Covariance of Larmor-Lienard, Point Charge Going in a Circle in Motional Rest Frame K

G.R.Dixon, 7/9/2006

In this article the covariance of the Larmor-Lienard formula for P_{Rad} (the rate at which radiation is emitted) is tested for a charge going in a circle in the xy-plane. The general formula for P_{Rad} is

, (1)

where

, (2)

and where u is the magnitude of the charge’s velocity:

. (3)

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

. (4)

This is twice the energy emitted per cycle for straight-line oscillation along either axis. In this circular motion case P_{Rad}(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

. (5)

This can be tested by ** transforming the K kinematic quantities** to K’ quantities and then applying Eq. 1. The result of this exercise is

. (6)

Thus the Larmor-Lienard formula for P_{Rad} is covariant for circular motion, quite as it is for straight-line oscillations.

By way of review, it was found that P_{Rad}’(t’) = P_{Rad}(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 P_{Rad}(t), and Fig. 2 plots P_{Rad}’(t’).

Figure 1

P_{Rad}(t)

Figure 2

P_{Rad}’(t’)