Area product and mass formula for Kerr–Newman–Taub–NUT spacetime

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 Lorentzian geometry. We observe that these thermodynamic products are 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 depend on mass, charge and NUT parameter of the background spacetime. Thus, we can conclude that the universal (mass-independence) behavior 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 first law of BH thermodynamics and 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 nontrivial NUT charge which makes the spacetime to be asymptotically non-flat, in contrast with KN BH. The other reason of the failure is that Lorentzian KNTN geometry contains Dirac–Misner type singularity, which is a manifestation of a nontrivial topological twist of the manifold. The BH mass formula and Christodoulou–Ruffini mass formula for KNTN BHs 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 play a crucial role in BH thermodynamics to understand the microscopic nature of BH entropy.