One conjecture and two observations on de Sitter space

We study the thermodynamic properties associated with the black hole horizon and cosmological horizon for the Gauss-Bonnet solution in de Sitter space. When the Gauss-Bonnet coefficient is positive, a locally stable small black hole appears in the case of spacetime dimension $d=5,$ the stable small black hole disappears, and the Gauss-Bonnet black hole is always unstable quantum mechanically when $d>~6.$ On the other hand, the cosmological horizon is found to be always locally stable independent of the spacetime dimension. But the solution is not globally preferred; instead, the pure de Sitter space is globally preferred. When the Gauss-Bonnet coefficient is negative, there is a constraint on the value of the coefficient, beyond which the gravity theory is not well defined. As a result, there is not only an upper bound on the size of black hole horizon radius at which the black hole horizon and cosmological horizon coincide with each other, but also a lower bound depending on the Gauss-Bonnet coefficient and spacetime dimension. Within the physical phase space, the black hole horizon is always thermodynamically unstable and the cosmological horizon is always stable; furthermore, as in the case of the positive coefficient, the pure de Sitter space is still globally preferred. This result is consistent with the argument that the pure de Sitter space corresponds to an UV fixed point of dual field theory.

Observers in de Sitter space can only access the space up to their cosmological horizon. Assuming thermal equilibrium, we use the quantum Ryu–Takayanagi or island formula to compute the entanglement entropy between the states inside the cosmological horizon and states outside, as a function of time. We obtain a Page curve that is bound at a value corresponding to the Gibbons–Hawking entropy. At this transition an ‘island’ forms, which is in a significantly different location as compared to when considering black hole horizons and even moves back in time. These differences turn out to be essential for non-violation of the no-cloning theorem in combination with entanglement wedge reconstruction. This consideration furthermore introduces the need for a scrambling time, the entropy dependence of which turns out to coincide with what is expected for black holes. The model we employ has classically pure three-dimensional de Sitter space as a solution. We dimensionally reduce to two dimensions in order to take into account semi-classical effects. Nevertheless, we expect the aforementioned qualitative features of the island to persist in higher dimensions.

We propose a novel prescription for computing the boundary stress tensor and charges of asymptotically de Sitter (dS) spacetimes from data at early or late time infinity. If there is a holographic dual to dS spaces, defined analogously to the AdS/conformal field theory correspondence, our methods compute the (Euclidean) stress tensor of the dual. We compute the masses of Schwarzschild--de Sitter black holes in four and five dimensions, and the masses and angular momenta of Kerr--de Sitter spaces in three dimensions. All these spaces are less massive than de Sitter space, a fact which we use to qualitatively and quantitatively relate de Sitter entropy to the degeneracy of possible dual field theories. Our results in general dimensions lead to a conjecture: Any asymptotically de Sitter spacetime with mass greater than de Sitter space has a cosmological singularity. Finally, if a dual to de Sitter space exists, the trace of our stress tensor computes the renormalized group (RG) equation of the dual field theory. Cosmological time evolution corresponds to RG evolution in the dual. The RG evolution of the c function is then related to changes in accessible degrees of freedom in an expanding universe.

Kerr-Newman black holes in a de Sitter (dS) space have the limit of rotating Nariai black holes with the near-horizon geometry of a warped dS3×S1/Z2 when the black hole horizon and the cosmological horizon coincide or approach close to each other. We study the rotation effect on the spontaneous emission of charges in the near-extremal rotating charged Nariai black hole and compare it to those from the near-extremal Nariai black hole in [C.-M. Chen , ] and near-extremal Kerr-Newman black hole in de Sitter space in [C.-M. Chen and S. P. Kim, ]. In strong contrast to the near-extremal Kerr-Newman black hole in dS space, the near-extremal rotating Nariai black hole also has an exponential amplification for the emission of high-energy charges, which becomes catastrophic regardless of angular momentum when two horizons coincide. The radius of rotating Nariai black holes monotonically increases as the angular momentum and charge of black holes increase, which gives a weaker electric field on the horizon than Nariai black holes. Thus, the angular momentum of black holes that drags particles on the horizon decreases the mean number of charges by a factor not by an order. We observe a catastrophic emission of boson condensation for charges with an effective energy equal to the chemical potential in the spacelike outer region of the cosmological horizon. Remarkably, the Schwinger emission of charges in the standard particle model may prevent the rotating Nariai black holes from evolving into spacetimes with a naked singularity when the angular momentum is close to the allowed maximum, which Nariai black holes cannot avoid.

