Abstract
Modern high-energy physics experiments commonly employ charged particles of very high velocities, frequently extremely close to the speed of light. To gain such ultrarelativistic velocities, the energies of the particles must be increased far beyond their rest masses. In addition, if one wants to confine the particles in a bounded region, for instance, in an accumulation ring, their trajectories must be bent. During both these processes the particles are subject to acceleration and so they emit electromagnetic radiation, thus dissipating part of the accumulated energy. The evaluation of the energy radiated by ultrarelativistic charged particles can no longer be based on the multipole expansion, legitimate only in the non-relativistic limit, and requires appropriate computational tools yielding exact results. One of the basic tools of this kind is the relativistic Larmor formula, which allows, in particular, to compute the energy dissipated via radiation in high-energy particle accelerators.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
If \(dP^\mu _\mathrm{rad}/ds\) would equal the total four-momentum loss of the particle at the proper time s, then surely these four quantities would form a four-vector. Actually, in Sect. 15.2.4 we will see that at the same time s the particle exchanges with the electromagnetic field a further fraction of four-momentum – the Schott term, see Eq. (15.44) – which is, however, separately Lorentz invariant. Therefore, the assumption made in the text is justified a posteriori.
- 2.
With a more accurate analysis, left as exercise, one can prove the general inequalities
$$ 1\le \left| {\mathbf {n}-\mathbf {v}\over 1-\mathbf {v}\cdot \mathbf {n}}\right| \le {1\over \sqrt{1-v^2}}. $$If \(\alpha \) denotes the angle between \(\mathbf {v}\) and \(\mathbf {n}\), the lower bound is attained for \(\alpha =0\) and \(\alpha =\pi \), while the upper bound is attained for \(\sin \alpha =\sqrt{1-v^2}\). Therefore, if \(v\approx 1\), the magnitude of the vector \((\mathbf {n}-\mathbf {v})/(1-\mathbf {v}\cdot \mathbf {n})\) becomes, actually, very large for \(\alpha \approx \sqrt{1-v^2}\). This means that the estimates (10.39) and (10.40) in reality are lower bounds.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Lechner, K. (2018). Radiation in the Ultrarelativistic Limit. In: Classical Electrodynamics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-91809-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-91809-9_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91808-2
Online ISBN: 978-3-319-91809-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)