Conformal Compactification of Asymptotically Flat Spacetimes

We first define a flat metric on Minkowski space, resembling the standard Minkowski metric, given in the spherical coordinates as:

\(d s^2=-d t^2+d r^2+2 r^2 \gamma_{z \bar{z}} d z d \bar{z}\)

Here, \(\gamma_{z \bar{z}}=\frac{2}{(1+z \bar{z})^2}\) is defined with polar coordinates \(z=e^{i \phi} \tan \frac{\theta}{2}\) and the spherical coordinates are projective coordinates for the \(S^{2}\) factor named the celestrial sphere \(CS^{2}\).

Next, let \(u, v \in(-\infty, \infty)\), where the finite range \(U,V \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) covers a given manifold. Next, we define new coordinates \((T,R)\), using the double null coordinates \((U,V)\) such that \(T=U+V\) and \(R=V-U\). The Minkowski metric \(ds^{2}\) relative to the coordinates \((T,R)\) is a conformal factor times the Lorentz metric on \(S^{3}\times\mathbb{R}\) and can be written as:

\(\begin{aligned} ds^{2} = & \left(\frac{L^2}{4 \cos ^2\left(\frac{R-T}{2}\right) \cos ^2\left(\frac{R+T}{2}\right)}\right) \\ & -d T^2+d R^2+2 \sin ^2 R \gamma_{z \bar{z}} d z d \bar{z} \\ = & \;\Omega^{-2} d \widetilde{s}^2 \end{aligned}\)

Here, our conformal factor is denoted as \(\Omega^{-2}\) and, as we can see from above, is defined as:

\(\Omega^{-2}=\frac{L^2}{4 \cos ^2\left(\frac{R-T}{2}\right) \cos ^2\left(\frac{R+T}{2}\right)}\)

An important property of this conformal factor is that it is positive, thus, preserving the casual structure for the rescaled metric \(d \widetilde{s}^2\). This is important, because preserving the casual structures in this compactification allows for curves that begin as timelike, null, or spacelike, to maintain as such with respect to the rescaled metric.

Next, we will attach a boundary to this compactification so that we can better understand the behavior of inifinity as it relates to Minkowski space and asymptotically flat spacetimes. These boundaries are given by the author, and are as folllows:

Massive particles following time-like trajectories will enter at past timelike infinity and will exit at future timelike infinity. Where past timelike infinity is denoted as \(i^{-}\) and paramterized by \((R,T)=(0,-\pi)\) and future timelike infinity is denoted as \(i^{+}\) and parameterized by \((R,T)=(0,\pi)\).

Massless particles enter along past null-infinity and exit along future null-infinity. Where past null-inifinity is denoted as \(\mathcal{I}^{-}\) and paramterized by \(U=-\frac{\pi}{2}, V \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) and future null-infinity is denoted as \(\mathcal{I}^{+}\) and paramterized by \(V=\frac{\pi}{2}, U \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).

Moving along any spacelike infinity, results in spacelike infinity. Where spacelike infinity is denoted as \(i^{0}\) and paramterized by \((R,T)=(\pi,0)\).

Image Source: https://tikz.net/relativity_penrose_diagram/

To continue.

Geometry of Conformal Inversions

Section Reference: Equating Extrapolate Dictionaries for Massless Scattering by Eivind Jørstad, Sabrina Pasterski and Atul Sharma from Perimeter Institute for Theoretical Physics, Dept. of Physics & Astronomy, University of Waterloo, Center for the Fundamental Laws of Nature & Black Hole Initiative, Harvard University \(^{[1]}\)

Four-dimensional Lorentzian space \(\mathbb{R}^{4}\) is know as Minkowski space and is denoted as \(\mathbb{M}=\mathbb{R}^{1,3}\), with the metric signature \((1,3)\) and the Cartesian coordinates \(X^\mu, \mu=0,1,2,3\). For this Lorentzian Space to be Minkowski Space, it must possess a type of metric tensor taken to be \(\eta_{\mu \nu}=\operatorname{diag}(-1,1,1,1)\), aptly named the Minkowski metric or Minkowski tensor. The light cone of the origin to null infinity is mapped using the conformal inversion map \(\mathbb{M}\). The reason being is the authors mention embeded space formalism is eaiser to study conformal transormations with conformal compactification of \(\mathbb{M}\).

Glossary

AdS/CFT Correspondence

Anti-de Sitter/conformal theory correspondence AdS/CFT in theoretical physics is a conjecture that describes the relationship between two kinds of physical theories. AdS used in quantum gravtiy and is formulated in terms of string theory of M-theory, while, CFT are quantum field theories that include theories such as Yang-Mills theories describing elementatry particles.