Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes (1301.5926v2)

Abstract: We consider the thermodynamics of rotating and charged asymptotically de Sitter black holes. Using Hamiltonian perturbation theory techniques, we derive three different first law relations including variations in the cosmological constant, and associated Smarr formulas that are satisfied by such spacetimes. Each first law introduces a different thermodynamic volume conjugate to the cosmological constant. We examine the relation between these thermodynamic volumes and associated geometric volumes in a number of examples, including Kerr-dS black holes in all dimensions and Kerr-Newman-dS black holes in D=4. We also show that the Chong-Cvetic-Lu-Pope solution of D=5 minimal supergravity, analytically continued to positive cosmological constant, describes black hole solutions of the Einstein-Chern-Simons theory and include such charged asymptotically de Sitter black holes in our analysis. In all these examples we find that the particular thermodynamic volume associated with the region between the black hole and cosmological horizons is equal to the naive geometric volume. Isoperimetric inequalities, which hold in the examples considered, are formulated for the different thermodynamic volumes and conjectured to remain valid for all asymptotically de Sitter black holes. In particular, in all examples considered, we find that for fixed volume of the observable universe, the entropy is increased by adding black holes. We conjecture that this is true in general.

• The paper derives three distinct first laws using Hamiltonian perturbation theory that connect variations in the cosmological constant to thermodynamic volumes between event and cosmological horizons.
• The paper establishes a reverse isoperimetric inequality in de Sitter spacetimes, contrasting the behavior observed in anti-de Sitter cases and suggesting that black holes enhance universe entropy at fixed volumes.
• The paper extends its analysis to Kerr-Newman-de Sitter and analytically continued Chong-Cvetic-Lu-Pope solutions, providing pivotal insights into black hole thermodynamics relevant for cosmic inflation models.

Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes

The paper "Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes" offers a comprehensive examination of the thermodynamics of asymptotically de Sitter (dS) black holes, with a rigorous application of Hamiltonian perturbation theory to derive various first law relations. These investigations include variation with respect to the cosmological constant, resulting in associated Smarr formulas. This paper extends existing knowledge in the field by focusing on the role of thermodynamic volumes associated with these black holes, particularly within the context of dS spacetimes where positive cosmological constants are considered as thermodynamic variables.

Key Insights and Contributions

The authors derive three distinct versions of the first law, each introducing different thermodynamic volumes that correspond to variations in the cosmological constant. These volumes are linked to the physical regions between the respective event and cosmological horizons of black holes. This work not only explores how these regions interact thermodynamically but also raises noteworthy conjectures on the behavior of entropy in dS spacetimes.

A notable proposition within the paper is the hypothesis that the entropy of the universe increases with the addition of black holes, holding for fixed thermodynamic volumes. This insight is derived from a detailed paper of thermodynamic inequalities, particularly a reverse isoperimetric inequality that stands in contrast with that of AdS spacetimes.

The paper comprehensively references the results in Kerr-Newman-de Sitter spacetimes, and extends the discussion to the Chong-Cvetic-Lu-Pope solutions analytically continued to a positive cosmological constant. In these cases, it affirms that the thermodynamic piece associated with the volume between the horizons equates to what can be understood as the "observable universe".

Practical and Theoretical Implications

The findings presented have significant implications both theoretically and practically. On a theoretical level, they confirm and solidify notions within black hole thermodynamics by extending these ideas to dS spacetimes, thereby expanding the understanding of black hole entropy in cosmologies with positive vacuum energies. Practically, these insights could inform models that speculate on entropy distribution in cosmologies that entertain inflationary epochs, potentially contributing to the broader discourse on the likelihood of inflationary processes in the context of black hole presence.

Speculation on Future Developments

The exploration of thermodynamic volumes in dS contexts, particularly as a function of black holes, opens avenues for further inquiry, especially in higher-dimensional spacetimes and possibly within different gravity theories. Notions such as compressibility and speed of sound in black hole thermodynamics, as briefly touched upon towards the end of the paper, could yield new insights if explored further.

Conclusion

This work offers a broad and technically detailed exploration of de Sitter black holes, navigated through the lens of thermodynamic volumes and isoperimetric inequalities. While rigorous in its mathematical formulations, it also conjectures on broader cosmological implications that carry forward the discourse within theoretical physics on the interplay between black holes, entropy, and the universe's expansion dynamics. The objective presentation and cross-comparison of dS with existing AdS insights render this work indispensable for forming a complete understanding of black hole thermodynamics amid different spacetime asymptotics.