Covariance of the Abraham-Lorentz Formula for Radiated Power, Oscillation Along the x-axis

G.R.Dixon, 7/4/2006

This article duplicates an exercise done in **"Covariance of the Larmor-Lienard Formula for Radiated Power."** The difference is that in the present case the relativistically correct form for the __Abraham-Lorentz__ formula is substituted for the Larmor-Lienard formula.

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

. (1)

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

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

If the charge’s motion is again

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

, (5)

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

As previously pointed out, 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)

As in the Larmor-Lienard case, we can determine whether Abraham-Lorentz (Eq. 1) produces this result in K’ by transforming the quantities in the table into K’ quantities. (See **"The Lorentz Transformation of da _{x}/dt"** for
the transformation of da

. (7)

Fig. 1a plots P(t), and Fig. 1b plots P’(t’), with v=.95c. Note the markedly different shapes in K and K’, and again the different scales on the time axes. (In the Larmor-Lienard case the shapes in K and K’ were identical.) 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

. (8)

(Larmor-Lienard produces 60312 joules.) Also,

. (9)

Figure 1a

P(t)

Figure 1b

P’(t’)

The Abraham-Lorentz formula is not enigmatic in the same sense that the Larmor-Lienard formula is. In the Abraham-Lorentz case F_{x}u_{x} (where F_{x} is the driving agent’s counteraction to the radiation reaction force) has its maximum magnitude at x=0, and is zero at x=__+__A. Also noteworthy is the Abraham-Lorentz result that a charge,
subjected to a constant
force, does not radiate (although it most assuredly accelerates). Thus here is a key difference between the two formulas that can be experimentally tested (and perhaps already has been). (General Relativists might prefer to point out that a charge, held at rest in a gravitational field, does not radiate.) Another notable difference between Larmor-Lienard and Abraham-Lorentz is that, in the case of
relativistic
oscillators, P_{Rad} may actually be negative for parts of each cycle. This is obviously so in Fig. 1b. (The oscillator __is__ relativistic, viewed from K’.) Perhaps the most dramatic difference between Larmor-Lienard and Abraham-Lorentz is the different shapes of P_{Rad} and P’_{Rad} (Figs. 1a and b) in the Abraham-Lorentz case. Indications are that an oscillating charge, radiating spherical waves in its motional rest frame, will radiate electromagnetic __pulses__ when it translates at relativistic speeds.