More on accreting black hole spacetime in equatorial plane

Spacetime around an accreting black hole is an interesting issue to study. The metric of an isolated black hole (rotating or non-rotating) spacetime has been well-known for decades. Although metrics of some spacetimes containing accreting black holes are known in some situations, the issue has some faces that are not well-known yet and need further investigation. In this paper, we construct a new form of metric which the effect of accretion disk on black hole spacetime is taken into account in the equatorial plane. We study motion and trajectories of massive particles and also photons falling from infinity towards black hole in equatorial plane around the black hole. We use an exponential form for the density profile of the accretion disk in equatorial plane as ρ=ρ0e−αr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho =\rho_{0}e^{-\alpha r}$\end{document}. We show that with this density profile, the disk is radially stable if α≤3×10−3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \leq 3 \times 10^{-3}$\end{document} (in units of length inverse). In order to study some important quantities related to the accretion disks such as locations of marginally stable circular orbits (rms\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{ms}$\end{document} or rISCO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{\mathit{ISCO}}$\end{document}), marginally bounded circular orbits (rmb)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(r_{mb})$\end{document}, and also photon orbits in equatorial plane, we use the effective potential approach. We show that in this spacetime metric the innermost stable circular orbit in equatorial plane is given by rISCO=4.03μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{\mathit{ISCO}}=4.03 \mu $\end{document} (where μ=MGc2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu =\frac{MG}{c ^{2}}$\end{document}) which is different, but comparable, with the Schwarzschild spacetime result, rISCO(Sch)=6μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r^{(Sch)}_{\mathit{ISCO}}=6 \mu $\end{document}. We show that the maximum radiation efficiency of the accretion disk, η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\eta $\end{document}, in equatorial plane is 8.6 percent which is greater than the corresponding value for Schwarzschild spacetime. Finally, we show that in this setup photons can have stable circular orbits in equatorial plane unlike the Schwarzschild spacetime.