A partition function for Schwarzschild-AdS and Kerr-AdS black holes and for quantized globally hyperbolic spacetimes with a negative cosmological constant

TL;DR

This paper develops a quantum statistical framework for Schwarzschild-AdS and Kerr-AdS black holes, deriving a partition function and related thermodynamic quantities, and explores their behavior as the cosmological constant varies, with implications for dark matter and dark energy.

Contribution

It introduces a novel quantum statistical approach to quantized black holes and hyperbolic spacetimes with a negative cosmological constant, including the definition of a partition function and thermodynamic analysis.

Findings

Partition function $Z$ is well-defined for these spacetimes.

Entropy $S$ and energy $E$ diverge as $ o 0$ and vanish as $| ext{cosmological constant}| o ext{infinity}.

Conjecture linking energy to dark matter and eigenvalues to dark energy density.

Abstract

We apply quantum statistics to our quantized versions of Schwarzschild-AdS and Kerr-AdS black holes and also to the quantized globally hyperbolic spacetimes having an asymptotically Euclidean Cauchy hypersurface by first proving, for the temporal Hamiltonian $H_0$, that $e^{-\beta H_0}$, $\beta>0$, is of trace class and then, that this result is also valid for the spatial Hamiltonian $H_1$, which has the same eigenvalues but with larger multiplicities. Since the lowest eigenvalue is strictly positive the extension of $e^{-\beta H_1}$ to the corresponding symmetric Fock space is also of trace class and we are thus able to define a partition function $Z$, the operator density $\rho$, the entropy $S$, and the average energy $E$. We prove that $S$ and $E$ tend to infinity if the cosmological constant $\Lambda$ tends to $0$ and vanish if $|\Lambda|$ tends to infinity. We also conjecture that $E$ is the source of the dark matter and that the dark energy density is a multiple of the eigenvalue of $\rho$ with respect to the vacuum vector which is $Z^{-1}$.