Matching between $i^+$ and ${\cal I}^+$

Question 1

I have a confusion about the paper"Asymptotic symmetries of QED and Weinberg’s soft photon theorem" by Campiglia & Laddha. They use the hyperbolic slicing of the region inside the ligthcone of the origin to resolve$i^+$. We have$t \geq r$ and we define$$\tau=\sqrt{t^2-r^2},\quad \rho=\dfrac{r}{\sqrt{t^2-r^2}}\tag{1}.$$

Now let$\mathbb{H}_3^\tau$ be the constant$\tau$ surfaces. One wants to identify$i^+$ with the$\tau \to \infty$ limit which we denote$\mathbb{H}_3^+$. Now I get the impression that$\partial \mathbb{H}_3^+$ is often matched with${\cal I}^+_+$ but I can't see where this comes from.

For finite$\tau$, the authors claim$\mathbb{H}_3^\tau$ intersect${\cal I}^+$ at$u = t-r = 0$. In that case to have a Cauchy surface one matches$\partial \mathbb{H}_3^\tau$ with the sphere at$u=0$ on${\cal I}^+$ and adjoins the$u < 0$ portion of${\cal I}^+$.

So for all$\tau$ the boundary$\partial\mathbb{H}_3^\tau$ matches with the sphere at$u=0$ of${\cal I}^+$. Then suddenly when$\tau \to \infty$ the boundary$\partial \mathbb{H}_3^+$ matches with${\cal I}^+_+$.

Indeed I'm able to argue that a point in$\mathbb{H}_3^\tau$ with coordinates$(\tau,\rho,z,\bar{z})$ if we take$\rho\to \infty$ and translate to retarded coordinates goes to a point on$\mathscr{I}^+$ with$u =0$ and the same$(z,\bar{z})$. But for infinite$\tau$ I really don't understand this matching.

Why does$\partial \mathbb{H}_3^+$ is matched with${\cal I}^+_+$ when for all finite$\tau$ the boundary$\partial \mathbb{H}_3^\tau$ is matched with the cut$u=0$ of${\cal I}^+$?

Question 2

I was thinking of something else. I will delete my comments (and this one in a few minutes). See my answer.

Question 3

As I recall, the idea is to take the hyperboloid${\mathbb H}_\tau$ to be the Cauchy slice for massive particles and the portion of${\cal I}^+$ described by$u<0$ as the Cauchy slice for massless particles. This can always be done by translating the origin of the lightcone sufficiently far into the future (essentially sending$u \to u + u_0$) so that all massless particles exit the spacetime at$u< 0$.

In this case, since there is no flux through the$u>0$ portion of${\cal I}^+$, the fields are frozen (i.e. the field strengths vanish) in the region$ u> 0$. It then follows that for any field$$\phi|_{\partial {\mathbb H}_\tau} = \phi|_{u=0} = \phi|_{{\cal I}^+_+}.$$They use this property to match the fields accordingly. Note that we are NOT asserting that$\partial {\mathbb H}_\tau = {\cal I}^+_+$, only that the field values on these boundaries are equal.

ASIDE - The same idea is utilized when spatial infinity is blown up into de Sitter slices. The boundary of the de Sitter slices is at$u=0$ and in that case one assumes that all massless particles exit the system at$u>0$. Fields are therefore frozen in the region$u<0$ so$\phi|_{u=0} = \phi|_{{\cal I}^+_-}$.

Question 4

That's quite neat and clears a lot of confusion. I always had the (wrong) impression that people were assuming that $\partial \mathbb{H}_\tau = {\cal I}^+_+$ as $\tau \to \infty$, but honestly it always felt wrong. But using this idea of sending $u\to u + u_0$ so that all massless particles exit at $u < 0$ and then only matching the values of the fields as you did makes total sense. Thanks a lot !

Prahar 29.8k2 gold badges67 silver badges138 bronze badges · Accepted Answer · 2021-02-22 03:07:17Z

As I recall, the idea is to take the hyperboloid${\mathbb H}_\tau$ to be the Cauchy slice for massive particles and the portion of${\cal I}^+$ described by$u<0$ as the Cauchy slice for massless particles. This can always be done by translating the origin of the lightcone sufficiently far into the future (essentially sending$u \to u + u_0$) so that all massless particles exit the spacetime at$u< 0$.

In this case, since there is no flux through the$u>0$ portion of${\cal I}^+$, the fields are frozen (i.e. the field strengths vanish) in the region$ u> 0$. It then follows that for any field$$\phi|_{\partial {\mathbb H}_\tau} = \phi|_{u=0} = \phi|_{{\cal I}^+_+}.$$They use this property to match the fields accordingly. Note that we are NOT asserting that$\partial {\mathbb H}_\tau = {\cal I}^+_+$, only that the field values on these boundaries are equal.

ASIDE - The same idea is utilized when spatial infinity is blown up into de Sitter slices. The boundary of the de Sitter slices is at$u=0$ and in that case one assumes that all massless particles exit the system at$u>0$. Fields are therefore frozen in the region$u<0$ so$\phi|_{u=0} = \phi|_{{\cal I}^+_-}$.

That's quite neat and clears a lot of confusion. I always had the (wrong) impression that people were assuming that $\partial \mathbb{H}_\tau = {\cal I}^+_+$ as $\tau \to \infty$, but honestly it always felt wrong. But using this idea of sending $u\to u + u_0$ so that all massless particles exit at $u < 0$ and then only matching the values of the fields as you did makes total sense. Thanks a lot !

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