Cardy–Verlinde formula and asymptotically de Sitter spaces

Rong-Gen Cai

doi:10.1016/s0370-2693(01)01457-5

Abstract

In this paper we discuss the question of whether the entropy of cosmological horizon in some asymptotically de Sitter spaces can be described by the Cardy-Verlinde formula, which is supposed to be an entropy formula of conformal field theory in any dimension. For the Schwarzschild-de Sitter solution, although the gravitational mass is always negative (in the sense of the prescription in hep-th/0110108 to calculate the conserved charges of asymptotically de Sitter spaces), we find that indeed the entropy of cosmological horizon can be given by using naively the Cardy-Verlinde formula. The entropy of pure de Sitter spaces can also be expressed by the Cardy-Verlinde formula. For the topological de Sitter solutions, which have a cosmological horizon and a naked singularity, the Cardy-Verlinde formula also works well. Our result is in favour of the dS/CFT correspondence.

Highlights

In a recent paper [1], Verlinde argued that the Cardy formula [2], describing the entropy of a certain conformal field theory (CFT) in 1 + 1 dimensions, can be generalized to any dimension, leading to the so-called Cardy-Verlinde formula
In the de Sitter (dS)/CFT correspondence, we have investigated the question of whether the entropy of cosmological horizon in asymptotically dS spaces can be described by the CardyVerlinde formula, which was established in the AdS/CFT correspondence [1]
For the Schwarzschild-dS solution, the gravitational mass, calculated in the prescription of Ref. [37], of the solution is always negative, we have found that the entropy of the cosmological horizon can be expressed in terms of a form [see (2.10)] of the Cardy-Verlinde formula

Summary

Introduction

In a recent paper [1], Verlinde argued that the Cardy formula [2], describing the entropy of a certain conformal field theory (CFT) in 1 + 1 dimensions, can be generalized to any dimension, leading to the so-called Cardy-Verlinde formula. According to the dS/CFT correspondence, it might be expected that as the case of AdS black holes [5], the thermodynamics of cosmological horizon in asymptotically dS spaces can be identified with. It is of great interest to see whether the entropy of the cosmological horizon can be described by the Cardy-Verlinde formula (1.3) This is the purpose of the present paper.. In this paper we will follow the prescription recently proposed in [37] to calculate the mass of gravitational field of asymptotically dS spaces (and the energy of corresponding CFTs). It is found that the entropy of cosmological horizon in some asymptotically dS spaces can be described in terms of the Cardy-Verlinde formula.

Schwarzschild-dS Solutions

Topological dS Solutions

Conclusions