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 $\Delta ϵ\le \frac{1}{2}(\sqrt{\frac{2}{1+\sqrt{\frac{1}{1+\alpha }-(\frac{a}{M}{)}^2}}-\frac{\alpha }{1+\alpha }-\frac{1}{{(1+\sqrt{\frac{1}{1+\alpha }-(\frac{a}{M}{)}^2})}^2}}-1)$ where a is spin parameter, α is MOG parameter and M is the Arnowitt–Deser–Misner (ADM) mass parameter. When α=0, we obtain the gain in energy for Kerr BH. For extremal Kerr-MOG BH, we determine the maximum gain in energy is Δ$ϵ$≤$\frac{1}{2}$$(\sqrt{\frac{\alpha +2}{1+\alpha }}-1)$. We observe that the MOG parameter has a crucial role in the energy extraction process and it is in fact diminishes the value of ΔE in contrast with extremal Kerr BH. Moreover, we derive the 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.