Area Products and Mass Formula for Kerr-Newman-Taub-NUT Space-time

Abstract: We derive area product, entropy product, area sum and entropy sum of the event horizon and Cauchy horizons for Kerr-Newman-Taub-NUT(Newman-Unti-Tamburino) black hole in four dimensional \emph{Lorentzian geometry}. We observe that these thermodynamic products are \emph{not} universal(mass-independence) for this black hole(BH), whereas for Kerr-Newman(KN) BH such products are universal (mass-independence). We also examine the entropy sum and area sum. It is shown that they all are depends on mass, charge and NUT parameter of the back ground space-time. Thus we can conclude that the universal(mass-independence) behaviour of area product and entropy product, area sum and entropy sum for Kerr-Newman-Taub-NUT(KNTN) BH fails and which is also quite different from KN BH. We further show that the KNTN BH do not possess \emph{first law of BH thermodynamics } and \emph {Smarr-Gibbs-Duhem } relations, and that such relations are unlikely in the KN case. 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 KN BH. The another reason of the failure is that Lorentzian KNTN geometry contains \emph{Dirac-Misner type singularity}, which is a manifestation of a non-trivial topological twist of the manifold. The BH \emph{mass formula} and \emph{Christodoulou-Ruffini mass formula} for KNTN black holes are also derived. Finally, we compute the area bound which is just Penrose like inequality for event horizon. From area bound we derive entropy bound. These thermodynamic products on the multi horizon playing a crucial role in BH thermodynamics to understand the microscopic nature of BH entropy.