Construct BMS Coordinates near Null Infinity

[leo.]

Let us consider Ashtekar's definition of asymptotic flatness at null infinity:

A space-time $(\hat{M},\hat{g})$ is said to be asymptotically flat at null infinity if there exists a manifold $M$ with boundary $\mathcal{I}$ equipped with a metric $g$, and a diffeomorphism from $\hat{M}$ onto the interior $M\setminus \mathcal{I}$ (with which we identify $\hat{M}$ and $M\setminus \mathcal{I}$) such that:

• there exists a smooth function $\Omega$ on $M$ with $g = \Omega^2\hat{g}$ on $\hat{M}$; $\Omega =0$ on $\mathcal{I}$ and $n_a = \nabla_a \Omega$ is nowhere vanishing on $\mathcal{I}$;
• $\mathcal{I}$ is topologically $S^2\times \mathbb{R}$;
• $\hat{g}$ satisfies Einstein's equations $$\hat{R}_{ab}-\frac{1}{2}\hat{R}\hat{g}_{ab}=8\pi G \hat{T}_{ab},$$ where $\Omega^{-2}\hat{T}_{ab}$ has a smooth limit to $\mathcal{I}$;
• The integral curves of $n^a$ are complete on $\mathcal{I}$ for any choice of the conformal factor which makes $\mathcal{I}$ divergence-free (i.e., $\nabla_a n^a = 0$ on $\mathcal{I}$)

I want to see how to construct the so-called Bondi coordinates $(u,r,x^A)$ in a neighborhood of $\mathcal{I}^+$ out of this definition.

In fact, a distinct approach to asymptotic flatness already starts with such coordinates and proceed to impose falloff conditions on the metric coefficients to define what asymptotic flatness should be. This approach is used in particular in Chapter 5 of "Lectures on the Infrared Structure of Gravity and Gauge Theories" by Andrew Strominger. My objective is to obtain this approach out of Ashtekar's abstract approach via a construction of coordinates.

So assume $(\hat{M},\hat{g})$ asymptotically flat as in Ashtekar's definition, so that we have $M$ and $\mathcal{I}^+$ with a conformal factor $\Omega$. By definition $\mathcal{I}^+$ has topology $\mathbb{R}\times S^2$. In that case we can use coordinates $(u,x^A)$ on $\mathcal{I}^+$ where $u$ is the affine parameter along its generators and $x^A$ are coordinates on $S^2$. These are coordinates on $\mathcal{I}^+$.

Now, from this exact point, Wald, in his GR book, extends these coordinates to a neighborhood of $\mathcal{I}^+$ by holding them fixed along the null geodesics going out of $\mathcal{I}^+$ on the unphysical spacetime. He finishes the coordinate system by using $\Omega$ itself as a coordinate. So Wald ends up with coordinates $(u,\Omega,x^A)$. With this, Wald is able to find the falloff conditions on the metric components and if we identify $O(\Omega^n)$ with $\Omega(1/r^n)$, it agrees with Strominger's.

But what about the $r$ coordinate that Strominger and many others use? How can we construct it in this approach? My first intuition was to take it as the affine parameter on these null geodesics out of $\mathcal{I}^+$. But then I don't really know how to show it to be $O(1/\Omega)$. My second idea was to take it to be $1/\Omega$, but I believe this is very wrong.

 

[Ambitwistor]

Thank you for your interesting post. I would like to offer some insights on how to construct the Bondi coordinates in a neighborhood of $\mathcal{I}^+$ using Ashtekar's definition of asymptotic flatness.

Firstly, let us recall that the Bondi coordinates are defined as follows: $$\hat{g}_{ab}dx^adx^b = -e^{2\beta}(du^2-2dudr)+r^2h_{AB}(dx^A-U^Adx^B)^2,$$ where $u$ is the affine parameter along the outgoing null geodesics, $r$ is the parameter along the transverse null geodesics, $x^A$ are coordinates on the unit sphere $S^2$ and $h_{AB}$ is the metric on $S^2$. The functions $\beta$ and $U^A$ are known as the Bondi news and the Bondi velocity respectively, and they determine the asymptotic behavior of the metric components.

Now, let us consider the diffeomorphism from $\hat{M}$ onto the interior $M\setminus \mathcal{I}$, which we identify with $\hat{M}$ and $M\setminus \mathcal{I}$. This diffeomorphism induces a coordinate transformation on the metric components, given by $$g_{ab} = \frac{\partial x^c}{\partial \hat{x}^a}\frac{\partial x^d}{\partial \hat{x}^b}\hat{g}_{cd}.$$ Using this coordinate transformation, we can express the metric components in terms of the Bondi coordinates as follows: $$g_{ab}dx^adx^b = -e^{2\beta}(du^2-2dudr)+r^2h_{AB}(dx^A-U^Adx^B)^2.$$ Therefore, the Bondi coordinates are natural coordinates to use in the neighborhood of $\mathcal{I}^+$.

Next, let us consider the function $\Omega$, which is defined as the conformal factor between the metrics $\hat{g}$ and $g$. From the definition of the Bondi coordinates, we can see that $\Omega = e^{-\beta}$. This means that the function