Charge and the Equivalence Principle
G.R.Dixon, July 10, 2009
1. The Relativistic Radiation Reaction Force
2. Non-Radiating, Accelerating Charge
The idea that a charge, accelerated by a constant force, does not radiate will play a pivotal role in the following discussion. To the extent the idea agrees with experience, the Larmor formula (that radiated power is proportional to acceleration squared) cannot be universally true. For according to Newton a charge subjected to any force should accelerate. Hence, according to Larmor, a charge subject to a constant force should radiate.
The two articles cited above as References, address the problem of charge subjected to a constant force.
The purpose of the present article is to discuss two constant force cases within the context of the equivalence principle. In each case the radiation emitted by a charge within a box is considered. In Case 1 the radiation emitted by (a) a charge free falling in a uniform gravitational field, and (b) the charge at rest in gravity-free space, is considered. In Case 2 the radiation emitted by (a) the charge held at rest in a uniform gravitational field, and (b) the charge being accelerated by a constant force in gravity-free space is considered.
According to the equivalence principle it should be impossible to tell, from the radiation emitted inside the box, whether the box is (a) in a gravitational field, or (b) in gravity-free space.
2. The Charge Free-Falls in a Uniform Gravitational Field.
Let m be the charge’s gravitational mass. Assuming m is independent of speed, the charge should experience a constant force in a uniform field of magnitude g:
Since F is constant, the charge does not radiate (even though it accelerates downward in the g field). And of course the charge does not radiate when it is permanently at rest in deep space. It is evidently impossible to tell, from the radiation emitted in the box, whether the charge is free-falling or is floating force-free in deep space.
3. A Charge Held at Rest in a Uniform Gravitational Field.
The constant force that must be applied, in order to hold the charge at rest in the g field, is
Since the charge is at rest, it does not radiate. Out in deep space the charge, subjected to the constant force, F, also doesn’t radiate. Here again it’s impossible to tell, from the radiation emitted inside the box, whether the charge is being held at rest in a gravitational field or is being accelerated by a constant force in gravity-free space.
4. Inertial Mass and Gravitational Mass.
In Reference 2 a charge, subjected to a constant force, does not uniformly accelerate. For the governing equation presumably reduces to
where m in this instance is the rest inertial mass. Governed by this equation, the charge’s velocity asymptotically approaches c as the acceleration approaches zero.
To the extent the free-falling charge in Sect. 2 does not radiate, and to the extent the product of the gravitational mass and the uniform g field results in a constant force, we seem obliged to recognize that the gravitational mass is constant and independent of the charge’s speed, whereas the inertial mass depends on speed in the familiar, relativistic way.
This being the case, we may conclude that the free falling charge’s speed will asymptotically approach v as its acceleration approaches zero. In general, every charge may be considered to have gravitational mass and inertial mass. When the speed is <<c these two masses are proportional (or equal). But when v~c the gravitational mass remains unchanged, whereas the inertial mass increases by the factor g = (1-v2/c2)-1/2.
5. A Caveat to Free Fall in Earth’s g Field.
Earth’s gravitational field may prove unsuitable for testing the ideas discussed above for two reasons: (1) the atmosphere, and (2) the fact that the field varies as 1/r2.
The atmosphere can of course slow particles down with a force that is a function of the particle’s speed. Consequently a charge free falling in Earth’s atmosphere may emit a type of bremstrahlung.
The variability of g with depth in the Earth’s field may also result in a free falling charge experiencing a varying force. In such an environment the charge may again emit radiation.
In this article the suggested uniform gravitational field is presumably the same at all depths/positions, and it exists in a vacuum.