Einstein field equations for Bondi-Sachs formalism

Question

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.

Answer 1

I'm the author of OGRe, the Mathematica package you mentioned.

You can certainly calculate the Christoffel symbols and curvature tensors of the Bondi-Sachs metric in OGRe. Simply define the coordinate system $(u, r, \theta, \phi)$ using TNewCoordinates and the metric using TNewMetric. The components should, of course, be entered as abstract Mathematica functions of the coordinates, i.e. functions of the form f[u, r, \[Theta], \[Phi]].

It is not possible (at the moment) to use abstract tensors in OGRe, they must have concrete components. Therefore, you must take the conformal 2-metric $h_{AB}$ to have specific components, such as those in Eq. (6) of the ScholarPedia article.

Once you defined the coordinates and the metric, you can use the TCalc* modules to calculate any curvature tensors you want.

As for $g_{AB}$, you will have to define it as a separate metric using a separate set of 2D coordinates. Unfortunately OGRe does not currently support defining tensors on hypersurfaces, although I plan to add that functionality in a future update.

Please let me know if you have any further questions!