where satisfies

(37) |

This Green's function is unique, and it is easy to show[#!HOW_TO_FIND_IT!#] that

This means that is non-zero only inside the future of the source event , and vanishes identically everywhere else. The function is defined on all four Rindler sectors. However, our interest is only in those of its coordinate representatives whose source events lie Rindler sectors or ,

and whose observation events lie in Rindler sector ,

For these coordinate restrictions the two coordinate representatives of , Eq.(38), are

and

These two coordinate representatives give rise to the corresponding two representatives of the unit impulse response, Eq.(36),

This integral expression is exactly what is needed to obtain the radiation field from bodies accelerated in and/or . However, in order to ascertain agreement with previously established knowledge, we shall use the remainder of this subsection to evaluate the sum and the integral in Eq.(41) explicitly.

It is a delightful property of Bessel harmonics that the sum over
can be evaluated in closed form[#!Sommerfeld!#]. This property is the Euclidean plane
analogue of what for spherical harmonics is the spherical addition
theorem. One has

whenever and zero otherwise. The spread-out amplitudes of this linear superposition interfere constructively to form a Dirac delta function response. Indeed, using the standard representation

for this function, one finds that

whenever is in the future of . This is the familiar causal response in due to a unit impulse event in or in .