Circular null geodesic orbits, in extremal Reissner-Nordstrom spacetimes, are examined with regard to their stability, and compared with similar orbits in the near-extremal situation. Extremization of the effective potential for null circular orbits shows the existence of a stable circular geodesic in the extremal spacetime, precisely on the event horizon, which coincides with its null geodesic generator. Such an orbit also emerges as a global minimum of the effective potential for circular timelike orbits. This type of geodesic is of course absent in the corresponding nearextremal spacetime, as we show here, testifying to differences between the extremal limit of a generic RN spacetime and the exactly extremal geometry.

We argue by explicit computations that, although the area product, horizon radii product, entropy product, and irreducible mass product of the event horizon and Cauchy horizon are universal, the surface gravity product, the surface temperature product and the Komar energy product of the said horizons do not seem to be universal for Kerr-Newman black hole spacetimes. We show the black hole mass formula on the Cauchy horizon following the seminal work by Smarr [Phys Rev Lett 30:71 (1973), Phys Rev D 7:289 (1973] for the outer horizon. We also prescribe the four laws of black hole mechanics for the inner horizon. A new definition of the extremal limit of a black hole is discussed.

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 KerrNewman(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 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 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 Dirac-Misner type singularity, which is a manifestation of a non-trivial topological twist of the manifold. The BH mass formula and ChristodoulouRuffini 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.

Abstract. In this paper we investigate the equatorial causal (time-like and null) circular geodesics of the Kerr-Newman-Taub-NUT(Newman-Unti-Tamburino) black hole in four dimensional Lorentzian geometry. The special characteristics of this black hole is that it is of Petrov-Pirani type D and the photon trajectories are doubly degenerate principal null congruence. We derive the conditions for existence of innermost stable circular orbit, marginally bound circular orbit and circular photon orbit in the background of Kerr-Newman-Taub-NUT(KNTN) space-time. The effective potential for both time-like case and null cases have been studied. It is shown that the presence of the NUT parameter deforms the shape of the effective potential in contrast with the zero NUT parameter. We further investigate the energy extraction by the Penrose process for this space-time. It is shown that the efficiency of this black hole depends on both the charge and NUT parameter. It is observed that the energy gain is maximum when NUT parameter goes to zero value and for the maximum spin value. When the value of NUT parameter is increasing the energy-gain is decreasing.

We examine the thermodynamic properties of inner and outer horizons in the background of Hořava Lifshitz black hole. We compute the horizon radii product, the surface area product, the entropy product, the surface temperature product, the Komar energy product and the specific heat product for both the horizons. We show that surface area product, entropy product and irreducible mass product are universal(mass-independent) quantities, whereas the surface temperature product, Komar energy product and specific heat product are not universal quantities because they all depend on mass parameter. We further study the stability of such black hole by computing the specific heat for both the horizons. It has been observed that under certain condition the black hole possesses second order phase transition.