Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes

TL;DR

This work extends black hole thermodynamics to asymptotically de Sitter spacetimes by treating the cosmological constant as a thermodynamic pressure $P$ and deriving three independent first-law relations with conjugate volumes $V_h$, $V_c$, and $V$ using a Hamiltonian perturbation framework. It shows that $V_h$ and $V_c$ obey reverse isoperimetric inequalities while the inter-horizon volume $V$ satisfies a true isoperimetric bound, with $V$ equaling the geometric volume $V'$ between horizons in rotating cases. The analysis covers Kerr-dS black holes in all dimensions, Kerr-Newman-dS in $D=4$, and a charged rotating $D=5$ Einstein-Chern-Simons-dS solution, providing both analytic results and numerical support for the general inequalities. A compressibility and speed-of-sound study is included for $D=4$, and the results support the conjecture that these thermodynamic relations and inequalities hold broadly for asymptotically de Sitter black holes, with potential implications for cosmological scenarios such as inflation.

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.