Scattering of massive scalars by Schwarzschild black holes

TL;DR

This paper analyzes how massive scalar waves scatter off Schwarzschild black holes, revealing a mass-dependent phase shift at infinity and near the horizon, bridging quantum wave behavior with classical scattering.

Contribution

It introduces a mass-dependent logarithmic phase shift in scalar wave scattering, extending previous models and matching classical cross-sections in the non-relativistic limit.

Findings

Discovered a mass-dependent logarithmic phase shift at infinity.

Calculated phase shift near the black hole horizon.

Connected quantum scattering results with classical Newtonian cross-sections.

Abstract

The Klein-Gordon equation for the wave function of a single massive scalar is written in spherical and parabolic coordinates in the presence of a Schwarzschild background, and some semi-classical techniques for deriving asymptotic results are considered. In addition to the well-known logarithmic phase shift due to the tortoise radial coordinate, it is found that there is a mass-dependent logarithmic phase shift at infinity. This is found to be necessary to match the Newtonian cross-section in the non-relativistic limit. Finally, by imposing suitable boundary conditions near the horizon, the phase shift is calculated.