Back reaction and Hawking radiation from a Vaidya black hole

Using Damour-Ruffini’s and Hamilton-Jacobi’s methods, Hawking radiation from a Vaidya black hole is investigated. Due to non-stationary black holes, the event horizon rH and the entropy S are all related to both the mass m(υ) and  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot{r}_{H}$\end{document} . When the back-reaction of particles’s energy to space-time is considered, we get the emission probability. It is found that the result is different from that of the stationary Schwarzschild black hole, because \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot{r}_{H}$\end{document} is the function of mass m(υ).