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On the Interaction of Charge and an External Gravitational Field

There are at least two mysterious quantities in physics. Their names are “mass” and “charge”. In our first course in physics we are introduced to these two, but rarely if ever are we taught “what” they are. Newton seemed to be aware of this enigma, and suggested that mass is the “quantity of matter”. An analogous “explanation” for charge might be that it is the “quantity of electricity”. Despite such limited explanations regarding the “whatness” or essence of mass and charge, these two variables have proven to be ubiquitous  in physical theory, thanks to the equations of Newton, Coulomb, Maxwell and many others.

Newton of course specified (1) how mass is affected by an applied force, and (2) what the magnitude of the attractive force between two spatially separated masses will be. Coulomb did the same (point no. 2) in his law of the interaction of separate charges. But there is scant literature on how an external mass (or its gravitational field) might interact with electric charge.

In this article we will attempt to discuss that, whereas every static distribution of charge has (at each point) a non-zero energy density (uelec=eoE2/2) proportional to the square of the electric field, it must also have an equivalent non-zero mass density, umass=uelec/c2. The key equation  is Einstein’s famous E=mc2, where E stands for energy and not for electric field. This famous equation implies that wherever there is a non-zero uelec there must be an equivalent, much smaller umass. Hence in an external gravitational field each increment of electric field should experience an incremental gravitational force. The integral of all these force increments sums to a net gravitational force on the electric field, and thence on the field’s source, q.

We can appreciate this force, and its effect on a given charge, q, by considering Einstein’s celebrated box accelerating in gravity-free space. Any charge entering the box through a hole in the top of a wall must, from the perspective of an observer in the box, travel across the interior of the box in a curved trajectory. And by the Equivalence Principle the same curved trajectory would be observed for a charge traveling normal through a stationary gravitational field.

We conclude that every electric field, E, has an energy density proportional to E2, and hence a mass density of umass=uelec/c2.  Thus every increment of electric field energy must experience a subtle gravitational force in the presence of an external gravitational field.

Now the same discussion can be applied to increments of a magnetic field. Every magnetic field has an energy density proportional to B2. The magnetic field, and its source magnet, must therefore experience a gravitational force in the presence of an ambient gravitational field. Thus both non-spinning and spinning charges experience a subtle gravitational force in the presence of an ambient gravitational field.