Study of energy extraction and epicyclic frequencies in Kerr-MOG~(Modified Gravity) black hole

We investigate the energy extraction by the Penrose process in Kerr-MOG black hole~(BH). We derive the gain in energy for Kerr-MOG as \begin{eqnarray} \Delta {\cal E} \leq \frac{1}{2}\left(\sqrt{\frac{2}{1+\sqrt{\frac{1}{1+\alpha}-\left(\frac{a}{{\cal M}}\right)^2}} -\frac{\alpha}{1+\alpha} \frac{1}{\left(1+\sqrt{\frac{1}{1+\alpha}-\left(\frac{a}{{\cal M}}\right)^2} \right)^2}}-1\right) \nonumber \end{eqnarray} Where $a$ is spin parameter, $\alpha$ is MOG parameter and ${\cal M}$ is the Arnowitt-Deser-Misner(ADM) mass parameter. When $\alpha=0$, we obtain the gain in energy for Kerr BH. For extremal Kerr-MOG BH, we determine the maximum gain in energy is $\Delta {\cal E} \leq \frac{1}{2} \left(\sqrt{\frac{\alpha+2}{1+\alpha}}-1 \right)$. We observe that the MOG parameter has a crucial role in the energy extraction process and it is in fact diminishes the value of $\Delta {\cal E}$ in contrast with extremal Kerr BH. Moreover, we derive the \emph{Wald inequality and the Bardeen-Press-Teukolsky inequality} for Kerr-MOG BH in contrast with Kerr BH. Furthermore, we describe the geodesic motion in terms of three fundamental frequencies: the Keplerian angular frequency, the radial epicyclic frequency and the vertical epicyclic frequency. These frequencies could be used as a probe of strong gravity near the black holes.Comment: Accepted in EPJ

Logarithmic Corrections to the Black Hole Entropy Product of H±{\cal H}^{\pm} via Cardy Formula

We compute the logarithmic corrections to black hole (BH) entropy product of ${\cal H}^{\pm}$ \footnote{ ${\cal H}^{+}$ and ${\cal H}^{-}$ denote outer (event) horizon and inner (Cauchy) horizons} by using \emph{Cardy prescription}. We particularly apply this formula for \emph{BTZ BH}. We speculate that the logarithmic corrections to entropy product of ${\cal H}^{\pm}$ when computed \emph{via Cardy formula} the product should be neither \emph{mass-independent (universal)} nor be \emph{quantized}.Comment: EPL Style, 4 page

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.Comment: Version accepted for publication in EP

Thermodynamic Properties of Kehagias-Sfetsos Black Hole \& KS/CFT Correspondence

We speculate on various thermodynamic features of the inner horizon~(${\mathcal H}^{-}$) and outer horizons~(${\mathcal H}^{+}$) of Kehagias-Sfetsos~(KS) black hole~(BH) in the background of Ho\v{r}ava Lifshitz gravity. We compute particularly the \emph{area product, area sum, area minus and area division} of the BH horizons. We find that they all are \emph{not} showing universal behavior whereas the product is a universal quantity~ [Pradhan P., \textit{Phys. Lett. B}, {\bf 747} (2015) {64}]. Based on these relations, we derive the area bound of all horizons. From the area bound we derive the entropy bound and irreducible mass bound for all the horizons~(${\mathcal H}^{\pm}$). We also observe that the \emph{First law} of BH thermodynamics and \emph {Smarr-Gibbs-Duhem } relations do not hold for this BH. The underlying reason behind this failure due to the scale invariance of the coupling constant. Moreover, we compute the \emph{Cosmic-Censorship-Inequality} for this BH which gives the lower bound for the total mass of the spacetime and it is supported by cosmic cencorship conjecture. Finally, we discuss the KS/CFT~(Conformal Field Theory) correspondence via a thermodynamic procedure.Comment: Version accepted in EP