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Relativistic generalization of Larmor fomula?

  1. Apr 29, 2015 #1
    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.

    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.
     
  2. jcsd
  3. Apr 29, 2015 #2

    Mentz114

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    To get the covariant form the author has introduced Lorentz scalar ##\frac{dp_{\mu} dp^{\mu}}{d\tau d\tau}## making the expression manifestly covariant. This is not affected by the restriction on ##\beta##.

    You say that ##m## is a function of ##t##. If ##m## is the rest mass then this is not true.
     
  4. Apr 29, 2015 #3
    Thanks for the reply!

    about the mass, I'm still confused by the author's notation. I did notice that he/she uses ##m## to represent rest mass some chapters before. However it's not mentioned that the derivation is carried out in the particle's frame, hence ##\textbf{v} = \frac{\textbf{p}}{m}## should apply to any frame here -- or, say that the observer is measuring in frame ##S##, then ##S## is not necessarily the particle's frame and all ##\textbf{v}, \textbf{p}, m## are with respect to ##S##.

    The introduction of Lorentz scalar is the last step of the derivation which is another way of saying "##(\frac{d\textbf{p}}{d\tau})^2 - (\frac{1}{c^2}\frac{dE}{d\tau})## in Minkowski metric" to me. It's not clear to me when it jumps out of the ##\beta \ll 1## context.
     
  5. Apr 29, 2015 #4

    Mentz114

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    Gold Member

    Rest mass is frame invariant in the ##\beta<<1## regime, however it is defined. But if you are using ##m'=\gamma m## or some kind of 'relativistic mass', I think that is wrong.

    To be covariant the quantity must be a Lorentz scalar. ##p^\mu p_\mu## is obviously invariant under coordinate transformation because ##\lambda p^\mu \lambda^{-1} p_\mu=p^\mu p_\nu = -mc^2##. ( note that ##m## here must be invariant ).
     
  6. Apr 29, 2015 #5
    I'd like to clarify my opinion for the mass first. Surely the rest mass is frame invariant, my point is that beginning with ##|\frac{d\textbf{v}}{dt}|^2##, the author did

    ##\textbf{v} = \frac{\textbf{p}}{m}## -- (3)

    now what I argued was that the ##m## in (3) is NOT rest mass, thus it should be followed by

    ##|\frac{d\textbf{v}}{dt}|^2 = |\frac{d(\textbf{p}/m)}{dt}|^2## -- (4)

    and the ##m## in (4) is not rest mass either so it's time dependent.
     
  7. Apr 30, 2015 #6
    I'm afraid I was totally wrong from the very beginning, problem solved now.

    @Mentz114 you're alright in your answers, thanks a lot but they didn't hit the key point of my confusion :)

    Here's what solves my problem (from http://www.cv.nrao.edu/course/astr534/LarmorRad.html):

     
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