The disjointed thermodynamics of rotating black holes with a NUT twist

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