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

摘要

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 ΔE≤1221+11+α-aM2-α1+α11+11+α-aM22-1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} arDelta {{mathcal {E}}} le rac{1}{2}left( sqrt{rac{2}{1+sqrt{rac{1}{1+lpha }-left( rac{a}{{{mathcal {M}}}}ight) ^2}} -rac{lpha }{1+lpha } rac{1}{left( 1+sqrt{rac{1}{1+lpha }-left( rac{a}{{{mathcal {M}}}}ight) ^2} ight) ^2}}-1ight) end{aligned}$$end{document}where a is spin parameter, αdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$lpha $$end{document} is MOG parameter and Mdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${{mathcal {M}}}$$end{document} is the Arnowitt–Deser–Misner (ADM) mass parameter. When α=0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$lpha =0$$end{document}, we obtain the gain in energy for Kerr BH. For extremal Kerr-MOG BH, we determine the maximum gain in energy is ΔE≤12α+21+α-1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$arDelta {{mathcal {E}}} le rac{1}{2} left( sqrt{rac{lpha +2}{1+lpha }}-1 ight) $$end{document}. We observe that the MOG parameter has a crucial role in the energy extraction process and it is in fact diminishes the value of ΔEdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$arDelta {{mathcal {E}}}$$end{document} 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.