Surface area products for Kerr-Taub-NUT space-time

We examine properties of the inner and outer horizon thermodynamics of TaubNUT (Newman-Unti-Tamburino) and Kerr-Taub-NUT (KTN) black hole (BH) in four-dimensional Lorentzian geometry. We compare and contrasted these properties with the properties of Reissner Nordstrom (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 nice quantization feature. Nor do they have any universal property. We further observe that the first law of BH thermodynamics and 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 the presence of the non-trivial NUT charge which makes the space-time be asymptotically non-flat, in contrast with RN BH and Kerr BH. Another reason for the failure is that Lorentzian TN and Lorentzian KTN geometries contain Dirac-Misner-type singularity, which is a manifestation of a non-trivial topological twist of the manifold. The black-hole mass formula and Christodoulou-Ruffini mass formula for TN and KTN BHs are also computed. These thermodynamic product formulae give us further understanding of the nature of inner as well as outer BH entropy at the microscopic level. Copyright (C) EPLA, 2016