[Paper Review] How far can the generalized second law be generalized?

P. C. W. Davies, T. M. Davis|arXiv (Cornell University)|Oct 18, 2003

Cosmology and Gravitation Theories4 citations

TL;DR

This paper investigates the validity of the generalized second law (GSL) of thermodynamics when extended to cosmological event horizons in Friedmann-Robertson-Walker (FRW) universes with a positive cosmological constant. Using analytical and numerical methods, it demonstrates that the total entropy—comprising black hole and cosmological horizon entropy—increases monotonically over time, confirming the GSL holds even in dynamic, expanding universes, provided black holes remain smaller than the cosmological horizon.

ABSTRACT

Jacob Bekenstein's identification of black hole event horizon area with entropy proved to be a landmark in theoretical physics. In this paper we trace the subsequent development of the resulting generalized second law of thermodynamics (GSL), especially its extension to incorporate cosmological event horizons. In spite of the fact that cosmological horizons do not generally have well-defined thermal properties, we find that the GSL is satisfied for a wide range of models. We explore in particular the case of an asymptotically de Sitter universe filled with a gas of small black holes as a means of casting light on the relative entropic 'worth' of black hole versus cosmological horizon area. We present some numerical solutions of the generalized total entropy as a function of time for certain cosmological models, in all cases confirming the validity of the GSL.

Motivation & Objective

• To extend the generalized second law (GSL) of thermodynamics from black hole event horizons to cosmological event horizons in expanding FRW universes.
• To investigate whether the total entropy—defined as the sum of black hole and cosmological horizon entropy—remains non-decreasing over time in models with a positive cosmological constant.
• To assess the relative entropic contributions of black holes versus cosmological horizons in asymptotically de Sitter universes.
• To provide numerical and analytical evidence supporting the GSL in non-stationary, dynamic cosmological spacetimes.
• To reconcile the GSL with the dynamics of black hole evaporation and horizon evolution in a universe dominated by dark energy.

Proposed method

• Derives the rate of change of cosmological horizon entropy using the area law: $ \dot{S}_{\rm c} = \dot{A}_{\rm c}/4 $, where $ A_{\rm c} $ is the area of the cosmological event horizon.
• Models the evolution of black hole entropy via the energy density $ \rho_{\rm bh} $ and number density $ n_{\rm bh} $, assuming a gas of small black holes in a comoving frame.
• Applies the Friedmann equations (Eqs. 11 and 12) to describe the time evolution of the scale factor and Hubble parameter in FRW models with $ \Omega_M = 0.3 $, $ \Omega_\Lambda = 0.7 $ or $ 1.4 $.
• Computes the total black hole area as $ A_{\rm bh}^{\rm total} = 64\pi^2 M_{\rm bh} \rho_{\rm bh} / (3H^3) $, and its time derivative to assess entropy loss.
• Imposes the GSL condition $ \dot{A}_{\rm bh}^{\rm total} + \dot{A}_{\rm c} \geq 0 $, leading to the inequality $ M_{\rm bh} \lesssim 1/H $, or $ r_{\rm bh} \lesssim r_{\rm c} $.
• Performs numerical simulations of entropy evolution over time for specific cosmological models, including a closed universe with $ \Omega_\Lambda = 1.4 $, to verify monotonic increase in total entropy.

Experimental results

Research questions

• RQ1 Does the generalized second law (GSL) hold for cosmological event horizons in expanding FRW universes with a positive cosmological constant?
• RQ2 Under what conditions does the total entropy—sum of black hole and cosmological horizon entropy—remain non-decreasing over time?
• RQ3 How do the entropic contributions of small black holes compare to those of the cosmological horizon in a de Sitter-like universe?
• RQ4 Can the GSL be satisfied dynamically in non-stationary spacetimes, such as those undergoing accelerated expansion?
• RQ5 What constraints does the GSL impose on the size of black holes relative to the cosmological horizon?

Key findings

• The generalized second law (GSL) is satisfied at all times in the studied FRW models, including those with significant deviations from de Sitter space, as total entropy increases monotonically with cosmic time.
• Numerical solutions for a radiation-dominated FRW universe with $ \Omega_M = 0.3 $, $ \Omega_\Lambda = 0.7 $ confirm that the increase in cosmological horizon entropy exceeds the loss of radiation entropy across the horizon, preserving the GSL.
• For a closed universe with $ \Omega_\Lambda = 1.4 $, the cosmological event horizon forms at a finite time, and the total entropy increases monotonically from the moment the horizon appears.
• The analytical condition $ M_{\rm bh} \lesssim 1/H $ (or $ r_{\rm bh} \lesssim r_{\rm c} $) is derived as a sufficient condition for the GSL to hold, indicating that small black holes do not violate the law.
• The GSL remains valid even when black holes evaporate and lose area, provided their size remains smaller than the cosmological horizon, as the increase in horizon area compensates for entropy loss.
• The results support and illustrate broader theorems on horizon area non-decrease in asymptotically de Sitter spacetimes, confirming the GSL holds not just asymptotically but dynamically at all times in the models studied.