Gravitationally induced electromagnetism at the Compton scale

TL;DR

This paper explores how Einstein gravity influences electromagnetic fields at the Compton scale, predicting specific modifications to electron and proton multipole moments and potential small corrections to hydrogen spectra.

Contribution

It demonstrates that Einstein-Maxwell equations predict measurable electromagnetic multipole moments and field modifications at the Compton wavelength, linking gravity and electromagnetism at quantum scales.

Findings

Predicted magnetic dipole matches Dirac g-factor g=2.

Electric dipole moment remains zero, consistent with experiments.

Electric quadrupole moment of the electron is approximately -124 barn.

Abstract

It is shown that Einstein gravity tends to modify the electric and magnetic fields appreciably at distances of the order of the Compton wavelength. At that distance the gravitational field becomes spin dominated rather than mass dominated. The gravitational field couples to the electromagnetic field via the Einstein-Maxwell equations which in the simplest model causes the electrostatic field of charged spinning particles to acquire an oblate structure relative to the spin direction. For electrons and protons, a pure Coulomb field is therefore likely to be incompatible with general relativity at the Compton scale. In the simplest model, the magnetic dipole corresponds to the Dirac g-factor, g=2. Also, it follows from the form of the electric field that the electric dipole moment vanishes, in agreement with current experimental limits for the electron. Quantitatively, the classical Einstein-Maxwell theory predicts the magnetic and electric dipoles of the electron to an accuracy of about one part in 10^{-3} or better. Going to the next multipole order, one finds that the first non-vanishing higher multipole is the electric quadrupole moment which is predicted to be -124 barn for the electron. Any non-zero value of the electric quadrupole moment for the electron or the proton would be a clear sign of curvature due to the implied violation of rotation invariance. There is also a possible spherical modification of the Coulomb force proportional to r^{-4}. However, the size of this effect is well below current experimental limits. The corrections to the hydrogen spectrum are expected to be small but possibly detectable.