Unit Impulse Response

The solution to Eq.(34) is the retarded Green's function, a unique scalar field, whose domain extends over all four Rindler sectors. One accommodates the cylindrical symmetry of the coordinate geometry by representing the scalar field in terms of the appropriate eigenfunctions, the Bessel harmonics $J_m(kr)e^{im\theta}$, for the Euclidean $(r,\theta)$-plane:

$${\mathcal{G}}(t,z,r,\theta;t',z',r',\theta')= \sum_{m=-\inft... ...J_m(kr)\frac{e^{im(\theta-\theta')}}{2\pi} J_m(kr')~k\, dk ~,$$ (36)

where $G$ satisfies

$$\left( -\frac{\partial ^2}{\partial t^2}+ \frac{\partial^2}{\partial z^2} - k^2 \right) G=-\delta(t-t')\delta(z-z')~,$$ (37)

and

$$G=0 \quad \textrm{whenever }t<t'~.$$

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

$$G=\left\{ \begin{array}{ll} \frac{1}{2}J_0(k\sqrt{(t-t')^2... ...textrm{ whenever } t-t'<\vert z-z'\vert~. \end{array} \right.$$ (38)

This means that $G$ is non-zero only inside the future of the source event $(t',z')$, and vanishes identically everywhere else. The function $G(t,z;t',z')$ is defined on all four Rindler sectors. However, our interest is only in those of its coordinate representatives whose source events lie Rindler sectors $I$ or $II$,

$$\left. \begin{array}{ll} t'&=\pm\xi'\sinh\tau'\\ z'&=\pm\xi... ...ign for }I\\ \textrm{ lower sign for }II~, \end{array}\right.$$

and whose observation events lie in Rindler sector $F$,

$$\begin{array}{ll} t&=\xi\cosh\tau\\ z&=\xi\sinh\tau ~. \end{array}$$

For these coordinate restrictions the two coordinate representatives of $G(t,z;t',z')$, Eq.(38), are

$$G_I(k\xi,\tau;\tau',k\xi') = \left\{ \begin{array}{ll} \frac{1}{2} J_0\left(k\sqrt{\xi^2-\xi'^... ...&\textrm{ whenever }\xi^2-\xi'^2+2\xi\xi'\sinh(\tau-\tau')<0 \end{array}\right.$$ (39)

and

$$G_{II}(k\xi,\tau;\tau',k\xi') = \left\{ \begin{array}{ll} \frac{1}{2} J_0\left(k\sqrt{\xi^2-\xi'^... ...&\textrm{ whenever }\xi^2-\xi'^2-2\xi\xi'\sinh(\tau-\tau')<0 \end{array}\right.$$ (40)

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

$${\mathcal{G}}_{I,II}(\xi,\tau,r,\theta;\tau',\xi',r',\theta'... ...J_m(kr)\frac{e^{im(\theta-\theta')}}{2\pi} J_m(kr')~k\, dk ~,$$ (41)

This integral expression is exactly what is needed to obtain the radiation field from bodies accelerated in $I$ and/or $II$. 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 $m$ 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

$$\sum_{m=-\infty}^\infty J_m(kr)\frac{e^{im(\theta-\theta')}... ...pi} J_0\left(k\sqrt{r^2+r'^2 -2rr'\cos(\theta-\theta')}\right)$$ (42)

Inserting this result, as well as Eqs.(39) or (40) into Eq.(41) yields the two unit impulse response functions with sources in $I$ (upper sign) and $II$ (lower sign)

$${\mathcal{G}}_{I,II}(\xi,\tau,r,\theta;\tau',\xi',r',\theta') = \frac{1}{4\pi}\int_0^\infty J_0\left(k\sqrt{\xi^2-\xi'^2\pm 2\xi\... ...\tau')}\right) J_0\left(k\sqrt{r^2+r'^2 -2rr'\cos(\theta-\theta')}\right) k\,dk$$

$$= \frac{1}{4\pi}\int_0^\infty J_0\left( k\sqrt{(t-t')^2-(z-z')^2}\right)J_0\left(k\sqrt{(x-x')^2-(y-y')^2}\right) \,k\,dk$$

whenever $t-t' \ge \vert z-z'\vert$ 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

$$\int_0^\infty J_0(ka)J_0(kb)~kdk=\frac{\delta(a-b)}{b}$$

for this function, one finds that

$${\mathcal{G}}_{I,II}(\xi,\tau,r,\theta;\tau',\xi',r',\theta') = \frac{1}{4\pi} \frac{\delta\left( \sqrt{\xi^2-\xi'^2\pm 2\xi\xi' ... ... -2rr'\cos(\theta-\theta')} \right)}{\sqrt{r^2+r'^2 -2rr'\cos(\theta-\theta')}}$$

$$= \frac{1}{2\pi} \delta\left( \xi^2-\xi'^2\pm 2\xi\xi' \sinh (\tau-\tau')- (r^2+r'^2 -2rr'\cos(\theta-\theta')) \right)$$ (43)

$$= \frac{1}{2\pi} \delta\left( (t-t')^2-(z-z')^2-(x-x')^2-(y-y')^2 \right)$$

whenever $(t,z,x,y)$ is in the future of $(t',z',x',y')$. This is the familiar causal response in $F$ due to a unit impulse event in $I$ or in $II$.