While reading an online tutorial (http://www.phy.duke.edu/~rgb/Class/phy319/phy319/node146.html) about deriving the relativistic generalization of Larmor Formula, I got some problems with the steps.(adsbygoogle = window.adsbygoogle || []).push({});

Basically with an assumption ##\beta \ll 1## the author gets

##Power = \frac{e^2}{6 \pi \epsilon_0 c^3} |\frac{d\textbf{v}}{dt}|^2## -- (1)

then by replacing ##\textbf{v} = \frac{\textbf{p}}{m}## he/she gets

##Power = - \frac{e^2}{6 \pi \epsilon_0 c^3} \frac{dp_{\mu} dp^{\mu}}{d\tau d\tau}## -- (2)

where ##p^{\mu}, p_{\mu}## are contravariant and covariant forms of the momentum 4-vector respectively. To my understanding it's using Minkowski metric.

According to wikipedia (http://en.wikipedia.org/wiki/Larmor_formula#Covariant_form), result (2) makes sense because when ##\beta## again goes to ##\beta \ll 1## (2) reduces to (1).

So here comes a problem, the reasoning for (1) makes use of the assumption ##\beta \ll 1##, thus the WHOLE CONTEXT is already non-relativistic, how come one can derive (2) from (1) in this context?

By the way there might be a mistake in the tutorial: when starting with (1), the author takes ##\textbf{v} = \frac{\textbf{p}}{m}##, then he/she directly applies

##|\frac{d\textbf{v}}{dt}|^2 = \frac{1}{m^2} |\frac{d\textbf{p}}{dt}|^2##

which doesn't seems right, in ##\frac{d(\textbf{p}/m)}{dt}## both ##\textbf{p}## and ##m## are functions of ##t##.

I did check other tutorials about Larmor formula like http://farside.ph.utexas.edu/teaching/em/lectures/node130.html, but the maths is taking much time to understand there :(

Any help will be appreciated.

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# Relativistic generalization of Larmor fomula?

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