Bose gas modeling of the Schwarzschild black hole thermodynamics

Black holes violate the third law of thermodynamics, and this gives rise to difficulties with the microscopic description of their entropy. Recently, it has been shown that the microscopic description of the Schwarzschild black hole thermodynamics in \(D = 4\) space–time dimensions is provided by the analytic continuation of the entropy of Bose gas with a nonrelativistic one-particle energy to \(d =-4\) negative spatial dimensions. In this paper, we show that the \(D=5\) and \(D=6\) Schwarzschild black holes thermodynamics can be modeled by the \(d\)-dimensional Bose gas, \(d=1,2,3,\dots\,\), with the one-particle energy \(\varepsilon(k)=k^\alpha\) under the respective conditions \(\alpha=-d/3\) and \(\alpha=-d/4\). In these cases, the free energy of the Bose gas has divergences, and we introduce a cut-off and perform the minimal renormalizations. We also perform renormalizations using analytic regularization and prove that the minimal cut-off renormalization gives the same answer as the analytic regularization by the Riemann zeta-function.