Density of states for the gravitational field in black-hole topologies

Using a previously developed formulation of black-hole thermodynamics for a system of finite size, we show that the partition function of the related canonical ensemble can be used to obtain the density of states in the microcanonical ensemble. We work with the partition function in a zero-loop approximation based on classical solutions with fixed boundary data and obtain the corresponding density of states by using an inverse Laplace transform. The computation requires the introduction of a uniformizing variable so that a path can be defined along which the classical action of a stable black hole is single valued in the integration over imaginary inverse temperatures. Although the zero-loop partition function is not a Laplace transform, its inversion integral yielding the density of states is nevertheless well defined, and we show that it is the exact inverse of a Fourier-Laplace integration over all real energies, positive and negative. We argue that (1) this need for all positive and negative energies and (2) a constraint on the boundary data for obtaining the partition function from classical solutions are both consequences of the zero-loop approximation that should be absent in a complete quantum theory. We also find that the single-valued action enables a discussion of negative temperatures for positive mean energies that appear to be essential for a satisfactory relation between the partition function and the density of states in the zero-loop approximation. The full significance of this observation must await further study.