Thermal Hawking Broadening and Statistical Entropy of Black Hole Wave Packet
34G~ A (1) constitutes a triple point in the phase of physical theories, touching gravity, even beyond general relativity, quantum mechanics, and statistical mechanics. However, despite its central role in physics, and several illuminating derivations [2], its statistical mechanical roots have not been fully revealed. There are only a few exceptions [3] where one can actually account, at some limit, for the number of micro states. As far as the prototype Schwarzschild black hole is concerned, we still do not know where the micro states are hiding and even what we are about to enumerate. A black hole is classically characterized by its event horizon, but once ~ is switched on, even an innocent looking question like ’where is this horizon located’ still lacks a meaningful answer in the quantum or even semi classical sense. In this paper, we examine the possibility that the Schwarzschild black hole states are apparently hidden simply because they degenerate into one single general relativistic state at the ~ → 0 limit. We carry out our analysis within the framework of the mini super spacetime (a variant of the mini superspace [4] limit of Wheeler-DeWitt formalism) without relying on string theory and/or loop quantum gravity. We do it by invoking the most classical, that is of minimal uncertainty, Hartle-Hawking wave packet solution. The variance of the emerging statistical mass distribution is then interpreted as thermal Hawking broadening, with the corresponding semi-classical picture resembling a Gaussian horizon profile with Compton width (horizon fluctuations were considered in [5, 6]). The quantum mechanically exposed mass spectrum is Hawking temperature dependent. The analytical calculation of the associated statistical entropy then calls for the Fowler prescription [7] which has been designated to deal with temperature dependent energy levels. Our starting point is the static spherically symmetric line element