Covariance of Larmor-Lienard, Point Charge Going in a Circle in Motional Rest Frame K
In this article the covariance of the Larmor-Lienard 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 formula for PRad is
and where u is the magnitude of the charge’s velocity:
For a circular radius of 1E-6 meters, an orbital speed of .0001c, and a charge of 1 coulomb, numerical integration of Eq. 1 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. In this circular motion case PRad(t) is single-valued in motional rest frame K.
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. 1. The result of this exercise is
Thus the Larmor-Lienard 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 both the x- and y-axes of frame K. (The factor of g in Eq. 5 is solely attributable to the longer "oscillation" period in K’.)
Fig. 1 plots the single-valued function PRad(t), and Fig. 2 plots PRad’(t’).