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## Main Question or Discussion Point

I'm trying to understand the BMS formalism in General Relativity and I'm in doubt with the so-called Bondi Coordinates.

In the paper

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

So he is saying that

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

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

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

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?