Entropy Variation of a Charged (2 + 1)-Dimensional BTZ Black Hole Under Hawking Radiation

Using the definition of a black hole’s volume introduced by Christodoulou and Rovelli, we calculate the interior volume of a (2 + 1)-dimensional charged Banados Teitelboim Zanelli (BTZ) black hole, and find that the volume increases linearly with time. Afterwards, the entropy of a massless scalar field inside the black hole is calculated and the result indicates that the entropy will be also increasing with time infinitely. Moreover, thinking about Hawking radiation, the ratio of variation of the scalar field’s entropy to the variation of Bekenstein–Hawking entropy is approximately a linear function of m, which is quite different from a RN black hole while m is relatively large. In the end, we extend the calculation above to a massive BTZ black hole and find that the relationship with different M~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde {M}$\end{document} between two kinds of entropy is similar to the previous result. But the difference is that the relationship function F(m,q,M~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F(m,q,\tilde {M})$\end{document} will tend to be a constant when the mass parameter M~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde {M}$\end{document} becomes big enough.