We first briefly discuss the relation between black hole thermodynamics and the entropy function formalism. We find that an equation which governs the relationship between Sen's entropy function and black hole entropy, can quickly give higher order corrections to entropy of pure (anti-) de Sitter space without knowing the corrected metric. We also show that near horizon geometry and the entropy function extremization is no longer required for pure (anti-)de Sitter space. The entropy of (anti-)de Sitter space and Schwarzschild-(anti-) de Sitter black holes together with Gauss-Bonnet terms, $R^2$ terms and $R^4$ terms are calculated as concrete examples.

Cardy–Verlinde formula and thermodynamics of black holes in de Sitter spaces

We study a close relationship between the topological anti-de Sitter (TAdS) black holes and topological de Sitter (TdS) spaces including the Schwarzschild–de Sitter (SdS) black hole in five dimensions. We show that all thermal properties of the TdS spaces can be found from those of the TAdS black holes by replacing k by −k. Also we find that all thermal information for the cosmological horizon of the SdS black hole is obtained from either the hyperbolic-AdS black hole or the Schwarzschild–TdS space by substituting m with −m. For this purpose we calculate thermal quantities of bulk (Euclidean) conformal field theory (ECFT) and moving domain wall by using the A(dS)/(E)CFT correspondences. Further, we compute logarithmic corrections to the Bekenstein–Hawking entropy, Cardy–Verlinde formula and Friedmann equation due to thermal fluctuations. It implies that in the thermal relation between the TdS spaces and TAdS black holes, the cosmological horizon plays the same role as the horizon of TAdS black holes. Finally we note that the dS/ECFT correspondence is valid for the TdS spaces in conjunction with the AdS/CFT correspondence for the TAdS black holes.

We study two-dimensional de Sitter universe which evolves and proliferates according to the Ginsparg-Perry-Bousso-Hawking mechanism, using Jackiw-Teitelboim gravity coupled to conformal matter. Black holes are generated by quantum gravity effects from pure de Sitter space and then evaporate to yield multiple disjoint de Sitter spaces. The back-reaction from the matter CFT is taken into account for the dilaton as a function of the temperature of the CFT. We discuss the evaporation of black holes and calculate the finite temperature entropy of an inflating region using the island formula. We find that the island moves towards the apparent horizon of the black hole as the temperature increases. The results are applied to the case of multiple evaporating black holes, for which we suggest multiple islands.

We consider black holes in 2d de Sitter JT gravity coupled to a CFT, and entangled with matter in a disjoint non-gravitating universe. Tracing out the entangling matter leaves the CFT in a density matrix whose stress tensor backreacts on the de Sitter geometry, lengthening the wormhole behind the black hole horizon. Naively, the entropy of the entangling matter increases without bound as the strength of the entanglement increases, but the monogamy property predicts that this growth must level off. We compute the entropy via the replica trick, including wormholes between the replica copies of the de Sitter geometry, and find a competition between conventional field theory entanglement entropy and the surface area of extremal “islands” in the de Sitter geometry. The black hole and cosmological horizons both play a role in generating such islands in the backreacted geometry, and have the effect of stabilizing the entropy growth as required by monogamy. We first show this in a scenario in which the de Sitter spatial section has been decompactified to an interval. Then we consider the compact geometry, and argue for a novel interpretation of the island formula in the context of closed universes that recovers the Page curve. Finally, we comment on the application of our construction to the cosmological horizon in empty de Sitter space.

