I'm trying to find the right way to calculate the point of intersection of the null hypersurface emitted by a point $S$ event in a spacetime manifold with metric $g_{\mu \nu}$, with another time-like geodesic $t^{\mu}(t)$. Intuitively speaking, one can imagine $S$ is a star emitting some burst of radiation, and $t^{\mu}$ is the telescope capturing the signal.
Valid simplifying assumptions is that we are in the linearized gravity regime, so $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$, with $\eta$ being the Minkowski metric, and $h$ a small deformation
This is what I've tried so far: I assume that I have solved the problem of finding the intersection point for flat spacetime (geodesics are simple lines, so finding intersections is a simple algebraic problem), and I want to find a perturbation on the geodesic path as the deformation $h$ becomes stronger
$$ \frac{d^2 x^{i}}{d \lambda^2} = - \Gamma^i_{pq}(\eta_{\mu \nu}) \frac{d x^p}{d \lambda} \frac{d x^q}{d \lambda} = 0 $$
This is the definition of $\frac{d x^q}{d \lambda}$, and it is assumed to be a line that intersects $S$ and the geodesic worldline $t^{\mu}(t)$ on flat spacetime
Now I propose a variation of the geodesic in a formal power series of the small deformation parameter $\epsilon$:
$$ \delta x^{\mu} = \sum_{i=1}^{\infty} \delta x^{\mu}_{(i)} \epsilon^i $$
and I consider the effect of the deformation on the proper distance of the path:
$$ ds^2 = (\eta_{\mu \nu} + \epsilon h_{\mu \nu} ) \frac{d(x^{\mu} + \delta x^{\mu})}{d\lambda} \frac{d(x^{\nu} + \delta x^{\nu})}{d\lambda} d\lambda^2 = 0 $$
Ordering terms by powers of $\epsilon$, the expression looks like
$$ ds^2 = \left\{ \eta_{\mu \nu} \frac{dx^{\mu}}{d\lambda} \frac{dx^{\nu}}{d\lambda} + \epsilon \Big [ 2 \eta_{\mu \nu} \frac{dx^{\mu}}{d\lambda} \frac{d \delta x^{\nu}_{(1)}}{d\lambda} + h_{\mu \nu} \frac{dx^{\mu}}{d\lambda} \frac{dx^{\nu}}{d\lambda} \Big ] + O(\epsilon^2) \right\} d\lambda^2 = 0 $$
The zero-th order term vanishes identically, and the first order term creates the constraint
$$ \frac{dx^{\mu}}{d\lambda} \left\{ 2 \eta_{\mu \nu} \frac{d \delta x^{\nu}_{(1)}}{d\lambda} + h_{\mu \nu} \frac{dx^{\nu}}{d\lambda} \right\} = 0 $$
As I understand it, this is the condition that $d \delta x^{\nu}_{(1)}/d\lambda$ must satisfy in order for the perturbed path to belong to the (deformed) null hypersurface
I now want to go back to the geodesic equation and use this information in some clever way in order to obtain the new point of intersection between the first-order correction of $t^{\mu}$ geodesic and the deformed null hypersurface
The equation for the deformed geodesic looks like
$$ \frac{d^2 (x^{i} + \delta x^{i})}{d \lambda^2} = - \Gamma^i_{pq}(\eta_{\mu \nu} + \epsilon h_{\mu \nu}) \frac{d (x^p + \delta x^{p})}{d \lambda} \frac{d (x^q + \delta x^{q})}{d \lambda} $$
On the linearized regime, $\Gamma$ behaves as a linear operator on the metrics, and it leaves us with an equation for the first-order correction to the geodesic. Since $\Gamma^i_{pq}(\eta_{\mu \nu}) = 0$ (flat spacetime) the remaining terms look like
$$ \frac{d^2 \delta x^{i}}{d \lambda^2} = - \Gamma^i_{pq}( \epsilon h_{\mu \nu}) \Big [ \frac{d x^p}{d \lambda} \frac{d x^q}{d \lambda} + 2 \frac{d x^p}{d \lambda} \frac{d \delta x^q}{d \lambda} + \frac{d \delta x^p}{d \lambda} \frac{d \delta x^q}{d \lambda} \Big ] $$
If we repeat the trick of writing $\delta x^q$ in powers of $\epsilon$ inside the geodesic equation, and proceed to group terms of the same perturbative order, we obtain a first order correction equation given by
$$\frac{d^2 \delta x^{i}_{(1)}}{d \lambda^2} = - \Gamma^i_{pq}(h_{\mu \nu}) \frac{d x^p}{d \lambda} \frac{d x^q }{d \lambda}$$
Which would invite the following integral:
$$ \frac{d \delta x^{i}_{(1)}}{d \lambda} = - \int_{S}^{\lambda = t_1'} \Gamma^i_{pq}(h_{\mu \nu}) \frac{d x^p}{d \lambda} \frac{d x^q }{d \lambda} d \lambda $$
The problem I have right now is that I don't have any reason to believe that this correction will intersect at all with the deformed path corrections to the $t^{\mu}$, since I still haven't used these anywhere in the calculation!
I am not sure how to formulate the problem from here in order to bridge the remaining gap (making sure the null deformed geodesic still intersects the $t^{\mu}$ time-like geodesic)