Some consequences of gravitationally induced electromagnetic fields in microphysics

Abstract

We discuss the relation between the gravitational and electromagnetic fields as governed by the Einstein-Maxwell field equations. It is emphasized that the tendency of the gravitational field to induce electromagnetic effects increases as the size of the system decreases. This is because the charge-to-mass ratio $Q/M$ is typically larger in smaller systems. For most astrophysical systems, $Q/M$ is $\ll 1$ while for a Millikan oil drop, $Q/M \sim 10^6$. Going all the way down to elementary particles, the value for the electron is $Q/M \sim 10^{21}$. For subatomic systems there is an additional phenomenon which comes into play. In fact, according to general relativity, the gravitational field tends to become dominated by the spin at distances of the order of the Compton wavelength.The relevant quantity which governs this behavior is the ratio $S/M^2$ where $S$ is the (spin) angular momentum. For an electron, $S/M^2 \sim 10^{44}$.As a consequence, the gravitational field becomes dominated by gravitomagnetic effects in the subatomic domain.This fact has important consequences for the electromagnetic fields of spinning charged particles.To analyze this situation we use the asymptotic structure in the form of the multipole fields.%Such an approach avoids the pitfalls should one try to use a near-field approach using some kind of semi-classical formulation of the Einstein-Maxwell equations for example. To obtain more exact results however, one must take quantum effects into account including radiative contributions. Although such effects are not considered in this work, the order of magnitude of the considered effects are not expected to change drastically when going to a quantum mechanical treatment. The most relevant solution of the Einstein-Maxwell equations in this context is the Kerr-Newman metric. It is the preferred solution which is in accord with all the four known multipole moments of the electron to an accuracy of one part in a thousand. Our main result is that general relativity predicts corrections to the Coulomb field for charged spinning sources. Experimentally verifiable consequences include a predicted electric quadrupole moment for the electron, possible quasi-bound states in positron-heavy ion scattering with sizes corresponding to observed anomalous peaks, as well as small corrections to energy levels in microscopic bound systems such as the hydrogen atom