In spherical coordinates $(t, r, \theta, \phi)$ general relativistic wave equation is given by
$\partial_t^2 U=\frac{1}{r^2}(1-\frac{a}{r})\partial_r\left[(1-\frac{a}{r})\partial_r (r^2U)\right]+\frac{1}{r^2\sin\theta}\partial_\theta \sin\theta \partial_\theta U+\frac{1}{r^2\sin^2\theta}\partial_\phi^2 U*emphasized text*\ ,$
where $a$ is small parameter, $a\leq 0.5$.
The solution of wave equation can sought as
$U=e^{-i\omega t}\frac{R_{\ell m}(r)}{r}Y_{\ell m}(\theta,\phi)$
and then we can obtain the following equation
$\frac{1}{r}(1-\frac{a}{r})\partial_r\left[(1-\frac{a}{r})\partial_r (rR)\right]+\left(\omega^2-\frac{\ell(\ell+1)}{r^2}\right)R=0\ ,$
which is difficult to solve analytically. Since $a$ is a small parameter we can use perturbation: $R(r,a)=R_0(r)+a R_1(r)+\frac{1}{2}a^2R_2(r)+...$, so finally we can get the following set of equations, $R_i(r)$
$\frac{1}{r}\partial_r^2(rR_0)+\left(\omega^2-\frac{\ell(\ell+1)}{r^2}\right)R_0=0\ ,$
$\frac{1}{r}\partial_r^2(rR_1)+\left(\omega^2-\frac{\ell(\ell+1)}{r^2}\right)R_1 -\frac{l^2+l-1}{r^3}R_0+\frac{1}{r^2}R_0'+\frac{2\omega^2}{r}R_0=0$
$......$
Solution for $R_0$ is easy and it can be expressed in terms of spherical Bessel or Hankel functions $R_0=C_0H_{\ell}(\omega r)$ and for $R_1$ function we have the following equation
$\frac{1}{r}\partial_r^2(rR_1)+\left(\omega^2-\frac{\ell(\ell+1)}{r^2}\right)R_1 - \frac{C_0}{r^3}\left[H_l(r) \left(l^2-1-2 r^2 \omega^2\right)+r H_{l+1}(r)\right]=0$
Solution of this equation is:
$R_1 = C_1 H_{\ell}(\omega r)+ {\cal R}_1$
HOW CAN WE FIND PARTICULAR SOLUTION ${\cal R}_1$???
$(\Delta+\omega^2)R_1 = \frac{C_0}{r^3}\left[H_l(r) \left(l^2-1-2 r^2 \omega^2\right)+r H_{l+1}(r)\right]$
Solution is
$R_1(\bf{r}) = -\frac{1}{4\pi}\int d{\bf r}'\frac{e^{-i\omega |{\bf r}-{\bf r}'|}}{|{\bf r}-{\bf r}'|}\frac{C_0}{r'^3}\left[H_l(r') \left(l^2-1-2 r'^2 \omega^2\right)+r' H_{l+1}(r')\right]$