Magnetic monopoles are hypothetical elementary particles that are isolated magnets (north or south) with only one magnetic pole. These are predicted by grand unified theories and superstring theories, but so far, only elementary particles with electric monopoles have been found. The Dirac quantization condition states that, if there exists any magnetic monopoles, then all electric charges in the universe must be quantized, as it is observed. The fact that electric charges are quantized does not imply that magnetic monopoles exist, but it is a theoretical suggestion to consider their existence, or to interpret them correctly.
The non-existance of magnetic monopoles leave Maxwell’s equations asymmetric, so that there is not symmetry in the exchange between the electric and magnetic fields. Due to a certain symmetry of Maxwell’s equations called duality transformation, one cannot uniquely decide whether a particle has an electric charge, a magnetic charge, or both, just by observing its behavior and comparing that to Maxwell’s equations. All particles ever observed have the same ratio of magnetic charge to electric charge, and duality transformations change the ratio to any arbitrary numerical value, but cannot change that all particles have the same ratio. The convention is that this ratio is zero for convenience, so that magnetic charges are zero. Although one could interpret that there is a way to account for magnetic charges by the duality transformations, there is another way to fill the “gap” left by them in Maxwell’s equations, which will be presented onwards.
Gravitoelectromagnetism equations were first written by Oliver Heaviside as an extension of Newtonian mechanics, and later derived as a weak field approximation of General Relativity and experimentally confirmed around 2008 by Gravity Probe B and others. Gravitoelectromagnetism equations are also asymmetric and have the same form of Maxwell’s equations, but with mass charges.
The earliest example of natural units are Stoney units, which allow to express the S.I. unit of mass in terms of a charge, the gravitational constant, and vacuum permittivity through a Stoney conversion factor. This factor, which also appears within the Plank voltage, is used in the construction of an stress-energy tensor out of the electromagnetic field for the Reissner-Nordstrom metric.
Newton’s law of gravity and Columb’s law of electricity can be identical if complex numbers are used through the Stoney conversion factor and choosing one charge (mass or electric charge) to be complex. A complex number provides a suitable data structure to store information about both mass and electric charge in a single entity, making it very natural to mathematically describe gravitoelectromagnetism and classical electromagnetism, in terms of two mutually orthogonal multivector fields.
Maxwells equations can be written in a full symmetric form with the inclusion of a variable for the density of magnetic charge, so that a “magnetic current density” variable also appears. Through complex numbers and the Stoney conversion factor, the geometric representation of the full set of gravitoelectromagnetism equations can be reached. But another way to reach this is through the generalized gravitoelectromagnetism equations, which do not contain monopoles, associating the magnetic part with classical gravitational phenomena and justifying the analysis under duality transformations.
In this way, not only generalized gravitoelectromagnetism becomes symmetric, but a magnetic monopole would look like a massive body and contrary to what we expect, magnetic monopoles passing through a coil of wire would generate pure imaginary forces leading to no electric current, which would show why classical magnetic monopoles are not observed.
Contribution from Marcelo Villarreal Fasanelli