Covariance of the Abraham-Lorentz Formula for Radiated Power, Charge Oscillating Along the y-axis

G.R.Dixon, 7/4/2006

The relativistic Abraham-Lorentz formula for the power radiated by a charge moving periodically along the y-axis of inertial frame K is

. (1)

For purposes of numerically integrating P_{Rad}(t), we again
build a table with columns:

t |
u |
g |
a |
da |
P |

If the charge’s motion is

, (2)

then consecutive values of t_{i} can be separated by increments dt, where

. (3)

Having populated the table, the energy radiated per cycle is:

. (4)

For q=1E-6 coul, A=1E-6 meters, w=.0001c/A, Eq. 4 yields

, (5)

which is the same result obtained using the Larmor-Lienard formula.

Given the motion in Eq. 2, q radiates a spherical wave. In frame K the net __momentum__ in one complete wave is zero. Thus from the perspective of inertial frame K’ (moving in the positive x-direction of K at speed v),

. (6)

We can determine whether Abraham-Lorentz (Eq. 1) produces this result in K’ by transforming the quantities in the table into K’ quantities.

Fig. 1a plots P(t), and Fig. 1b plots P’(t’), with v=.95c. Note that, for this oscillation along the __y__-axis, the K and K’ plots have the same shape, differing only in the scales on the time axes. (When motion was along the x-axis, the shapes were distinctly different for relativistic values of v.) Also noteworthy in frame K is the phase shift of p/2 from the Larmor-Lienard plot. When applied to the K’ values, the sum in Eq. 4 produces

. (7)

(Larmor-Lienard produces 60312 joules.) Also,

. (8)

Figure 1a

P(t)

Figure 1b

P’(t’)