The width of a black hole atmosphere

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.