The disjointed thermodynamics of rotating black holes with a NUT twist Original Research Article

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 image, and thereby renders the first law true, false, meaningless, or tautological, depending on how one approaches it.