold lines, respectively. The Nariai solution has the same Penrose diagram except for the nature of the boundaries. The Penrose diagram for a multiple Schwarzschild-de Sitter solution is obtained by joining several copies of this diagram before identifying the ends.
• Figure 5: Fragmentation. Nariai space is a product $S^1 \times S^2$. The $S^2$ is represented as an $S^1$ here; the time direction is suppressed in the drawings and indicated only through the arrows evolving the snapshots of the spatial geometry. Upon a higher-mode perturbation (here, $n=2$), Nariai space can evolve into a sequence of Schwarzschild-de Sitter universes. Black hole (b) and cosmological (c) horizons are indicated. When the black holes evaporate, the geometry pinches in several places, and only disconnected de Sitter portions remain.