Question 12.72: The natural relativistic generalization of the Abraham-Loren...

(a) It’s inconsistent with the constraint \eta_{\mu} K^{\mu}=0 (Prob. 12.39(d)).

(b) We want to find a 4-vector b^{\mu} with the property that \left(\frac{d \alpha^{\mu}}{d \tau}+b^{\mu}\right) \eta_{\mu}=0 . How about b^{\mu}=\kappa\left(\frac{d \alpha^{\nu}}{d \tau} \eta_{\nu}\right) \eta^{\mu} ? Then \left(\frac{d \alpha^{\nu}}{d \tau}+b^{\mu}\right) \eta_{\mu}=\frac{d \alpha^{\mu}}{d \tau} \eta_{\mu}+\kappa \frac{d \alpha^{\nu}}{d \tau} \eta_{\nu}\left(\eta^{\mu} \eta_{\mu}\right) . But \eta^{\mu} \eta_{\mu}=-c^{2} , so this becomes \left(\frac{d \alpha^{\mu}}{d \tau} \eta_{\mu}\right)-c^{2} \kappa\left(\frac{d \alpha^{\nu}}{d \tau} \eta_{\nu}\right) , which is zero, if we pick \kappa=1 / c^{2} . This suggests K_{ rad }^{\mu}=\frac{\mu_{0} q^{2}}{6 \pi c}\left(\frac{d \alpha^{\mu}}{d \tau}+\frac{1}{c^{2}} \frac{d \alpha^{\nu}}{d \tau} \eta_{\nu} \eta^{\mu}\right) . Note that \eta^{\mu}=(c, v ) \gamma , so the spatial components of b^{\mu} vanish in the nonrelativistic limit v \ll c , and hence this still reduces to the Abraham-Lorentz formula. [Incidentally, \alpha^{\nu} \eta_{\nu}=0 \Rightarrow \frac{d}{d \tau}\left(\alpha^{\nu} \eta_{\nu}\right)=0 \Rightarrow \frac{d \alpha^{\nu}}{d \tau} \eta_{\nu}+\alpha^{\nu} \frac{d \eta_{\nu}}{d \tau}=0, \text { so } \frac{d \alpha^{\nu}}{d \tau} \eta_{\nu}=-\alpha^{\nu} \alpha_{\nu} , and hence b^{\mu} can just as well be written \left.-\frac{1}{c^{2}}\left(\alpha^{\nu} \alpha_{\nu}\right) \eta^{\mu} .\right]