We examine properties of the inner and outer horizon thermodynamics of Taub-NUT (Newman-Unti-Tamburino) and Kerr-Taub-NUT (KTN) black hole (BH) in four dimensional \emph{ Lorentzian geometry}. We compare and contrasted these properties with the properties of Reissner Nordstr{ \o}m (RN) BH and Kerr BH. We focus on "area product", "entropy product", "irreducible mass product" of the event horizon and Cauchy horizons. Due to mass-dependence, we speculate that these products have no beautiful quantization feature. Nor does it has any universal property. We further observe that the \emph{ First law} of BH thermodynamics and \emph { Smarr-Gibbs-Duhem} relations do not hold for Taub-NUT (TN) and KTN BH in Lorentzian regime. The failure of these aforementioned features are due to presence of the non-trivial NUT charge which makes the space-time to be asymptotically non-flat, in contrast with RN BH and Kerr BH. The another reason of the failure is that Lorentzian TN and Lorentzian KTN geometry contains \emph{ Dirac-Misner type singularity}, which is a manifestation of a non-trivial topological twist of the manifold. The black hole \emph{ mass formula} and \emph{ Christodoulou-Ruffini mass formula} for TN and KTN BHs are also computed. This thermodynamic product formulae gives us further understanding to the nature of BH entropy (inner and outer) at the microscopic level.
Categories: gr-qc math-ph math.MP