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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 PRad(t) and P’Rad(t’), as computed using Abraham-Lorentz and the Lorentz transformations. (Here PRad(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 PRad(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 PRad(t). The other figures show P’Rad(t’) for a range of v values.

Figure 1

                                                                     

PRad(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