Microcanonical Quantum Statistics of Schwarzschild Black Holes

It is shown that a quantized Schwarzschild black hole, if described by a square root energy spectrum with exponential multiplicity, can be treated as a microcanonical ensemble without problem leading to the expected thermodynamical properties. PACS: 04.60.Kz; 04.70.Dy; 97.60.Lf Recently H.A.Kastrup [1-3] made the interesting observation that a Schwarzschild black hole with energy spectrum En = σ √ nEP , (n = 1, 2 . . . , EP =Planck energy) and multiplicity g n [4] leads to the expected thermodynamical properties if the divergent canonical partition function Z is analytically continued [2]. The analytic continuation in the complex g-plane gives a complex Z, the desired thermodynamics is obtained from its imaginary part. This situation has some similarity with Langer’s nucleation theory [5]. Therefore, it is argued [3] that the complex canonical partition function might signal the instability of the black hole. On the other hand, it is known that the statistical ensembles are not always equivalent. In this case it is necessary to return to the microcanonical ensemble which has the widest range of validity. For a black hole this is even more appealing because it is hard to imagine a thermal bath coupled to it. We will quickly see that our system can be treated microcanonically without problem. This is even the simplest microcanonical calculation I know because it uses mathematics from high school, only. Let energy eigenvalues En with multiplicities ν(n) be En = b √ n, ν(n) = g, n = 1, 2, . . . (1) where b = σEP , σ =O(1), g > 1 and EP = √ hc5/G is Planck’s energy. The microcanonical partition function Ω(E) is equal to the number of eigenvalues below the energy E, hence