From the Lorentz Group to the Celestial Sphere

TL;DR

The notes establish a deep link between four-dimensional Lorentz symmetry and two-dimensional conformal symmetry on celestial spheres: specifically, the connected Lorentz group $L_+^{\uparrow}$ is isomorphic to $\mathrm{SL}(2,\mathbb{C})/\mathbb{Z}_2$, and the action of Lorentz transformations on the celestial sphere at null infinity reduces to Möbius (conformal) transformations $z'=(az+b)/(cz+d)$. The text builds this bridge by first detailing SR and Lorentz transformations, then proving the $L_+^{\uparrow}\cong \mathrm{SL}(2,\mathbb{C})/\mathbb{Z}_2$ isomorphism through the adjoint action on Hermitian matrices, and finally showing that conformal transformations of the sphere are precisely given by $\mathrm{SL}(2,\mathbb{C})/\mathbb{Z}_2$-maps via stereographic coordinates. In the celestial-sphere picture, boosts act as angle-dependent scalings on $u$ and as contractions on the sphere, yielding an optical interpretation of relativistic effects (e.g., the Milennium Falcon example). The discussion touches on broader implications for holography, BMS symmetry, and two-dimensional conformal field theories, highlighting how four-dimensional Lorentz invariance can be encoded in two-dimensional conformal structure on null infinity. Overall, the work clarifies how a classical symmetry in spacetime translates into a conformal symmetry on a lower-dimensional manifold with potential relevance to holography and quantum gravity.

Abstract

In these lecture notes we review the isomorphism between the (connected) Lorentz group and the set of conformal transformations of the sphere. More precisely, after establishing the main properties of the Lorentz group, we show that it is isomorphic to the group SL(2,C) of complex 2 by 2 matrices with unit determinant. We then classify conformal transformations of the sphere, define the notion of null infinity in Minkowski space-time, and show that the action of Lorentz transformations on the celestial spheres at null infinity is precisely that of conformal transformations. In particular, we discuss the optical phenomena observed by the pilots of the "Millenium Falcon" during the jump to lightspeed.

Figures (12)

• Figure 1: Alice (drawn in black) and Bob (drawn in red) in space-time, with their respective reference frames (including clocks); for simplicity, only two spatial dimensions are represented. Looking at the same event, Alice and Bob will typically associate different coordinates with it. In general, there is no way to relate the coordinates of Alice's frame to those of Bob's frame.
• Figure 2: Two inertial frames $A$ and $B$ related by a rotation of their spatial axes (the third space direction is omitted). The clocks of $A$ and $B$ are synchronized.
• Figure 3: The frame $B$ (in red) is boosted with respect to $A$ (in black) with velocity $v$ along the $x^1$ axis. The coordinates $x^2$ and $x^3$ coincide with $x'^2$ and $x'^3$. In principle, the clocks of $A$ and $B$ (not represented in this figure) need not tick at the same rate.
• Figure 4: A manifold ${\cal M}$ embedded in $\mathbb{R}^N$. The point $p$ belongs to the manifold, and the plane tangent to ${\cal M}$ at $p$ is the tangent space $T_p{\cal M}$. In this drawing, we take $N=3$ and the manifold is two-dimensional. The grid was added to emphasize the fact that the manifold looks, locally, like a plane $\mathbb{R}^2$.
• Figure 5: The plane $\mathbb{R}^2$ and the Cartesian coordinates used to label its points. A point with coordinates $(x,y)$ is also represented, and $v$, $w$ are two vectors at that point. Their scalar product with respect to the Euclidean metric (\ref{['Eu']}) is given by (\ref{['Euclideans']}).

Theorems & Definitions (8)

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