A quantum bound on the thermodynamic description of gravity

The seminal works of Bekenstein and Hawking have revealed that black holes have a well-defined thermodynamic description. In particular, it is often stated in the physics literature that black holes, like mundane physical systems, obey the first law of thermodynamics: ΔS=ΔE/T BH \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \Delta S=\Delta E/T_{\rm BH}$\end{document}, where T BH \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ T_{\rm BH}$\end{document} is the Bekenstein-Hawking temperature of the black hole. In the present paper we test the regime of validity of the thermodynamic description of gravity. In particular, we provide compelling evidence that, due to quantum effects, the first law of thermodynamics breaks down in the low-temperature regime T BH ×rH≲(ℏ/rH)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{\rm BH} \times r_{\rm H} \lesssim (\hbar/r_{\rm H})^{2}$\end{document} of near-extremal black holes (here rH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ r_{\rm H}$\end{document} is the radius of the black-hole horizon).