I'm trying to re-derive the results of Bondi-Sachs formalism. The metric is given in the form
\begin{array}{c}g_{a b} d x^{a} d x^{b}=-\frac{V}{r} e^{2 \beta} d u^{2}-2 e^{2 \beta} d u d r+r^{2} h_{A B}\left(d x^{A}-U^{A} d u\right)\left(d x^{B}-U^{B} d u\right) \\ g_{A B}=r^{2} h_{A B} \quad \text { with } \quad \operatorname{det}\left[h_{A B}\right]=\mathfrak{q}\left(x^{A}\right),\end{array} where where $\mathfrak{q}\left(x^{A}\right)$ is the determinant of the unit sphere metric $q_{A B}$ associated with the angular coordinates $x^{A}$, e.g. $q_{A B}=\operatorname{diag}\left(1, \sin ^{2} \theta\right)$ for standard spherical coordinates $x^{A}=(\theta, \phi)$
The Bondi-Sachs coordinates $x^{a}=\left(u, r, x^{A}\right)$ are based on a family of outgoing null hypersurfaces $u=$ const (you can see a better description here Bondi-Sachs Formalism )
My problem is: I don't know where to start? I'm using this bject-oriented general relativity package since it's more intuitive. But I don't know how to incorporate yet another summed-part, namely $g_{AB}$, into my definition? Besides, functions $U^A$, $\beta$ , and $V$ are general functions of the coordinates which I want to derive the form from EFEquations. But how should I represent them to the machine so that it can evaluate Christoffle symbols and curveture tensors?
Forgive me for my lack of experience
EDIT: I haven't started to code it because I don't know where to start. We have the explicit matrix form of $h_{AB}$ which is messy and not needed to be represented explicitly. My $ds^2$ becomes unnessecarily messy too if I want to insert every single term. I'm looking for a way to avoid that.