Study of energy extraction and epicyclic frequencies in Kerr-MOG (modified gravity) black hole

Abstract 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 $$\begin{aligned} \varDelta {{\mathcal {E}}} \le \frac{1}{2}\left( \sqrt{\frac{2}{1+\sqrt{\frac{1}{1+\alpha }-\left( \frac{a}{{{\mathcal {M}}}}\right) ^2}} -\frac{\alpha }{1+\alpha } \frac{1}{\left( 1+\sqrt{\frac{1}{1+\alpha }-\left( \frac{a}{{{\mathcal {M}}}}\right) ^2} \right) ^2}}-1\right) \end{aligned}$$ ΔE≤1221+11+α-aM2-α1+α11+11+α-aM22-1 where a is spin parameter, $$\alpha $$ α is MOG parameter and $${{\mathcal {M}}}$$ M is the Arnowitt–Deser–Misner (ADM) mass parameter. When $$\alpha =0$$ α=0 , we obtain the gain in energy for Kerr BH. For extremal Kerr-MOG BH, we determine the maximum gain in energy is $$\varDelta {{\mathcal {E}}} \le \frac{1}{2} \left( \sqrt{\frac{\alpha +2}{1+\alpha }}-1 \right) $$ ΔE≤12α+21+α-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 $$\varDelta {{\mathcal {E}}}$$ 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.