The Schwarzschild–de Sitter (SdS) metric is the simplest spacetime solution in general relativity with both a black hole event horizon and a cosmological event horizon. Since the Schwarzschild metric is the most simple solution of Einstein’s equations with spherical symmetry and the de Sitter metric is the most simple solution of Einstein’s equations with a positive cosmological constant, the combination in the SdS metric defines an appropriate background geometry for semi-classical investigation of Hawking radiation with respect to past and future horizons. Generally, the black hole temperature is larger than that of the cosmological horizon, so there is heat flow from the smaller black hole horizon to the larger cosmological horizon, despite questions concerning the definition of the relative temperature of the black hole without a measurement by an observer sitting in an asymptotically flat spacetime. Here we investigate the accelerating boundary correspondence of the radiation in SdS spacetime without such a problem. We have solved for the boundary dynamics, energy flux and asymptotic particle spectrum. The distribution of particles is globally non-thermal while asymptotically the radiation reaches equilibrium.

Logarithmic corrections to three-dimensional black holes and de Sitter spaces

We propose a new information transfer protocol for de Sitter space, using black holes as energy reservoirs. We consider antipodal observers in pure de Sitter space in the Bunch-Davis state. They can store Hawking modes from the cosmological horizon in a box. Alternatively, due to thermal fluctuations in de Sitter space, black holes formed through a pair-creation process can be used as energy reservoirs. We focus on the Nariai black hole case, which corresponds to an equilibrium state. Once the black hole is produced, energy pulses can be released into its interior, opening a traversable wormhole. We provide bounds for the amount of information that can be transferred. Specializing in (1+1)-dimensions, we explore how the teleportation protocol leads to an explicit geometric description of the information transmitted through an island region. The protocol uncovers quantum information aspects of de Sitter space, independently of any particular realization of de Sitter space holography.

We compute and clarify the interpretation of the on-shell Euclidean action for Reissner-Nordström black holes in de Sitter space. We show the on-shell action is minus the sum of the black hole and cosmological horizon entropy for arbitrary mass and charge in any number of dimensions. This unifying expression helps to clear up a confusion about the Euclidean actions of extremal and ultracold black holes in de Sitter, as they can be understood as special cases of the general expression. We then use this result to estimate the probability for the pair creation of black holes with arbitrary mass and charge in an empty de Sitter background, by employing the formalism of constrained instantons. Finally, we suggest that the decay of charged de Sitter black holes is governed by the gradient flow of the entropy function and that, as a consequence, the regime of light, superradiant, rapid charge emission should describe the potential decay of extreme charged Nariai black holes to singular geometries.

We combine the well-known Beltrami–Klein model of non-Euclidean geometry on a two-dimensional disk, where the geodesics are the chords of the disk, with the two—dimensional de Sitter space. The geometry of the de Sitter space is defined on the complement of the Beltrami–Klein disk in the plane, with the de Sitter metric being the unique Lorentzian Einstein metric whose light cones form cones tangent to the disk in this complement. This leads to a Beltrami–de Sitter model on the plane R 2 , which is endowed with the Riemannian Beltrami metric on the disk and the Lorentzian de Sitter metric outside the disk in R 2 . We explore the relevance of this model for Penrose’s conformal cyclic cosmology, first in the two-dimensional setting and subsequently in higher dimensions, including the physically significant case of four dimensions. In this context, we define a Radon-like transform between the de Sitter and Beltrami spaces, facilitating the purely geometric transformation of physical fields from the Lorentzian de Sitter space to the Riemannian Beltrami space. In the two –, and three-dimensional cases, we also uncover a hidden G 2 symmetry associated with the de Sitter spaces in these dimensions, which is related to a certain vector distribution naturally defined by the geometry of the model. We suggest the potential for discovering similar hidden symmetries in the n -dimensional Beltrami–de Sitter model.

We study the wave equation for a massive scalar field in three-dimensional anti-de Sitter (AdS) black hole and de Sitter (dS) spaces to find what are the differences and similarities between the two spaces. Here the AdS black hole is provided by the J = 0 BTZ black hole. To investigate its event (cosmological) horizons, we compute the absorption cross section, quasinormal modes and study the AdS(dS)/CFT correspondences. Although there remains an unclear point in defining the ingoing flux near infinity of the BTZ black hole, quasi-normal modes are obtained and the AdS/CFT correspondence is confirmed. However, we do not find quasi-normal modes and thus do not confirm the assumed dS/CFT correspondence. This difference between AdS black hole and dS spaces is very interesting, because their global structures are similar to each other.