The electromagnetic radiation whose decay violates the inverse-square law: detailed mathematical treatment of an experimentally realized example

Abstract

I analyse and numerically evaluate the radiation field generated by an experimentally realized embodiment of an electric polarization current whose rotating distribution pattern moves with linear speeds exceeding the speed of light in vacuum. I find that the flux density of the resulting emission (i) has a dominant value and is linearly polarized within a sharply delineated radiation beam whose orientation and polar width are determined by the range of values of the linear speeds of the rotating source distribution, and (ii) decays with the distance d from the source as d −α in which the value of α lies between 1 and 2 (instead of being equal to 2 as in a conventional radiation) across the beam. In that the rate at which boundaries of the retarded distribution of such a source change with time depends on its duration monotonically, this is an intrinsically transient emission process: temporal rate of change of the energy density of the radiation generated by it has a time-averaged value that is negative (instead of being zero as in a conventional radiation) at points where the envelopes of the wave fronts emanating from the constituent volume elements of the source distribution are cusped. The difference in the fluxes of power across any two spheres centred on the source is in this case balanced by the change with time of the energy contained inside the shell bounded by those spheres. These results are relevant not only to long-range transmitters in communications technology but also to astrophysical objects containing rapidly rotating neutron stars (such as pulsars) and to the interpretation of the energetics of the multi-wavelength emissions from sources that lie at cosmological distances (such as radio and gamma-ray bursts). The analysis presented in this paper is self-contained and supersedes my earlier works on this problem. †

A heuristic account of the radiation by the superluminally moving current sheet in the magnetosphere of a neutron star

“…The enhanced focusing of the emitted waves that takes place as a result of the diminishing separation between the nearly coincident stationary points of a phase function gives rise, in turn, to a lower rate of decay of the flux density of the radiation with distance (Section 4). That the decay of the present radiation with distance disobeys the inverse-square law in certain directions is not incompatible with the requirements of the conservation of energy because the radiation process discussed here is intrinsically transient: the difference in the fluxes of power across any two spheres centred on the star is balanced by the change with time of the energy contained inside the shell bounded by those spheres (see [15], Appendix C).…”

“…8, 10-13) or multiply (Figs. [14][15][16], can have both narrow (Figs. 8, 9, 13 and 16) and wide (Figs.…”

“…Note that, as indicated by the last column of Table 1, the pulse profiles depicted in Figs. [8][9][10][11][12][13][14][15][16] have to be plotted on considerably shorter longitudinal scales before their peaks assume the shapes shown in Fig. 17 and the maximum values of their dimensionless intensity Î can be discerned graphically.…”

“…Temporal rate of change of the energy density of the radiation generated by this process has a time-averaged value that is negative (instead of being zero as in a conventional radiation) at points where the envelopes of the wave fronts emanating from the constituent volume elements of the source distribution are cusped. The difference in the fluxes of power across any two spheres centred on the star is thus balanced by the change with time of the energy contained inside the shell bounded by those spheres (see [15], Appendix C, where this is demonstrated for each high-frequency Fourier component of a superluminally rotating source distribution).…”

Gamma-ray spectra of the Crab, Vela and Geminga pulsars fitted with SED of the emission from their current sheet

“…The radiation from a rigidly-rotating extended source whose distribution pattern moves with linear speeds exceeding the speed of light in vacuum is intrinsically transient: temporal rate of change of its energy density has a time-averaged value that is negative (instead of being zero as in the case of a steady-state radiation) at points where the envelopes of the wave fronts emanating from the constituent volume elements of the source distribution are cusped. As a result, there is a difference between the fluxes of power that cross two different spheres centred on the source at any given time, a difference that is balanced by the change with time of the energy contained inside the shell bounded by those spheres (see appendix C of [24]). That is how this radiation conserves energy at the same time as decaying more slowly with distance than predicted by the inverse-square law in certain directions [23,24].…”