Gravity and the Superposition Principle

TL;DR

This work proposes a nonrelativistic bridge between gravity and quantum mechanics by replacing the Compton wavelength with the Michell-Laplace radius $r_g = \frac{G m}{c^2}$ for masses $m \ge m_P$, and by rewriting the Schrödinger framework so the wave-packet width $\sigma(t)$ is governed by $G$ and $c$ instead of $\hbar$. It shows that for $m \ge m_P$ the stationary equation effectively uses $\hbar \to \frac{G m^{2}}{c}$, and derives a modified wave-packet spreading in spherical coordinates with $\sigma(t) = \sigma_0 \sqrt{1 + \frac{1}{c^{2}} \left(\frac{G m}{\sigma_0^{2}}\right)^{2} t^{2}}$, introducing the acceleration $a = \frac{G m}{\sigma_0^{2}}$. The analysis considers two observers (static on the surface and free-falling inside) and shows observer-dependent expansion rates of the gravitational source, suggesting gravity operates as a quantum superposition in scale. Appendices connect the framework to the de Broglie–Bohm quantum potential, the Mita energy, and the Generalized Uncertainty Principle, demonstrating how gravity can influence quantum localization and energy in this scheme. The results offer a conceptual route to quantum-gravitational phenomenology, highlighting the role of $G$ and $c$ as fundamental constants in macroscopic quantum behavior and their potential experimental consequences in precise free-fall settings.

Abstract

The relation between gravity and quantum mechanics is investigated in this work. The link is given by the wave packet expansion process, rooted from the Uncertainty Principle. The basic idea is to express the de Broglie wavelength used by Schrodinger for a massive particle in terms of the associated Compton wavelength which is replaced by the Michell-Laplace radius $Gm/c^{2}$ of the spherical object of mass $m\geq m_{P}$, where $m_{P}$ is the Planck mass. The wave packet spreading is studying in spherical coordinates, having the width $σ(t)$, expressed in terms of $G$ and $c$ instead of $\hbar$. Therefore, for masses larger than the Planck mass, a faster dispersion rate of $σ(t)$ is obtained, compared to the standard case. The dispersion of the wave packet is observed only by a free falling observer and the process breaks down once the observer hits the surface of the object. Different freely falling observers notice different rates of expansion of the wave packet and the source of gravity is in a quantum superposition. We further confront the Mita formula for the mean energy of the wave packet with the de Broglie-Bohm quantum potential energy when the Schrodinger equation is expressed in the Madelung form.