Black Holes: Fermions at the Extremal Limit?

I present exact results matching Kerr-Newman Black Hole thermodynamics in the extremal limit to the two-dimensional Fermi Gas. Two dimensions are consistent with the membrane paradigm of black holes. Key in the analysis is the thermodynamic Riemannian curvature scalar R, negative for most ordinary thermodynamic systems, including those near the critical point, but mostly positive for the Kerr-Newman Black Hole and the Fermi Gas. The Kerr-Newman Black Hole (KNBH), with mass M , angular momentum J , and charge Q, has a well-established thermodynamic structure originated by Bekenstein [1] and Hawking [2]. Logically, this should be supported ∗Electronic address: [email protected] 1 ar X iv :0 71 1. 43 28 v1 [ gr -q c] 2 7 N ov 2 00 7 by some underlying microscopic picture, a ”statistical mechanics” of black holes. But, since matter is expected to rapidly collapse to the central singularity, what entities could drive this statistics? I argue that insight results from comparing known KNBH thermodynamics to standard statistical mechanical models. A major element here [3] is the thermodynamic Riemannian curvature scalar R, first evaluated for the KNBH by Åman, Bengtsson, and Pidokrajt [4]. In the sign convention of Weinberg [5], R is mostly positive for the KNBH, but mostly negative for ordinary thermodynamic systems, including most critical point models. An exception is the three-dimensional (3D) Fermi Gas, which Janyszek and Mruga la [6] found to have positive R. These authors also emphasized the general importance of the sign of R. Here, I work out the 2D Fermi Gas and find some exact correspondences with the KNBH in the extremal limit, where the temperature goes to zero. The KNBH entropy is [7] S(M,J,Q) = 1 8 ( 2M −Q + 2 √ M4 − J2 −M2Q2 ) , (1) in geometrized units [8], where S and J are in cm, and M and Q are in cm. The temperature T is given by 1 T ≡ ( ∂S ∂M ) J,Q = (K + 2K + L)M 4K , (2) and the heat capacity CJ,Q ≡ T ( ∂S ∂T ) J,Q = MK(K + L + 2K) 4(L2 − 2K) . (3) Here, the dimensionless simplifying variables {K,L} ≡ { √ 1− α− β, √ 1 + α}, (4) 2 with α ≡ J/M and β ≡ Q/M. To be in the physical regime with real, positive S and T requires