Recovering semiclassical Einstein’s equation using generalized entropy

In this letter, we show that the Semiclassical Einstein’s Field Equation can be recovered using the generalized entropy Sgen\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{gen}$$\end{document}. This approach is reminiscent of non-equilibrium thermodynamics. Furthermore, contrary to the entanglement equilibrium approach of deriving the semiclassical Einstein’s equation, this approach does not require any such assumptions and still recovers its correct form. Therefore, in a sense, we also show the validity of the semiclassical approximation, a crucial approach for establishing a number of important ideas such as the Hawking effect.