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Upper bound for entropy in asymptotically de Sitter space-time

Kengo Maeda, Tatsuhiko Koike, Makoto Narita, Akihiro Ishibashi

TL;DR

The paper addresses horizon thermodynamics in asymptotically de Sitter space-times with a black hole, focusing on the cosmological and black hole horizons. It introduces a Λ-adjusted quasi-local energy and proves that the CEH area is non-decreasing toward future timelike infinity, and that the combined BEH+CEH area satisfies $A_B + A_C \le \frac{12\pi}{\Lambda}$, yielding a bound on total entropy $S_B + S_C + \sqrt{S_B S_C} \le \frac{3\pi}{\Lambda}$. The results rely on energy conditions and a generalized area theorem, using a Hayward-type energy $E({\cal S})$ to establish monotonicity and positivity. These findings support a generalized second law for total horizon-plus-matter entropy and connect to the cosmic no-hair scenario, with de Sitter space expected to maximize total entropy for fixed $\Lambda$.

Abstract

We investigate nature of asymptotically de Sitter space-times containing a black hole. We show that if the matter fields satisfy the dominant energy condition and the cosmic censorship holds in the considering space-time, the area of the cosmological event horizon for an observer approaching a future timelike infinity does not decrease, i.e. the second law is satisfied. We also show under the same conditions that the total area of the black hole and the cosmological event horizon, a quarter of which is the total Bekenstein-Hawking entropy, is less than $12π/Λ$, where $Λ$ is a cosmological constant. Physical implications are also discussed.

Upper bound for entropy in asymptotically de Sitter space-time

TL;DR

The paper addresses horizon thermodynamics in asymptotically de Sitter space-times with a black hole, focusing on the cosmological and black hole horizons. It introduces a Λ-adjusted quasi-local energy and proves that the CEH area is non-decreasing toward future timelike infinity, and that the combined BEH+CEH area satisfies , yielding a bound on total entropy . The results rely on energy conditions and a generalized area theorem, using a Hayward-type energy to establish monotonicity and positivity. These findings support a generalized second law for total horizon-plus-matter entropy and connect to the cosmic no-hair scenario, with de Sitter space expected to maximize total entropy for fixed .

Abstract

We investigate nature of asymptotically de Sitter space-times containing a black hole. We show that if the matter fields satisfy the dominant energy condition and the cosmic censorship holds in the considering space-time, the area of the cosmological event horizon for an observer approaching a future timelike infinity does not decrease, i.e. the second law is satisfied. We also show under the same conditions that the total area of the black hole and the cosmological event horizon, a quarter of which is the total Bekenstein-Hawking entropy, is less than , where is a cosmological constant. Physical implications are also discussed.

Paper Structure

This paper contains 6 sections, 10 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Asymptotically de Sitter space-time and the area law for a cosmological event horizon
  3. Quasi-local energy in space-times with $\Lambda$
  4. Upper bound for the area
  5. Conclusions and Discussion
  6. Proof of Proposition 1

Key Result

Lemma 1

(Each component of) any sufficiently nice cut of the cosmological event horizon $H^-({\cal I}^+)$ is a topological two-sphere.

Figures (2)

  • Figure 1: The mass parameter $m$ of a Schwarzschild-de Sitter solution for a fixed $\Lambda$ is related to the area $A$ of event horizons as $m =(A/16\pi)^{1/2}(1-\Lambda A/12\pi)$. $A_b$, $A_c$ are the areas of a BEH and a CEH, respectively.
  • Figure 2: Penrose diagrams of Schwarzschild-de Sitter space-times with mass parameters (a)$m>0$, and (b)$m<0$, respectively.

Theorems & Definitions (10)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4: Limit Curve Lemma
  • Lemma 5
  • Theorem 1: Area law for a CEH
  • Proposition 1
  • Lemma 6
  • Lemma 7
  • Theorem 2