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In the paper

*Lectures on the Infrared Structure of Gravity and Gauge Theories*Andrew Strominger points out in section 5.1 the following:

*In the previous sections, flat Minkowski space in retarded coordinates near ##\mathscr{I}^+## was described by the metric $$ds^2=-du^2-2dudr+2r^2\gamma_{z\bar{z}}dzd\bar{z}. \tag{5.1.1}$$ We would now like to study gravitational theories in which the metric is asymptotic to, but not exactly equal to, the flat metric. We will work in Bondi coordinates ##(u,r,z,\bar{z})##, and we abbreviate ##\Theta^A = (z,\bar{z})##. In this gauge, the most general four-dimensional metric takes the form $$ds^2=-Udu^2-2e^{2\beta}dudr+g_{AB}\left(d\Theta^A + \frac{1}{2}U^A du\right)\left(d\Theta^B + \frac{1}{2}U^B du\right) \tag{5.1.2}$$*

where $$\partial_r \det \left(\frac{g_{AB}}{r^2}\right)=0 \tag{5.1.3}$$Equation (5.1.3) implies that ##r## is the luminosity distance.

where $$\partial_r \det \left(\frac{g_{AB}}{r^2}\right)=0 \tag{5.1.3}$$

So far we have not imposed any sort of assymptotic flatness condition.

**Any geometry can be described locally by the metric (5.1.2).**

*any metric tensor*can be written like this, so in particular, coordinates like these can be defined on

*any spacetime*.

So why is this true? Why can such coordinates be defined on any spacetime? What is the construction that gives rise to the coordinates ##(u,r,\Theta^A)## on any spacetime?

I've tried reading the original paper by Sachs from 1962 where he apparently introduces this, but I found quite confusing how this works out.

He picks any ##u \in C^\infty(M)## such that the level sets are null hypersurfaces with normals ##k_a = \nabla_a u##. Next, he imposes two conditions on such ##u##: first ##\rho = \frac{1}{2}\nabla_a k^a \neq 0## and second ##|\sigma|^2 = \frac{1}{2} \nabla_b k_a \nabla^b k^a -\rho^2 \neq \rho^2##.

Next he says we should pick

*any two functions ##\theta,\phi##*with the property that they are constant along the generators of the hypersurface, namely ##k^a \nabla_a \theta = k^a \nabla_a \phi = 0##. He says that ##\rho\neq 0## implies that $$D = \nabla_a \phi \nabla^a \phi \nabla_b \theta \nabla^b \theta - (\nabla^a \phi \nabla_a \theta)^2 \neq 0.$$ Finally he claims such ##\theta,\phi## to be angles and defines ##r^4 = (D\sin^2\theta)^{-1}## and says ##r## is a luminosity distance.

This seems to be the construction Strominger has in mind. Still I can't understand the construction: (1) I don't get the two weird conditions imposed on ##u## in terms of ##\rho,|\sigma|^2##, (2) how could

*two arbitrary functions constant along the generators of the surace*be angles? (3) What is the matter with that ##D## function (which is clearly a determinant?

Finally, I should say that nowhere in the construction the domain of validity of the chart on spacetime was pointed out, nor the ranges of the so-defined coordinates. Nor a proof that ##(u,r,\theta,\phi)## indeed provides a chart (a homeomorphism from spacetime to an open set of ##\mathbb{R}^4##).

So how to

*truly understand*the construction of Bondi coordinates on a general spacetime? How to understand the intuition behind the steps, and how actually rigorously construct such coordinate system?