The disjointed thermodynamics of rotating black holes with a NUT twist

We study the solutions of the Euclidean-signature Einstein equations (gravitational instantons) whose line-element takes the Kerr–Bolt form, characterized by three real parameters: a size, a NUT charge, and a spin rate. The exclusion of singularities eliminates most combinations of these parameters, leaving only a disconnected family of solution-manifolds between which continuous transitions are impossible. (The angular velocity divided by the temperature is forced to be a rational multiple of 2π  .) This “quantization” prevents the free variations presupposed by an equation like TdS=dM+ΩdJ, and thereby renders the first law true, false, meaningless, or tautological, depending on how one approaches it.