I am having a problem with gravitational lensing question where we are interested in deflection angle of light traveling in potential of galactic cluster, described with tensor
$h_{00}=\frac{a}{\sqrt{1+\left(\frac{r}{r_0}\right)^2}}$, which makes
$h_{\mu \nu}= \begin{bmatrix} h_{00} & 0 & 0 & 0 \\ 0 & h_{00} & 0 & 0 \\ 0 & 0 & h_{00} & 0 \\ 0 & 0 & 0 & h_{00} \end{bmatrix}$.
The idea is to use Lagrangian function and solve the differential equation derived from it to get the orbit of photons. Lagrangian function is
$L=\frac{1}{2}m\left[h_{\mu \nu} + \eta_{\mu \nu}\right]\dot{x}^{\mu}\dot{x}^{\nu}$,
so we get
$L = \frac{1}{2}m \left[ -(1-h_{00})\dot{t}^2 + (1+h_{00})(\dot{x}^2 + \dot{y}^2 +\dot{z}^2)\right]$.
Since we're working with photons, we can rewrite that as
$L' = \frac{L}{2m} = \left[ -(1-h_{00})\dot{t}^2 + (1+h_{00})(\dot{x}^2 + \dot{y}^2 +\dot{z}^2)\right]$. This is supposedly right, but I am having trouble converting it to spherical coordinates. I've used
$x = r\sin{\theta}\cos{\phi}$, $y = r\sin{\theta}\sin{\phi}$, $z = r\cos{\theta}$,
but I was told the result
$L' = \left[ -(1-h_{00})\dot{t}^2 + (1+h_{00})\left(\dot{r}^2 + r^2\left( \dot{\theta}^2 +\sin^2{\theta}\dot{\phi}^2 \right)\right)\right]$
is not correct.
What would be correct version of Lagrangian function in spherical coordinates? Could it be treated as 2D motion and reduced to polar coordinates? Is there maybe a better way to approach this problem than through Lagrangian? How would one calculate deflection angle for this case? By the way, I did look for help in literature. I think I understand basic derivation of gravitational lensing for point mass. I have a problem with this particular example of potential and answers in form of general definitions don't help me.
