It is a fact surprisingly rarely mentioned in relativity or

electrodynamics textbooks that the electric field of a single point

charge familiar from elementary electrostatics becomes deformed when

the charge is supported within a gravitational field. At least in part,

this apparent lack of 'popular appeal,' may be due to the fact that,

whereas a sophisticated mathematical treatment has been developed at

different times in the highly specialized literature after the

introduction of the general relativity theory, the first realistic

framework for experimental confirmation of this important prediction

has appeared only very recently (see Physics, July 5, 2005). This

fascinating phenomenon, however, offers even more than previously

unsuspected experimental approaches to test field theory in curved

space-time since its intriguing history is an example of a problem

discovered and rediscovered several times over by researchers who gave

their contribution often without being aware of those who had come

before them.

It appears that the earliest publication on the problem of

electrostatics in curved space was that by Enrico Fermi, published when

he was a third year student at the Scuola Normale Superiore at Pisa. In

this paper, Fermi discussed the correction to the electric field of a

single point charge held at rest within a gravitational field to first

order in the gravitational acceleration. The problem of the single

charge seems to have been completely forgotten until a few years later,

when Edmund T. Whittaker solved it exactly both in the homogeneous

gravitational field and in the Schwarzschild geometry cases with no

mention of Fermi's previous work. His analysis was further developed by

E. T. Copson, who produced an expression still used today.

Fermi's early goal was not only to obtain the electric field of a

single charge held at rest in a gravitational field but to also prove

that, to within the adopted approximations, the magnitude and

orientation of the needed external force are but a manifestation of the

gravitational equivalent of the electrostatic potential energy of the

interacting charges. For instance, in the case of a simple dipole made

of two charges +q and -q separated by a distance s, Fermi's argument

would state that an effective lifting self-force is expected, equal to

+gq^2/s c^2 (hat indicates power operation), corresponding to an

effective decrease in the gravitational mass of the system due to its

negative potential energy and produced by the interaction of each

charge with the distorted electric field of the other in curved space.

This force manifests itself as a decrease in the magnitude of the

external force permanently holding the dipole at rest in the

gravitational field.

In recent years, Fermi's original result that the gravitational mass

correction one expects from energy considerations does coincide with

the electrostatic self-force on a system of supported charges has been

rediscovered, again to first order and in the particular case of a

dipole perpendicular to the gravitational acceleration. Interestingly,

an attempt to generalize this important example to the case of a dipole

accelerating in any direction, for instance longitudinally, has been

unsuccessful and the problem is presently characterized in the

literature as an "unsolved paradox.". These latter authors have

compared the ``energy-derived'' mass of an accelerated dipole to the

inertia offered by such a system under the action of an external force,

referred to as the 'self-force derived' mass. Their result that the two

masses coincide only if the dipole is accelerating perpendicularly to

its orientation certainly defies the very reasonable expectation that

this should instead occur regardless of the geometrical distribution of

the charges and it also contradicts Fermi's earlier results.

Now, in a paper to appear in the May 15, 2006 issue of the Physical

Review D, Dr. Fabrizio Pinto employs Whittaker's field equation to show

that no paradox exists and suggests that its appearance was due to

coordinate transformation errors. He then generalizes his findings to

more complex charge distributions, including interacting dipoles and

shows that Fermi's prediction is always fully consistent with

Einstein's energy-mass equivalency.

Why is electrostatics in curved space important beyond its intrinsic

interest? The reason is that these results provide the foundation for

realistic observation of these phenomena by means of trapped atom

interferometry. As recently shown by Dr. Fabrizio Pinto (see Physics,

July 5, 2005), it will be possible to observe the change in the

effective weight of charge distributions predicted by Fermi by

manipulating the van der Waals forces between cold atoms in optical

traps thus finally confirming Einstein's mass-energy equivalency

equation in gravitating systems.

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