Klein Gordon equation in Schwarzschild spacetime (spherical harmonic mode expansion) My Question: In his GR text, Robert Wald claims that solutions, $\phi$, to the Klein-Gordon equation, $\nabla_a\nabla^a \phi = m^2 \phi$, in Schwarzschild spacetime can be expanded in modes of the form $\phi(r,t,\theta,\varphi) = r^{-1}f(r,t)Y_{lm}(\theta,\varphi)$ to obtain 
\begin{equation}
\frac{\partial^2 f}{\partial t^2}-\frac{\partial^2f}{\partial r_*}+\left(1-\frac{2M}{r}\right)\left[\frac{l(l+1)}{r^2}+\frac{2M}{r^3}+m^2 \right]f=0, \label{eq 1} \tag{Wald 14.3.1}
\end{equation}
where we have the Regge-Wheeler tortoise coordinate $r_{*}\equiv r+2M\ln(r/2M-1)$, with $r$ the usual Schwarzschild "radial" coordinate, and $Y_{lm}(\theta,\varphi)$ the spherical harmonics.
I simply want to verify \eqref{eq 1}, using the definition of the tortoise coordinate and spherical harmonics.  
My Attempt: I first note that the Schwarzschild metric in its usual "radial" coordinate, $r$,  is 
\begin{equation}
ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2\left(d\theta^2+\sin^2\theta d\varphi^2\right).
\end{equation} 
Now, the definiton of the tortoise coordinate $r_*$ gives
\begin{equation}
\frac{dr_*}{dr}=\left(1-\frac{2M}{r}\right)^{-1}, \tag{*} \label{dr}
\end{equation}
so one can express the Schwarzschild metric instead as 
\begin{equation}
ds^2=\left(1-\frac{2M}{r}\right)\left(-dt^2+dr_*^2\right)+r^2d\Omega^2.
\end{equation} 
Therefore, the operator $\nabla_a\nabla^a$ in Schwarzschild is given by 
\begin{align}
g^{ab}\nabla_a\nabla_b &= g^{tt} \partial_t^2 + g^{r_*r_*} \partial^2_{r_*} + g^{\theta \theta} \partial^2_{\theta} +g^{\varphi \varphi} \partial^2_{\varphi} \\
&= \left(1-\frac{2M}{r}\right)^{-1}\left(-\frac{\partial^2}{\partial t^2}+\frac{\partial^2}{\partial r_* ^2}\right)+\frac{1}{r^2}\frac{\partial^2}{\partial\theta^2}+\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\varphi^2}\\
\end{align}
Hence, plugging this back into the K-G equation and multiplying through by $-(1-2M/r)$ yields an equation which seems on the right track toward \eqref{eq 1}: 
\begin{equation}
\frac{\partial^2\phi}{\partial t^2}-\frac{\partial^2\phi}{\partial r_*^2}+\left(1-\frac{2M}{r}\right)\left[-\frac{1}{r^2}\frac{\partial^2}{\partial\theta^2}-\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\varphi^2}+m^2\right]\phi^2=0.\tag{**}\label{promising}
\end{equation}
However, two things aren't working out quite right for me: 1) When I plug in the mode expansion $\phi=r^{-1}f(r,t)Y_{lm}(\theta,\varphi)$, I would like to invoke the relation
\begin{equation}
\Delta Y_{lm}(\theta,\varphi)=-\frac{l(l+1)}{r^2}Y_{lm}(\theta,\varphi),
\end{equation}
where $\Delta$ is the Laplacian on the 2-sphere,
\begin{equation}\Delta\equiv
\frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\varphi^2}.
\end{equation}
BUT, this looks to differ from the angular part of \eqref{promising} by a term proportional to $\partial/\partial\theta$.
