The following setup is from Rendall's 1990 paper regarding the characteristic initial value problem for Einstein's equations, and also from material on null geometry. Given a 2-dimensional spacelike slice $S \subset M$ ($(M, g)$ a Lorentzian 4-manifold) we have two future-pointing null vector fields $L$ and $\underline{L}$. Given coordinates $x^3, x^4$ (locally) on $S$, we consider the null geodesics generated by $L$ and $\underline{L}$, and let $x^1, x^2$ (respectively) be the affine parameters along these null geodesics so that $x^1 = x^2 = 0$ on $S$. Furthermore, letting $N_1$ be the null hypersurface generated by $L$, and set $x^2 = 0$ on $N_1$. Do the analogous thing to define $N_2$ and let $x^1 = 0$ on $N_2$. Define $x^3, x^4$ along these hypersurfaces by requiring them to be constant along the generating null geodesics.
In his paper, Rendall completes these to a harmonic coordinate system in some neighborhood of spacetime; I don't know whether or not that is needed to answer my question, which is what we can say about the values of certain metric components along the hypersurfaces $N_1$ and $N_2$. Let's just talk about $N_1$, since everything is the same/flipped for $N_2$. I'll bold my specific questions below.
First, on $S$, we know that $g_{11} = g_{22} = 0$ since these are null directions on $S$. On $N_1$, since it is generated by null geodesics with parameter $x^1$, we still have $g_{11} = 0$, but do we still have $g_{22} = 0$? (Answer: not in general). Does it depend on how we complete to a global coordinate system? On $S$, we also have $g_{2A} = g_{1A} = 0$ for $A = 3, 4$. On $N_1$ we have that $g_{1A} = 0$ for $A = 3, 4$. I believe a linear argument shows then that $g^{2\alpha} = 0$ on $N_1$ for $\alpha \neq 1$. Do any other components have a nice description? What about $g^{11}$; is $g^{11}$ on $N_1$?
Other questions: In general, can we get any nice identities for the metric or its inverse along $N_1$?
More context/motivation/questions: The reason I'm asking is that I'm going through Rendall's paper and trying to justify some of the computations he doesn't go through. In particular, he claims that in the coordinates constructed above (completed to a harmonic coordinate system), where we have $\Gamma^2 = g^{\alpha\beta}\Gamma^2_{\alpha\beta} = 0$, this condition is equivalent to $g_{12,1} = \frac{1}{4}g^{\alpha\beta}g_{\alpha\beta,1}g_{12}$. The only way I see that this is true is by assuming certain components of the inverse metric involved in computing $\Gamma^2$ are zero, and so this is why I'm asking this in the first place.
Any help is much appreciated!