# A new length scale, and modified Einstein-Cartan-Dirac equations for a point mass

## TL;DR

This paper introduces a modified set of Einstein-Cartan-Dirac equations incorporating a new length scale, exploring the interplay between gravity, torsion, and quantum mechanics, and proposing gravity as an emergent phenomenon linked to wave function collapse.

## Contribution

It formulates new field equations with a length scale $L_{CS}$, connecting quantum and gravitational regimes, and suggests gravity may emerge from quantum collapse processes.

## Key findings

- Large mass limit recovers Einstein equations with wave function collapse assumptions.
- Small mass limit yields Dirac equation with torsion sourced by spin density.
- Modified coupling constant affects torsion's role in quantum gravity.

## Abstract

We have recently proposed a new action principle for combining Einstein equations and the Dirac equation for a point mass. We used a length scale $L_{CS}$, dubbed the Compton-Schwarzschild length, to which the Compton wavelength and Schwarzschild radius are small mass and large mass approximations, respectively. Here we write down the field equations which follow from this action. We argue that the large mass limit yields Einstein equations, provided we assume wave function collapse and localisation for large masses. The small mass limit yields the Dirac equation. We explain why the Kerr-Newman black hole has the same gyromagnetic ratio as the Dirac electron, both being twice the classical value. The small mass limit also provides compelling reasons for introducing torsion, which is sourced by the spin density of the Dirac field. There is thus a symmetry between torsion and gravity: torsion couples to quantum objects through Planck's constant $\hbar$ (but not $G$) and is important in the microscopic limit. Whereas gravity couples to classical matter, as usual, through Newton's gravitational constant $G$ (but not $\hbar$), and is important in the macroscopic limit. We construct the Einstein-Cartan-Dirac equations which include the length $L_{CS}$. We find a potentially significant change in the coupling constant of the torsion driven cubic non-linear self-interaction term in the Dirac-Hehl-Datta equation. We speculate on the possibility that gravity is not a fundamental interaction, but emerges as a consequence of wave function collapse, and that the gravitational constant maybe expressible in terms of Planck's constant and the parameters of dynamical collapse models.