Classical Electrodynamics · Problem 14.16

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · synchrotron-radiation-lorentz-invariant-form

Problem

Exploiting the fact that k0d3N/d3kk_0 d^3N/d^3k, the number of quanta per invariant phase-space element d3k/k0d^3k/k_0, is a Lorentz-invariant quantity, show that the energy radiated per unit frequency interval per unit solid angle, (14.79), can be written in the invariant and coordinate-free form ℏωd3Nd3k=4e23π2m2[(p⋅k)2[d2(p⋅k)/dτ2]2(d(ϵ1⋅p)dτ)2K2/32(ξ)+(p⋅k)(ϵ2⋅p)22[d2(p⋅k)/dτ2]K1/32(ξ)]\hbar\omega\frac{d^3N}{d^3k} = \frac{4e^2}{3\pi^2 m^2}\left[\frac{(p\cdot k)^2}{[d^2(p\cdot k)/d\tau^2]^2}\left(\frac{d(\epsilon_1\cdot p)}{d\tau}\right)^2 K_{2/3}^2(\xi) + \frac{(p\cdot k)(\epsilon_2\cdot p)^2}{2[d^2(p\cdot k)/d\tau^2]} K_{1/3}^2(\xi)\right] where dτd\tau is the proper time interval of the particle of mass mm, pμp^\mu is the 4-momentum of the particle, kμk^\mu is the 4-wave vector of the radiation, and ϵ1\epsilon_1, ϵ2\epsilon_2 are polarization vectors parallel to the acceleration and in the direction ϵ1×k\epsilon_1 \times \mathbf{k}, respectively. The parameter is ξ=223m⋅(p⋅k)3/2(∣d2(p⋅k)/dτ2∣)1/2\xi = \frac{2\sqrt{2}}{3m}\cdot\frac{(p\cdot k)^{3/2}}{(|d^2(p\cdot k)/d\tau^2|)^{1/2}}

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