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

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 $\hslash$ (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 $\hslash$), 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.

12 pages; this article is a follow-up of arXiv:1704.00747v2 [gr-qc]; remarks on `cubic Dirac fermions in Dirac semi-metals' added on page 7; paragraph added on page 8 to quantitatively compare spin density and mass density; references added; v3: clarifying remarks added after Eqn. (13), conclusions unchanged, references updated, published in Int. J. Mod. Phys

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• A duality between curvature and torsion
• On the Role of Einstein-Cartan Gravity in Fundamental Particle Physics
• Dimensionally-dependent uncertainty relations, or why we (probably) won't see micro-black holes at the LHC, even if large extra dimensions exist
• Generalized uncertainty principle for a Dirac fermion in a torsion field