In our studies of radiation from charged particles moving at velocities approaching that of light, a number of interesting phenomena are observed, such as the searchlight effect wherein radiation from the charged particle is constrained to a very narrow forward radiation cone. Furthermore, the calculation of detailed angular radiation patterns, in the frame of reference moving with the charged particle, and wavelength distributions are readily accomplished. The results can then be transformed back to the laboratory, or observer, frame of reference. For instance, the calculation of undulator radiation reduces to use of the well-known formula for so-called dipole radiation from a simple oscillating electron. With this approach we need solve Maxwell's equations for only the simplest radiating system, a small amplitude oscillating electron. This approach is not only simple to follow, but gives valuable physical insights to the radiation process and the parameters that characterize it.

In order to relate calculations in one frame of reference to those in another frame of reference when the relative speed between the two approaches that of light, we must make use of the Lorentz space-time transformations, which provide relationships between spatial and temporal scales in the two frames of reference, and are consistent both with Einstein's postulates of special relativity and with all known experiments (see Ref. 2 for a discussion of the Lorentz transformations and their reduction to Galilean transformations as v/c → 0).