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 $\mathcal{M}$ 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 $\mathrm{\Delta }\mathcal{E}$ 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.