Covariance of Abraham-Lorentz, Point Charge Going in a Circle in Motional Rest Frame K

G.R.Dixon, 7/8/2006

In this article the covariance of the Abraham-Lorentz 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 (vector) formula for the radiation reaction force is

, (1)

where

, (2)

and where u is the magnitude of the charges velocity:

. (3)

We note in passing that

. (4)

In the case of periodic motion, the driving agent must counteract the radiation reaction force:

. (5)

Thus

. (6)

Or in component notation,

, (7a)

. (7b)

The rate at which radiation is emitted is then

. (8)

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

. (9)

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

. (10)

This can be tested by ** transforming the K kinematic quantities to K** quantities and then applying Eq. 6. The result of this exercise is

. (11)

Thus the Abraham-Lorentz 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

Since the circular motion considered in this article entails velocity components along the x-axis, we might again expect P_{Rad}(t) to have a different shape than P_{Rad}(t), notwithstanding the fact that Eq. 10 is satisfied. This indeed turns out to be the case. Fig. 1 plots the single-valued function P_{Rad}(t), and Fig. 2 plots P_{Rad}(t). As in the oscillation along the x-axis case, P_{Rad}(t) is negative part of each cycle time in K.

Figure 1

P_{Rad}(t)

Figure 2

P_{Rad}(t)