Surface Area Products for Kerr-Taub-NUT Space-time
  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. 
Parthapratim Pradhan
gr-qc math-ph math.MP