Aging of an electromagnetic wave group

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Physical Review D

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A long electromagnetic (EM) wave group with an effective finite length and cross section has been constructed from cylindrical waves to simulate some aspects of a photon beam believed significant for producing the extragalactic red shift. In the (r,φ,z) cylindrical coordinates used the wave group propagates in the axial or +z direction at essentially the velocity of light. To create the wave group initially, EM radiation is presumed to be provided by some radiating electron located at the origin (r=0, z=0). After this radiation has been assumed to take place, then the EM wave group has presumably come into existence in the form constructed here. The resulting wave group radiates energy in both the axial (+z) and radial (+r) directions. As desired, the axial radiation propagates along the +z direction to move the wave group forward, away from the initial position of the hypothetical radiating electron. A finite (even if small) amount of radial radiation is necessary to produce a red shift by means of the mechanism suggested here. As a mathematical artifice, it is convenient to assume that the radial radiation, coming from the cylinder axis at r=0, is provided by a finite cylindrical "black box" centered on the z axis. This black box moves with the center of the wave group in the +z direction, and so necessarily moves at essentially the velocity of light. After the wave group propagates outside the region where it was supposedly radiated, the radial radiation continues. For our application here, this is an awkward property of the EM eigenfunctions, since now no radiating electron is supposed to be present at the new location to provide this radial radiation. Hence, the propagation of the wave group outside the region of influence of the radiating electron is simulated by turning off the radial power flow supplied by the black box. The black box has been introduced expressly for the purpose of switching off this undesired radial radiation in a convenient manner. Once the radial power flow is turned off at the source, the enitre EM energy of the wave group is gradually lost by the radial power flow outward into infinitely large values of r. The EM wave group considered here is nonmonochromatic, consisting of a superposition of frequencies with a central dominant frequency. The eigenfunctions at the higher frequencies radiate away EM energy radially at a faster rate than the eigenfunctions at the lower frequencies. This gradually produces a shift in the central dominant frequency of the wave group towards the red. This shift resembles the extragalactic red shift. An explicit formula for the red shift is derived in terms of the parameters of the wave group and the elapsed time. No red shift is demonstrated for any type of monochromatic EM wave. © 1971 The American Physical Society.