differential geometry – Einstein field equations for Bondi-Sachs formalism

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 uright)left(d x^{B}-U^{B} d uright) \ 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. $q_{A B}=operatorname{diag}left(1, sin ^{2} thetaright)$ 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