A maximum force perspective on black hole thermodynamics, quantum pressure, and near-extremality

Abstract

I re-examined the notion of the thermodynamic force constructed from the first law of black hole thermodynamics. In general relativity, the value of the charge (or angular momentum) at which the thermodynamic force equals the conjectured maximum force F=1/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F=1/4$$\\end{document} is found to correspond to Q2/M2=8/9\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q^2/M^2=8/9$$\\end{document} (respectively, a2/M2=8/9\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a^2/M^2=8/9$$\\end{document}), which is known in the literature to exhibit some special properties. This provides a possible characterization of near-extremality. In addition, taking the maximum force conjecture seriously amounts to introducing a pressure term in the first law of black hole thermodynamics. This resolves the factor of two problem between the proposed maximum value F=1/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F=1/4$$\\end{document} and the thermodynamic force of Schwarzschild spacetime F=1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F=1/2$$\\end{document}. Surprisingly it also provides another indication for the instability of the inner horizon. For a Schwarzschild black hole, under some reasonable assumptions, this pressure can be interpreted as being induced by the quantum fluctuation of the horizon position, effectively giving rise to a diffused “shell” of characteristic width M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{M}$$\\end{document}. The maximum force can therefore, in some contexts, be associated with inherently quantum phenomena, despite the fact that it is free of ħ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar $$\\end{document}. Some implications are discussed as more questions are raised.