Hawking radiation: black hole vs de Sitter

TL;DR

This paper compares black hole and de Sitter thermodynamics, focusing on Hawking radiation and entropy, and examines the validity of the Gibbons-Hawking entropy in higher dimensions.

Contribution

It introduces a modified entropy formula for the Hubble volume in de Sitter space, extending the Gibbons-Hawking entropy to arbitrary dimensions.

Findings

The local de Sitter temperature is universal and twice the Gibbons-Hawking temperature.

The entropy of the Hubble volume scales as (d-1)A/8G, generalizing the Gibbons-Hawking entropy.

The original Gibbons-Hawking entropy is only valid in 3+1 dimensions.

Abstract

We discuss the difference between the thermodynamics of black holes and thermodynamics of the de Sitter expansion. Both systems experience the Hawking radiation, but its impact on thermodynamics is different. As distinct from the thermodynamics of black holes, which are finite compact objects, the de Sitter state is the infinite and homogeneous state. The presence of the cosmological horizon provides two sides of the de Sitter thermodynamics: the local thermodynamics and the thermodynamics related to the cosmological horizon. We discuss the connection between these two sides considering the entropy of the Hubble volume in de Sitter spacetime -- the region inside the horizon. On one hand there is the Gibbons-Hawking entropy $S_{\rm GH}=A/4G$ associated with the cosmological horizon. On the other hand this entropy can be obtained by integrating the local entropy density over the Hubble volume. In (3+1) spacetime, these two entropies coincide. This provides physical meaning and a natural explanation to the Gibbons-Hawking entropy -- it is the entropy in the volume $V_H$ bounded by the cosmological horizon. Here we consider whether the Gibbons-Hawking conjecture remains valid for the de Sitter state in general $d+1$ spacetime. To do this, we use the local de Sitter thermodynamics, characterized by a local temperature $T_{\rm dS}=H/\pi$. This temperature is not related to the horizon: it is the temperature of local activation processes, such as the ionization of an atom in the de Sitter environment, which occur deep within the cosmological horizon. This local temperature is universal and does not depend on dimension $d$, it is twice the Gibbons-Hawking temperature $T_{\rm GH}=H/2\pi$. We found that the entropy of the Hubble volume is $S_H=(d-1)A/8G$, which modifies the Gibbons-Hawking entropy of horizon. The original form of the Gibbons-Hawking entropy is valid only for $d=3$.