Bound states of the Dirac equation in Schwarzschild spacetime: an exploration of intuition for the curious student

TL;DR

This paper investigates the theoretical possibility of quantum bound states in Schwarzschild spacetime by extending the Dirac equation from flat to curved spacetime, aiming to connect classical gravitational potentials with quantum mechanics for curious students.

Contribution

It provides an accessible exploration of deriving quantum bound states in Schwarzschild spacetime using relativistic quantum mechanics, bridging undergraduate concepts with advanced general relativity.

Findings

Bound states are theoretically plausible in Schwarzschild spacetime.

The effective potential in Schwarzschild resembles a Coulomb-like potential with relativistic corrections.

Educational pathway outlined for students to explore quantum gravity concepts.

Abstract

In this work we explore the possibility of quantum bound states in a Schwarzschild gravitational field leveraging the analogy of the elementary derivation of bound states in the Coulomb potential as taught in an undergraduate course in Quantum Mechanics. For this we will also need to go beyond non-relativistic quantum mechanics and utilize the relativistic Dirac equation for a central potential as taught in an advanced undergraduate or first year graduate (special) relativistic quantum mechanics course. Finally, the special relativistic Dirac equation must be extended to the general relativistic version for curved spacetime. All these disparate component pieces exist in excellent, very readable textbooks written for the student reader, with sufficient detail for a curious student to learn and explore. We pull all these threads together in order to explore a very natural question that a student might ask: "If the effective $1/r$ radial potential of the Schwarzschild metric (with angular momentum barrier), as taught in elementary GR courses for undergraduates, appears Newtonian-like (with a $1/r^3$ correction), then is it possible to derive quantum bound states in the Schwarzschild spacetime by simply changing the radial potential $V(r)$ from $V_C(r)=-e^2/r$ to $V_{Schw}=-G M m/r$?"