Abraham-Lorentz Plots of P’_{Rad}(t’) for Several Values of v,
Oscillation Along the x-axis

G.R.Dixon, 7/4/2006

In a previous article a fundamental difference was noted between P_{Rad}(t) and P’_{Rad}(t’), as computed using Abraham-Lorentz and the Lorentz transformations. (Here P_{Rad}(t) is the rate
at which a non-relativistically oscillating charge emits radiant energy in frame K, the motional rest frame. And P’_{Rad}(t’) is the same function when the translating oscillator is viewed from frame K’, moving to the right relative to K at speed v.)

In this article P’_{Rad}(t’) is plotted for several values of v, ranging from non-relativistic to relativistic ones. The purpose is to show how, at non-relativistic values of v, P’_{Rad}(t’) has the same shape as P_{Rad}(t). (This was found to be the case for all values of v in the Larmor-Lienard case.) At higher values of v the onset of a transition is obvious, and at relativistic values the transition is complete. The user can duplicate the results on his/her own PC (with
Visual Basic) for arbitrary values of v. Fig. 1 shows P_{Rad}(t). The other figures show P’_{Rad}(t’) for a range of v values.

Figure 1

P_{Rad}(t)

Figure 2

P’_{Rad}(t’), v=.0000005c

Figure 3

P’_{Rad}(t’), v=.000005c

Figure 4

P’_{Rad}(t’), v=.00005c

Figure 5

P’_{Rad}(t’), v=.0005c

Figure 6

P’_{Rad}(t’), v=.5c