Topology of the Vavilov-Cherenkov Radiation Field

Abstract

The Vavilov–Cherenkov effect is a well established phenomenon widely used in physics and technology. A nice exposition of it may be found in the book by Frank [222]. In most text books and scientific papers the Vavilov-Cherenkov effect is considered in terms of Fourier components. To obtain an answer in physical space the inverse Fourier transform should be performed. The divergent integrals which occur obscure the physical picture. As far as we know, there are only a few attempts in which the Vavilov-Cherenkov effect is treated without performing a Fourier transform. First of all we should mention Somerfeld’s paper [223] in which the hypothetical motion of an extended charged particle in a vacuum with velocity v > c was considered. Although the relativity principle prohibits such a motion in vacuum, all the equations of [223] are valid in the medium if we identify c with the velocity of light in the medium. Unfortunately, because of the finite dimensions of the charge, equations describing the field strengths are so complicated that they are not suitable for physical analysis. The other reference treating the Vavilov-Cherenkov effect without recourse to the Fourier transform is the book by Heaviside [48] where the super-luminal motions of a point charge both in the vacuum and in an infinitely extended medium were considered. Yet Heaviside was not aware of Somerfeld’s paper [223] just as Tamm and Frank [224] did not know about Heaviside’s investigations. It should be noted that Frank and Tamm formulated their results in terms of Fourier components. The results of Heaviside (without referring to them) were translated into modern physical language in [225]. A similar motion of a charge of finite dimensions has been considered in [226]. The charge had zero sizes in the direction normal to the velocity and a Gaussian distribution along the direction of its velocity. It was shown there that a singular Cherenkov shock wave did not arise in this case. Instead, the field strengths had a finite maximum at the Cherenkov angle.