Covariance of Abraham-Lorentz, Point Charge Going in a Circle in Motional Rest Frame K
In this article the covariance of the Abraham-Lorentz formula for PRad (the rate at which radiation is emitted) is tested for a charge going in a circle in the xy-plane. The general (vector) formula for the radiation reaction force is
and where u is the magnitude of the charges velocity:
We note in passing that
In the case of periodic motion, the driving agent must counteract the radiation reaction force:
Or in component notation,
The rate at which radiation is emitted is then
For a circular radius of 1E-6 meters, an orbital speed of .0001c, and a charge of 1 coulomb, numerical integration of Eq. 8 over an orbital period produces an emitted energy per cycle of
This is twice the energy emitted per cycle for straight-line oscillation along either axis.
The net momentum in any complete "wave" is zero. Viewed from frame K, traveling in the positive x-direction of frame K at speed v, we thus expect
This can be tested by transforming the K kinematic quantities to K quantities and then applying Eq. 6. The result of this exercise is
Thus the Abraham-Lorentz formula for PRad is covariant for circular motion, quite as it is for straight-line oscillations.
By way of review, it was found that PRad(t) = PRad(t) in the case of oscillation along the y-axis of frame K. (The factor of g in Eq. 10 is solely attributable to the longer "oscillation" period in K.) For oscillation along the x-axis, on the other hand, PRad(t) does not equal PRad(t) (although Eq. 10 is still satisfied). The two distinguishing features of PRad(t) are (a) its apparent pulsed nature, and (b) the fact that it is negative part of the time.
Since the circular motion considered in this article entails velocity components along the x-axis, we might again expect PRad(t) to have a different shape than PRad(t), notwithstanding the fact that Eq. 10 is satisfied. This indeed turns out to be the case. Fig. 1 plots the single-valued function PRad(t), and Fig. 2 plots PRad(t). As in the oscillation along the x-axis case, PRad(t) is negative part of each cycle time in K.