__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 (u_{elec}=e_{o}E^{2}/2)
proportional to the square of the electric field, it must also have an
equivalent non-zero *mass* density, u_{mass}=u_{elec}/c^{2}. The
key equation is Einstein’s famous E=mc^{2},
where E
stands for energy and not for electric field. This famous equation implies that
wherever there is a non-zero u_{elec} there must be an equivalent, much
smaller u_{mass}. 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 E^{2},
and hence a mass density of u_{mass}=u_{elec}/c^{2}. 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 B^{2}. 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.