Given such a Schwarzschild metric, the covariant Klein-Gordon equation for a mass $m$ takes the form $$\left[\frac{1}{g_{00}} \frac{\partial^2}{\partial t^2 }-\frac{1}{r^2} \frac{\partial}{\partial r} \left(\frac{r^2}{g_{rr}} \frac{\partial}{\partial r}\right)+\frac{L^2}{r^2} +m^2 \right]\psi=0$$
where did the above equations come from?
Given any spacetime $(M,g)$ with metric tensor $g_{\mu\nu}$ the scalar Laplace operator is given by
$$\Box \phi=\dfrac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}g^{\mu\nu}\partial_\nu \phi)\tag{1}$$
Where $\sqrt{-g}=\sqrt{-\det g_{\mu\nu}}$. Given this equation you can plug any metric you want, in particular the Schwarzschild metric, in any coordinate system you want, and write down the scalar wave equation
$$(\Box-m^2)\phi=0.\tag{2}$$
