# 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

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