Thermality of horizon through near horizon instability: a path integral approach

Recent investigations revealed that the near horizon Hamiltonian of a massless, chargeless outgoing particle, for its particular motion in static as well as stationary black holes, is effectively ∼xp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim xp$$\end{document} kind. This is unstable by nature and has the potential to explain a few interesting physical phenomena. From the path integral kernel, we first calculate the density of states. Also, following the idea of Singh and Padmanabhan (Phys Rev D 85:025011, 2012. https://doi.org/10.1103/PhysRevD.85.025011. arXiv:1112.6279 [hep-th]) here, in the vicinity of the horizon, we calculate the effective path corresponding to its Schrodinger version of Hamiltonian through the path integral approach. The latter result appears to be complex in nature and carries the information of escaping the probability of the particle through the horizon. In both ways, we identify the correct expression of Hawking temperature. Moreover, here we successfully extend the complex path approach to a more general black hole like Kerr spacetime. We feel that such a complex path is an outcome of the nature of near horizon instability provided by the horizon and, therefore, once again bolstered the fact that the thermalization mechanism of the horizon may be explained through the aforesaid local instability.