Source as a Sum of Multipoles

The circumstance of long wave lengths is expressed by the inequality

$$ka\ll 1 ~,$$

where $a$ is the radius of the cylinder surrounding the source. This circumstance allows us to set

$$J_m(kr')\approx \left\{ \begin{array}{ll} \displaystyle\fra... ...right)^{\vert m \vert} & m=0,-1,-2,\cdots \end{array} \right.$$ (46)

throughout the integration region where the source is non-zero, and it allows us to introduce the $(m+1)$st multipole moment (per unit length $d\xi$)

$$\frac{i^m}{\vert m\vert !} \int_0^\infty r'dr' \int_0^{2\pi}... ...rm{length}} \times(\textrm{length})^{\vert m\vert +1}\right]$$ (47)

for the double integral on the right hand side of Eq.(45). This multipole density [#!factor_of_two!#] is complex. However, the reality of the master source $S_{I,II}(\tau',\xi',r',\theta')$ implies and is implied by

$$\overline{S^m_{I,II}(\tau',\xi')}=S^{-m}_{I,II}(\tau',\xi')$$

In terms of this multipole density the full scalar radiation field in $F$ is

$${ \psi_F(\xi,\tau,r,\theta)= \sum_{m=-\infty}^\infty \int_0^\infty dk \, k \,k^{\vert m \vert}e^{im\theta} J_m(kr) \times }$$

$$\int_{-\infty}^\infty d\tau' \int_0^\infty d\xi' \xi' \, \frac{2}... ...{\xi^2-\xi'^2-2\xi\xi' \sinh (\tau-\tau')}\right) 2S^m_{II}(\tau',\xi')\right\}$$ (48)

The evaluation of the mode integral $\int_0^\infty dk\,k \cdots$is now an easy two step task. First recall the $m$th recursion relation

$$e^{im\theta} J_m(kr)= \frac{(-1)^m}{k^{\vert m\vert}} \left... ...l \theta} \right) \right]^m J_0(kr),\quad m=0,\pm1,\pm2,\cdots$$ (49)

where for negative $m$ one uses

$$\left[ e^{i\theta} \left(\frac{\partial}{\partial r} +\frac{... ...c{\partial}{\partial \theta} \right) \right]^{\vert m\vert} ~.$$

This recursion relation is a consequence of consolidating two familiar contiguity relations for the Bessel functions. Introduce Eq.(49) into the integrand of Eq.(48).

Second, use the standard expression

$$\int_0^\infty J_0(kr)J_0(k\sqrt{\cdots})k\, dk= \frac{\delta(r-\sqrt{\cdots})}{\sqrt{\cdots}}$$

for the Dirac delta function. Apply this equation to Eq.(48). Consequently, the full scalar radiation field in Rindler sector $F$ reduces to the following multipole expansion

$$\psi_F(\xi,\tau,r,\theta)= \sum_{m=-\infty}^\infty (-1)^m \left[ ... ...tial \theta} \right) \right]^m \psi_m(\xi,\tau,r) \quad \quad [\textrm{charge}]$$ (50)

where

$$\psi_m(\xi,\tau,r) = \int_{-\infty}^\infty \int_0^\infty \frac{1}{2} \left\{ \frac{2S^... ...}} \delta\left(r-\sqrt{\xi^2-\xi'^2+2\xi\xi' \sinh (\tau-\tau')}\right)+\right.$$

$$~ ~~~~~~~~~~~~~~~~\left.\frac{2S^m_{II}(\tau',\xi')}{\sqrt{\xi^2-\x... ...t{\xi^2-\xi'^2-2\xi\xi' \sinh (\tau-\tau')}\right) \right\}d\tau'\xi'\,d\xi' ~.$$

Doing the $\tau'$-integration yields

$$\psi_m(\xi,\tau,r) = 2\int_0^\infty \frac{\left[S^m_I(\tau',\xi')\right]_I - \left[S^m... ...tau',\xi')\right]_{II}}{\sqrt{(\xi^2-\xi'^2-r^2)^2+(2\xi\xi')^2 }} \xi' d\xi'~.$$ (51)

Here $[~~]_I$ and $[~~]_{II}$ mean that the source functions are evaluated in compliance with the Dirac delta functions at $\tau'=\tau +\sinh^{-1}\frac{\xi^2-\xi'^2-r^2}{2\xi\xi'}$ and $\tau'=\tau -\sinh^{-1}\frac{\xi^2-\xi'^2-r^2}{2\xi\xi'}$ respectively. Recall that $\tau'$ is a strictly timelike coordinate in Rindler sector $I$, while in $F$ the coordinate $\tau$ is strictly spacelike. Consequently, one should not be tempted to identify $[~~]_I$ and $[~~]_{II}$ with what in a static inertial frame corresponds to evaluations at advanced or retarded times. Instead, one should think of the observation event $(\xi,\tau,r)$ in $F$ and the source event $(\tau',\xi',r )$ in $I$ as lying on each other's light cones

$$(t-t')^2 -(z-z')^2 =r^2~,$$

both of which cut across the future event horizons $t=\vert z\vert$ of $I$ and $II$. More explicitly, one has

$$\left[S^m_I(\tau',\xi')\right]_I\equiv S^m_I\left( \tau+\sinh^{-1} \frac{\xi^2-\xi'^2-r^2}{2\xi\xi'},\xi' \right)~,$$

which means that the source $S^m_I(\tau',\xi')$ has been evaluated on the past light cone

$$(t-t')^2-(z-z')^2\equiv\xi^2-\xi'^2+ 2\xi\xi' \sinh (\tau-\tau')=r^2$$

of $(\xi,\tau,r)$ at $(\tau',\xi',0)$ in Rindler sector $I$. Similarly,

$$\left[S^m_{II}(\tau',\xi')\right]_{II}\equiv S^m_{II}\left( \tau-\sinh^{-1} \frac{\xi^2-\xi'^2-r^2}{2\xi\xi'} ,\xi'\right)~,$$

which means that the source $S^m_{II}(\tau',\xi')$ has been evaluated on the past light cone

$$(t-t')^2-(z-z')^2\equiv\xi^2-\xi'^2- 2\xi\xi' \sinh (\tau-\tau')=r^2$$

of $(\xi,\tau,r)$ at $(\tau',\xi',0)$ in Rindler sector $II$. The expression, Eq.(51), for the full scalar radiation field is exact within the context of wavelengths large compared to the size of the source. Furthermore, one should note that even though there is only one $\xi'$-integral, $S^m_{I}$ and $S^m_{II}$ are source functions with distinct domains, namely, Rindler sectors $I$ and $II$ respectively.