Nonrelativistic Spinless Particle in Vicinity of Schwarzschild-like Black Hole

TL;DR

This paper derives and solves the Schrödinger equation for a non-relativistic spinless particle near a Schwarzschild-like black hole, revealing quantized energy levels and confirming the absence of non-relativistic Hawking radiation.

Contribution

It introduces a covariant Schrödinger equation in Schwarzschild-like spacetime and analytically solves it, providing new insights into particle behavior near black holes.

Findings

Energy levels are quantized near the black hole

Particles do not escape to infinity

No non-relativistic Hawking radiation observed

Abstract

In this article we analyze the behavior of a non-relativistic spinless particle near the event horizon of a Schwarzschild-like black hole. In this way, the Schr\"odinger covariant equation that describes the particle is obtained from the Galilean covariance technique. The Schr\"odinger equation in a Schwarzschild-like spacetime is solved analytically and its solutions are given in terms of the confluent Heun function. As a relevant result, we discovered that the energy levels of the particles are quantized and that the particle does not escape to infinity. We obtain the existing transmission and reflection coefficients for a particle and anti-particle pair at the event horizon. We thus verify that there is no non-relativistic equivalent of Hawking radiation.