Sidney Coleman's argument that a hyperbolically accelerated electron radiates

Question

On page 35 of Sidney Coleman's Classical Electron Theory from a Modern Standpoint, he writes:

If we remember that in Maxwellian dynamics, the radiation field is given by the difference of the advanced and retarded fields, it is easy to see that the electron radiates.

The figure shows the world line of an electron undergoing hyperbolic acceleration. The past light cone from the point P does not interact with the electron worldline, so the retarded potential vanishes; the future light cone intersects it once, so the advanced potential doesn't. Thus there is a radiation field at P.

The retarded potential at P is undefined, so how is Coleman able to conclude that it vanishes?

Answer 1

Strictly speaking hyperbolic motion exists in infinite past, and thus there is no finite retarded time $t_r$ at $P$. We can stop now and say $t_r$ is meaningless. Or, we can assign it value $-\infty$.

Coleman is closer to the second option. Then retarded field is defined at $P$ and it is zero.

One rationale for this could be as follows. Infinite (in past) hyperbolic motion is (so far) an unrealizable unobserved extreme, obeying some rule for all times in the past. Real motion can approximate it in some finite time window $t_A,t_B$, before which ($t<t_A$) the motion is no longer hyperbolic. For this real motion, at point $P$, retarded time $t_r$ exists and is finite and if acceleration is not too high, retarded potential exists and is finite. If $t_A$ is sufficiently far in the past, then this retarded potential is very small and can be replaced by zero for practical purposes.