Covariance of the Abraham-Lorentz Formula for Radiated Power, Oscillation Along the x-axis
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
For purposes of numerically integrating PRad(t), we again build a table, this time with columns:
If the charge’s motion is again
then consecutive values of ti can be separated by increments dt, where
Having populated the table, the energy radiated per cycle is:
For q=1 coul, A=1E-6 meters,w=.0001c/A, Eq. 4 yields
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),
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 dax/dt" for the transformation of dax/dt.) Since the ti’ will not be evenly spaced in K’ time, we define dti’ to be
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 ofp/2 from the Larmor-Lienard plot. When applied to the K’ values, the sum in Eq. 4 produces
(Larmor-Lienard produces 60312 joules.) Also,
The Abraham-Lorentz formula is not enigmatic in the same sense that the Larmor-Lienard formula is. In the Abraham-Lorentz case Fxux (where Fx 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, PRad 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 PRad 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.