The speed-of-light catastrophe

Abstract

It is shown that the retarded potential for an extended source, whose density depends on the azimuthal angle ᵠ and the time t in the combination ᵠ – ωt only, has a discontinuous gradient at points within the source that lie on the cylindrical surface r P = c/ω , and that the radial component of this gradient diverges like ( r P – c/ω ) –¼ as r P ω / c → 1 +, where r P is the radial position of the observation point in cylindrical polar coordinates, c is the propagation speed of the waves described by the retarded potential, and ω is a constant angular frequency. So, for an extended source moving on a curved path whose speed is in parts smaller and in parts greater than the speed of the disturbances it generates, there is a sudden and infinitely large change in the strength of the field across the surface (within the source) at which the speed of the source and the speed of the waves are equal. The divergence in the field strength does not arise from any long-term transfer of energy to the field; it occurs instantaneously and so is to be understood either as signifying the breakdown of the governing wave equation or as disallowing the type of source in question. Some implications of this non-elementary catastrophe – in the contexts of gas-dynamics, electrodynamics and general relativity – are also pointed out.