Suppose, this is the line element of a spherically symmetric FLRW metric, $$ ds^2 = -[1 + 2ψ(t,x)]dt^2 + a^2(t) [1 - 2ψ(t,x)]dx^2 $$ and the geodesic equation is, $$ \frac{d^2x^α}{dλ^2} = - Γ_{βγ}^α \frac{dx^β}{dλ} \frac{dx^γ}{dλ} $$ where λ is the affine parameter. Again, for null geodesic we can write
$$ ds^2 = 0 $$ Now, if I want to solve this null geodesic equation I know that I need to convert the four 2nd order differential equation into eight first order equation first and then easily anyone can solve this by using any mathematical tools. Actually, my concern is about the initial conditions, how can I set initial conditions to solve this null geodesic equation, i.e. what will be the initial conditions here for x, y, z and for the four velocities, \$$ \frac{dt}{dλ}, \frac{dx}{dλ}, \frac{dy}{dλ}, \frac{dz}{dλ}$ respectively, suppose, the initial condition for time is today that means I want to integrate the geodesic equation from today i.e, the redshift is zero.
Actually I have asked the same question here, please see the link
but did not get the answer, that's why I'm asking here again.
First of all, notice that a geodetic preserve the metric of the tangent vector, since:
$\nabla\left\langle \dot{\gamma}(t),\dot{\gamma}(t)\right\rangle=2 \left\langle\nabla\left(\dot{\gamma}(t)\right),\gamma(t)\right\rangle=0$
Thus, if you want a null geodesic, it is enough for the initial vector to be null, and the condition (that is, for the geodesic to be light-like) will follow. we are imposing one condition, which is going to give a relation between x and t (notice that, since you are talking about squared differentials, the solutions is not going to be unique, but you will have to decide their signs).
The geodesic then follows from the point you choose in the spacetime from which to start, in this case from today, on a certain point.