Constructing Bondi Coordinates on General Spacetimes

[leo.]

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 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.
So far we have not imposed any sort of assymptotic flatness condition. Any geometry can be described locally by the metric (5.1.2).
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So he is saying that 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?

 

[martinbn]

This is not something I am knowledgeable enough to give you an answer, so take it with the appropriate amount of salt. My impression is, and it may be far from true, that the work in the 60's, although mathematically rigorous, didn't not prove the existence of such space-times. Even the notion of asymptotically flat space-time wasn't fully understood. So you can take any of those papers as studying space-times with such properties, and you can take it for granted that they exists. I think, and again this may be completely wrong, that to see things done fully you need to look at the stability of Minkowski space-time work of Christodoulou and Klainerman. I believe that is where for the first time it was shown that such space-times exist and a fully rigorous construction of some of the notions was given, such as future null infinity, the optical function, the null foliation..

(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?

(1) These are the expansion and shear. The conditions simply exclude some unwanted cases, intuitively cylinder like and cone like cases. (2) I think the idea is that the intersections of $u=const$ and $r=const$ is a topological two-sphere, so a choice of coordinates can be called angles. (3) That is guaranteeing that they form a coordinate chart i.e. this is the condition that the Jacobian is not zero.

 

[leo.]

@martinbn I thought the same when I've read section 2 of Sachs' paper the first time. But notice that Strominger points out that any geometry can be locally written in these coordinates with that metric tensor. I actually have the impression that it is true. My problem is that if any geometry admits such local description, then there must be a construction in an arbitrary geometry which yields these coordinates.

Take the Riemann normal coordinates for example. It is locally available on any geometry. And there is such a cosntruction: take one point $z\in M$, build one orthonormal tetrad $e_a$ on the point. There is a neighborhood of the origin on which the exponential map is a diffeomorphism, let $N$ be the image, so that $\exp_z^{-1}$ is well defined on $N$. Let $q\in N$ then $\exp_z^{-1}(q)$ can be expanded on $e_a$. Define $\exp_z^{-1}(q) = x^{a}(q)e_a$. The map $x(q) = (x^a(q))$ is a coordinate chart because $\exp_z$ is a diffeomorphism on the region of interest.

So you see: there is mathematical proof that we can construct the system, and the procedure which yields the coordinate functions.

Another example would be angular coordinates on a spherically symmetric spacetime like Schwarzschild spacetime. There is a derivation that shows how the existence of the angular coordinates follows from spherical symmetry.

I'm looking for what yields these Bondi coordinates on any spacetime. A construction which shows what is the domain of validity of the coordinates and the respective ranges. Unfortunately, up to this point I didn't find this.

 

[martinbn]

You might be right that the coordinates exist locally.

 

[martinbn]

Intuitively locally it should be possible. In Mikowski space-time the construction is clear. Any space-time is locally very close to Minkowski so it should be possible. Or you could do something similar to radar coordinates. Use light pulses to define your $u$ coordinate and the spheres, which in turn will define the angle coordinates and the areal radius.