A charged particle at rest in a gravitational field, such as on the surface of the Earth, must be supported by a force to prevent it from falling.
One of the first to study this problem was Max Born in his 1909 paper about the consequences of a charge in uniformly accelerated frame.
[1] Earlier concerns and possible solutions were raised by Wolfgang Pauli (1918),[2] Max von Laue (1919),[3] and others, but the most recognized work on the subject is the resolution of Thomas Fulton and Fritz Rohrlich in 1960.
[4][5] It is a standard result from Maxwell's equations of classical electrodynamics that an accelerated charge radiates.
This principle states that it is impossible to distinguish through any local measurement whether one is in a gravitational field or being accelerated.
An elevator out in deep space, far from any planet, could mimic a gravitational field to its occupants if it could be accelerated continuously "upward".
They cancel out when equated, with the result discovered by Galileo Galilei in 1638, that all bodies fall at the same rate in a gravitational field, independent of their mass.
Just as with acceleration versus gravity, no experiment should be able to distinguish the effects of free fall in a gravitational field, and being out in deep space far from any forces.
Then a free-falling observer could distinguish free fall from the true absence of forces, because a charged particle in a free-falling laboratory would begin to be pulled upward relative to the neutral parts of the laboratory, even though no obvious electric fields were present.
(This is similar to how the concept of mechanics in an inertial frame is not applicable to the surface of the Earth even disregarding gravity due to its rotation - cf.
Therefore, in this case, we cannot apply Maxwell's equations to the description of a falling charge relative to a "supported", non-inertial observer.
Maxwell's equations can be applied relative to an observer in free fall, because free-fall is an inertial frame.
So the starting point of considerations is to work in the free-fall frame in a gravitational field—a "falling" observer.
In the free-fall frame, Maxwell's equations have their usual, flat-spacetime form for the falling observer.
In this case the gravitational field is fictitious because it can be "transformed away" by appropriate choice of coordinate system in the falling frame.
Unlike the total gravitational field of the Earth, here we are assuming that spacetime is locally flat, so that the curvature tensor vanishes.
Rohrlich emphasizes that this charge remains at rest in its free-fall frame, just as a neutral particle would.
As observed from the freefalling frame, the supported charge appears to be accelerated uniformly upward.
has a radiation rate given by the Lorentz invariant: The corresponding electric and magnetic fields of an accelerated charge are also given in Rohrlich.
So although the Coulomb law was discovered in a supporting frame, general relativity tells us that the field of such a charge is not precisely
The radiation from the supported charge viewed in the freefalling frame (or vice versa) is something of a curiosity: one might ask where it goes.
David G. Boulware (1980)[9] finds that the radiation goes into a region of spacetime inaccessible to the co-accelerating, supported observer.
Camila de Almeida and Alberto Saa (2006)[10] have a more accessible treatment of the event horizon of the accelerated observer.