[Paper Review] Depressing de Sitter in the Frozen Future

A. P. Lundgren, R. Bondarescu+1 more|arXiv (Cornell University)|Jan 5, 2012

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This paper re-evaluates black hole evaporation in de Sitter space, arguing that the cosmological horizon's entropy is depressed by the presence of a black hole due to geometric mean scaling. As the black hole evaporates, entropy increases via heat flow across the temperature gradient, culminating in maximum entropy de Sitter space—challenging the use of flat Minkowski space as a thermodynamic limit due to its infinite entropy in this context.

ABSTRACT

In this paper we focus on the gravitational thermodynamics of the far future. Cosmological observations suggest that most matter will be diluted away by the cosmological expansion, with the rest collapsing into supermassive black holes. The likely future state of our local universe is a supermassive black hole slowly evaporating in an empty universe dominated by a positive cosmological constant. We describe some overlooked features of how the cosmological horizon responds to the black hole evaporation. The presence of a black hole depresses the entropy of the cosmological horizon by an amount proportional to the geometric mean of the entropies of the black hole and cosmological horizons. As the black hole evaporates and loses its mass in the process, the total entropy increases obeying the second law of thermodynamics. The entropy is produced by the heat from the black hole flowing across the extremely cold cosmological horizon. Once the evaporation is complete, the universe becomes empty de Sitter space that (in the presence of a true cosmological constant) is the maximum entropy thermodynamic equilibrium state. We propose that flat Minkowski space is an improper limit of this process which obscures the thermodynamics. The cosmological constant should be regarded not only as an energy scale, but also as a scale for the maximum entropy of a universe. In this context, flat Minkowski space is indistinguishable from de Sitter with extremely small cosmological constant, yielding a divergent entropy. This introduces an unregulated infinity in black hole thermodynamics calculations, giving possibly misleading results.

Motivation & Objective

• To re-express gravitational thermodynamics in the far future of our universe, where matter dilutes and only supermassive black holes remain.
• To address the overlooked thermodynamic effects of black holes on the cosmological horizon in de Sitter spacetime.
• To challenge the conventional use of flat Minkowski space as a thermodynamic limit, arguing it introduces unregulated infinities.
• To establish that de Sitter space with a positive cosmological constant is the true maximum entropy equilibrium state.
• To clarify how entropy is produced during black hole evaporation via heat flow across the cosmological horizon.

Proposed method

• Models a non-spinning supermassive black hole slowly evaporating in a de Sitter background with a fixed positive cosmological constant.
• Uses the area law for entropy, where horizon entropy is one-quarter the horizon area in Planck units.
• Applies the standard Hawking temperature formula for both black hole and cosmological horizons, with T = (1/2π)√(Λ/3) for the de Sitter horizon.
• Calculates the entropy change of the cosmological horizon due to the black hole's presence, showing it is depressed by an amount proportional to the geometric mean of the two horizon entropies.
• Analyzes heat flow from the hotter black hole horizon to the colder cosmological horizon, computing entropy production via δS = δQ/T.
• Considers the flat space limit by taking Λ → 0, showing it leads to divergent entropy and thus an unregulated infinity in thermodynamic calculations.

Experimental results

Research questions

• RQ1 How does the presence of a black hole affect the entropy of the cosmological horizon in de Sitter spacetime?
• RQ2 Why is the flat Minkowski space limit problematic for black hole thermodynamics in the context of a positive cosmological constant?
• RQ3 What is the role of heat flow across the temperature difference between the black hole and cosmological horizons in entropy production?
• RQ4 Does the final state of black hole evaporation in de Sitter space correspond to maximum entropy, and if so, how is this entropy quantified?
• RQ5 Can the cosmological constant be interpreted not only as an energy scale but also as a regulator of maximum entropy in a universe?

Key findings

• The cosmological horizon's entropy is depressed by an amount proportional to the geometric mean of the black hole and cosmological horizon entropies, due to the black hole's gravitational influence.
• As the black hole evaporates, the total entropy of the system increases, satisfying the second law of thermodynamics.
• Entropy production arises primarily from heat flowing from the hotter black hole horizon to the colder cosmological horizon, with the latter absorbing radiation and increasing its entropy.
• The final state of the universe after complete evaporation is empty de Sitter space, which has the maximum possible entropy for a given cosmological constant.
• The flat Minkowski space limit is thermodynamically ill-defined because it leads to infinite entropy, making it an improper limit for black hole thermodynamics.
• The cosmological constant acts not only as an energy scale but also as a regulator of the maximum entropy of the universe, making de Sitter space the true equilibrium state.