Relativistic corrections to the Di{\'o}si-Penrose model

TL;DR

This paper extends the Di{\'o}si-Penrose gravitational collapse model to include relativistic effects using a novel approach, analyzing its behavior at the Planck scale and applying it to a particle in a box.

Contribution

It introduces a relativistic correction to the Di{\'o}si-Penrose model using the Grave de Peralta approach, providing new insights into the model's behavior at high energies.

Findings

Relativistic effects cause the matter field width to sharply drop at the Planck mass.

The model's breakdown occurs at the Planck scale, indicating limits of its applicability.

Application to a particle in a box shows good agreement with expected relativistic corrections.

Abstract

The Di{\'o}si-Penrose model is explored in a relativistic context. Relativistic effects were considered within a recently proposed Grave de Peralta approach [L. Grave de Peralta, {\em Results Phys.} {\bf 18} (2020) 103318], which parametrize the Schr{\"o}dinger-like hamiltonian so as to impose that the average kinetic energy of the system coincide with its relativistic kinetic energy. As a case of study, the method is applied to a particle in a box with good results. In the Di{\'o}si-Penrose model we observed that the width of a quantum matter field confined by its own gravitational field [L. Di{\'o}si, {\em Phys. Lett}. {\bf 105A} (1984) 199], sharply drop to zero for a mass of the order of the Planck mass, indicating a breakdown of the model at the Planck scale.