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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.

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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 $dx^0/dt = dt/dt = 1$.)

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