The width of a black hole atmosphere

TL;DR

The paper investigates quantum behavior near the Schwarzschild horizon by solving the Klein-Gordon equation in the quasi-classical regime and finds exponentially time-dependent near-horizon modes with a complex phase, implying absorption or reflection rather than horizon crossing. This leads to the concept of a thin black hole atmosphere with width $\epsilon \sim 10^{-22} r_s$, derived from quasi-classical geodesics and aligning with brick-wall entropy results. The work links horizon boundary conditions to a dynamic quantum layer around the horizon and discusses potential implications for black hole thermodynamics, quantum gravity, and the information paradox, while outlining future observational and theoretical directions. Overall, the analysis provides a concrete mechanism for horizon-reflected near-horizon matter and quantifies the atmospheric width within a semi-classical framework.

Abstract

Black holes, as classical solutions of General Relativity, are expected to exhibit quantum properties near their horizons. In this paper, we examine the behavior of quantum particles near the Schwarzschild horizon by solving the Klein-Gordon equation in the quasi-classical approximation. Our analysis shows that, rather than the periodic-in-time solutions typically associated with black hole exteriors or interiors, particles near the horizon exhibit exponentially decaying (or growing) time-dependent solutions with a complex phase. This suggests that particles are unlikely to cross the Schwarzschild horizon; instead, they are either absorbed by the black hole or reflected back, forming a thin atmospheric layer around the horizon. Using geodesic equations derived from the Klein-Gordon equation, we estimate the width of this atmosphere.