Suppose you have a spacetime $(M,g)$, i.e. a $4$-dimensional Lorentzian ($-+++$) manifold. Let $S$ be a spacelike hypersurface. If $p\in M$, in a neighborhood of $p$ you can find a smooth timelike vector $V$ field orthogonal to $S$. If you restrict to a small enough neighborhood of $p$, you obtain coordinates $(t,x_1,x_2,x_3)$ for a point $q$ as follows: through $q$ there exists a unique timelike geodesic such that the tangent vector at the point of intersection between this geodesic and $S$ (call it $r$) is given by $V$. Then $t$ is given in the difference between the affine parameter along that geodesic between $q$ and $r$, and $(x_1,x_2,x_3)$ are the coordinates of $q$ on $S$.
My question is why is this a well-defined coordinate system?
They resemble Riemannian normal coordinates, where you use the inverse mapping theorem to make the exponential map a diffeomorphism (and you choose an orthonormal basis of vectors), but I don't really know how to prove that the coordinates defined above (which in Wald are called Gaussian normal coordinates) are indeed well-defined.
The argument is basically the same strategy as used for normal coordinates: use ODE theorems to show that the map is smooth, and then use the differential at the poit of interest to show that it is invertible.
We can call the coordinate map map $\Phi:S\times \mathbb{R}\to M$ satisfying $\Phi(s,t)=\gamma_s(t)$, where $\gamma_s$ is a geodesic satisfying $\gamma_s(0)=s$ and $\dot{\gamma}_s(0)=V(s)$.
Choose $p\in S$. Note that $\Phi(s,t)$ is the solution of a smooth second order ODE with initial conditions given by $s,V(s)$. By existence, uniqueness, smoothness, and smooth dependence on initial conditions of solutions of smooth ODEs, $\Phi$ is well defined and smooth for sufficiently small neighborhood $(p,0)\in S\times\mathbb{R}$. It only remains to show that it is a diffeomorphism onto its image. This is accomplished by noting that the image of $d_{(p,0)}\Phi$ is $T_pS+\text{span}(V(p))=T_pM$ which is surjective, and also injective by dimension counting. Aplying inverse function theroem gives a neighborhood of $(p,0)$ where $\Phi$ is a diffeomorphism onto its image as well as smooth.
Composing this map with a coordinate chart around $p$ in $S$ gives the desired chart.