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

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 $Z^{-1}$.