= Acceleration correction for an instantaneous boosted kink
{title2=$\Delta L=\frac{\pi^2\dot\gamma^2}{3\gamma^3}$}
The actual derivative of $\theta_K(\gamma(t)(x-X(t)))$ is $[\dot\gamma(x-X)-\gamma\dot X]\theta_K'$. The omitted contribution is first order in acceleration. Its cross term with the translation velocity integrates to zero because $\int y(\theta_K'(y))^2dy=0$. The square adds $\dot\gamma^2\int y^2(\theta_K')^2dy/(2\gamma^3)=\pi^2\dot\gamma^2/(3\gamma^3)$ to the spatially integrated density, since $\theta_K'=2\operatorname{sech}y$ and $\int y^2(\theta_K')^2dy=2\pi^2/3$. Thus the <instantaneous-boost effective action for a sine-Gordon kink> is an adiabatic truncation rather than the exact action restricted to that accelerating configuration-space profile.
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