Covariant Formulation of the Transition Radiation Energy Spectrum of an Electron Beam at a Normal Angle of Incidence onto a Round Metallic Screen

In the transition radiation emission from a N electron bunch hitting at a normal angle of incidence a metallic screen, the transverse and the longitudinal spatial coordinates of the electron bunch play different roles in determining the N single electron radiation field amplitudes and their relative phases in relation to the different physical constraints which an electromagnetic radiative mechanism by a charged beam must meet: i.e., temporal causality and covariance. The distribution of the N electron longitudinal coordinates determines indeed the sequence of the N electron collisions onto the metallic screen and, on the basis of the temporal causality principle, it also determines the distribution function of the relative emission phases of the N single electron field amplitudes from the metallic surface. The distribution of the transverse coordinates of the N electrons contributes as well to determine the relative phase distribution of the N electron field amplitudes at the observation point - located on the longitudinal axis of the reference frame - providing a further phase information that accounts for the transverse displacement of the N electrons with respect to the beam axis. The distribution of the transverse coordinates of the N electrons is a relativistic invariant under a Lorentz transformation with respect to the direction of motion of the beam and, consequently, it is expected to leave a covariant mark on the N single electron amplitudes composing the radiation field. The covariant imprinting of the N electron transverse density on the radiation field affects both the temporal coherent and incoherent parts of the transition radiation energy spectrum. Such a dependence of the N single electron radiation field amplitudes on the electron density in the transverse plane manifests itself as....(abstract partially missed because of lack of space)