“Nernst theorem” and black hole thermodynamics

The Nernst formulation of the third law of ordinary thermodynamics (often referred to as the “Nernst theorem”) asserts that the entropy $S$ of a system must go to zero (or a “universal constant”) as its temperature $T$ goes to zero. This assertion is commonly considered to be a fundamental law of thermodynamics. As such, it seems to spoil the otherwise perfect analogy between the ordinary laws of thermodynamics and the laws of black hole mechanics, since rotating black holes in general relativity do not satisfy the analogue of the “Nernst theorem.” The main purpose of this paper is to attempt to lay to rest the “Nernst theorem” as a law of thermodynamics. We consider a boson (or fermion) ideal gas with its total angular momentum $J$ as an additional state parameter, and we analyze the conditions on the single-particle density of states, $g(\epsilon ,j)$, needed for the Nernst formulation of the third law to hold. (Here, $\epsilon$ and $j$ denote the single-particle energy and angular momentum.) Although it is shown that the Nernst formulation of the third law does indeed hold under a wide range of conditions, some simple classes of examples of densities of states which violate the “Nernst theorem” are given. In particular, at zero temperature, a boson (or fermion) gas confined to a circular string (whose energy is proportional to its length) not only violates the “Nernst theorem” also but reproduces some other thermodynamic properties of an extremal rotating black hole.