Necessary steps to calculate photon's path by using null geodesic equation Can anyone please give me an explanation on how to calculate photon's path by using the null geodesic equation? N.B. I know all of the non-zero values of Christoffel symbols.
The geodesic equation below is a second-order partial differential equation for the spacetime coordinates $x^\mu$, $\mu = 0 \ldots n$, $${d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0 \;,$$
where $\Gamma ^{\mu }{}_{\alpha \beta }$ are the Christoffel symbols, and the proper time $s$ is the parameter of the curve (i.e. how ``far along'' you are on the path). You need to solve these $n+1$ coupled equations as any such system, e.g. numerically.

If the parameter $s$ is confusing, you can reparametrise the curve as a function of coordinate time $t = x^0$, as explained e.g. on Wikipedia's page on geodesics in general relativity, $${d^{2}x^{i} \over dt^{2}}=-\Gamma ^{i}{}_{{\alpha \beta }}{dx^{\alpha } \over dt}{dx^{\beta } \over dt} + \Gamma ^{0}{}_{{\alpha \beta }}{dx^{\alpha } \over dt}{dx^{\beta } \over dt}{dx^{i} \over dt} \;,$$
which gives you $n$ partial differential equations for the spatial coordinates $i = 1 \ldots n$. (Note that there are remaining greek indices, for which $dx^0/dt = dt/dt = 1$.)