Comment on "Application of the three-dimensional telegraph equation to cosmic-ray transport" (arXiv:1606.08272)

TL;DR

The Comment addresses the correctness of the Green's function for the 3D telegraph equation used in cosmic-ray transport, showing that the previously published form in Ref. [Tautz2016] omits the ballistic wavefront and contains an inaccurate wake. It presents a rigorous Fourier-transform derivation leading to the correct retarded Green's function $G(R,T)=\left(\frac{\delta(T-R)}{4\pi R}+\frac{\beta}{4\pi} \frac{\theta(T-R) I_1(\beta \sqrt{T^2 - R^2})}{\sqrt{T^2 - R^2}}\right) e^{-\beta T}$, which explicitly includes both the Dirac delta front and the damped wake. The work confirms that in the limit $\beta \to 0$ the familiar delta-front is recovered and the long-time behavior approaches diffusion, $G\sim t^{-3/2}$, thereby preserving causality and correct asymptotics. The discrepancy is traced to neglecting the distributional derivative of a Heaviside function during differentiation, a mistake that led to an incorrect wake term and the loss of the ballistic component.

Abstract

In a recent publication [R. C. Tautz and I. Lerche, Res. Astron. Astrophys. 16, 162 (2016); arXiv:1606.08272], the authors present a derivation of the Green's function for the three-dimensional telegraph equation (also known as the heat wave equation, or relativistic heat conduction equation). We demonstrate that the closed-form expression derived in their Appendix A is incorrect. Specifically, the published solution lacks a Dirac delta term representing the ballistic wavefront and contains an algebraic error in the prefactor of the wake term. These omissions arise from the neglect of distributional derivatives when differentiating a Heaviside step function. We provide a rigorous derivation of the Green's function using the Fourier transform method, verify the correct limiting behavior as the damping vanishes, and pinpoint the exact mathematical step where the original derivation failed.