And 2) My second issue is in the derivative with respect to the tortoise coordinate.  In particular, I find (using the inverse of \eqref{dr} when necessary) 
\begin{align}
\frac{\partial^2}{\partial r_*^2}\left[\frac{f(r,t)}{r}\right] &=\frac{\partial}{\partial r_*}\left[-\frac{f}{r^2}\frac{\partial r}{\partial r_*}+\frac{1}{r}\frac{\partial f}{\partial r_*}\right]\\
&=\left[2\left(1-\frac{2M}{r}\right)\left(\frac{1}{r^3}-\frac{3M}{r^4}\right)-\frac{2}{r^2}\left(1-\frac{2M}{r}\right)\frac{\partial }{\partial r_*}+\frac{1}{r}\frac{\partial^2}{\partial r_*^2}\right]f,\\
\end{align}
which differs from terms in \eqref{eq 1} that I needed to come from the tortoise derivatives by terms proportional to both $f$ and $\partial f/\partial r_*$ (and even overall factors on terms with correct derivative number and $r$ order).
 A: The resolution to both my issues is that I was not correctly applying the covariant derivative in $\nabla_a \nabla^a$ (as was mentioned in a comment by user2309840).  More specifically, my expression should have been 
\begin{equation}
\nabla_a\nabla^a\phi= \partial_a\nabla^a\phi+\Gamma^a_{\;ab}\nabla^b\phi \tag{*}\label{1}
\end{equation}  
Now, the contracted Levi-Civita can be expressed succinctly (Wald 3.4.9) in terms of the metric determinant $g\equiv \det{g_{\mu\nu}}$ as 
\begin{equation}
\Gamma^a_{\;ab}=\frac{\partial}{\partial x^\mu}\ln{\sqrt{|g|}}.
\end{equation}
Thus, \eqref{1} becomes
\begin{align}
\nabla_a\nabla^a&=\sum_{\mu,\nu}\frac{1}{\sqrt{|g|}}\partial_\mu\left[\sqrt{|g|}g^{\mu\nu}\partial_\nu \phi\right] \\
&= g^{tt}\partial^2_t\phi+\frac{1}{\sqrt{|g|}}\partial_{r_*}\left[\sqrt{|g|}g^{r_*r_*}\partial_{r_*} \phi\right]+\frac{g^{\theta\theta}}{\sqrt{|g|}}\partial_\theta\left[\sqrt{|g|}\partial_\theta\phi\right]+g^{\varphi\varphi}\partial^2_\varphi\phi, \label{2}\tag{**}\\
\end{align}
where the square root of the metric determinant here is 
\begin{equation}
\sqrt{|g(r,\theta)|}=\sqrt{-g_{tt}g_{r_*r_*}g_{\theta\theta}g_{\varphi\varphi}}=\left(1-\frac{2M}{r}\right)r^2\sin{\theta}
\end{equation}
Therefore, the angular part of \eqref{2} is now precisely the Laplacian on the 2-sphere as I sought to show
\begin{equation}
\Delta\equiv\frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left[\sin\theta\frac{\partial}{\partial\theta}\right]+\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\varphi^2}.
\end{equation}
And we note that  
\begin{align}
\partial_{r_*}\left[\sqrt{|g|}g^{r_*r_*}\partial_{r_*}\phi\right]=\frac{\partial}{\partial r_*}\left[r^2\frac{\partial\phi}{\partial r_*}\right]&=2r\frac{\partial r}{\partial r_*}\frac{\partial\phi}{\partial r_*}+r^2\frac{\partial^2\phi}{\partial r^2_*}\\
&=rY_{lm}\frac{\partial^2 f}{\partial r_*^2}-\frac{2M}{r^2}\left(1-\frac{2M}{r}\right)Y_{lm}f, \label{3}\tag{***}
\end{align}
where in the second line I have taken $\phi(r,t,\theta,\varphi)=r^{-1}f(r,t)Y_{lm}(\theta,\varphi)$ and applied $\partial r/\partial r_*=(1-2M/r)$ .
Wald 14.3.1 now follows directly from \eqref{2} and \eqref{3} (plus the definition of the spherical harmonics, $r^2\Delta Y=l(l+1)Y$).  
