Is the celestial sphere we actually see the Riemann Sphere?

Question

I've been watching a few lectures by R. Penrose where he seems to say that what we see around us is the Riemann sphere. He usually gives the example of an observer floating in deep space or if the earth was transparent, so stars all around. Some other academics have said the fact that Twistor theory is based on what we 'actually see around us' is its unique appeal. Penrose notes that as one moves, the transformation of this celestial sphere is equivalent to the Lorentz transformations of relativity, that stars squash in the direction of the motion, circles (of stars) get mapped to circles, albeit skewed within the circle, and it's conformal/angle-preserving even if distances and shapes are distorted. This is a compelling idea, since of course the complex line is also the space of quantum mechanics.

I guess my first question is, did I get his idea right? My second is, isn't that a bit odd to say that we 'see' a line of numbers 'around' us? Most people I think feel like they see at least a 2D plane /interior of a sphere around them, though I realize there are a lot of hidden assumptions in that view. Or perhaps he means something like we see a live action stereographic projection of the Riemann sphere, out and then back on itself, or something more complicated? attached is an image of one of his slides. I had always thought of the complex numbers as a 2d cartesian plane, since that's the model used on beginners, before I heard him describe how formally the complex numbers are actually just a line just like the reals, and seeing this image, how all the arrows of a sphere kind of 'move together' kind of clicked for me.

Answer 1

Yes, OP understood correctly.

1. The celestial (anti-celestial) sphere $\mathbb{S}^2_{\mp}$ is the past (future) light-cone ${\cal V}^{\mp}/\mathbb{R}_{>0}$ modulo a positive scale factor, which in turn can be identified with the past (future) light-cone ${\cal V}^{\mp}$ at some fixed past (future) time $x^0$, respectively.

2. The Minkowski spacetime $(\mathbb{R}^{1,3},||\cdot||^2)$ is isometric to the space of $2\times 2$ Hermitian matrices $(u(2),\det(\cdot))$.

   Moreover, $SL(2,\mathbb{C})$ is the double cover of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$, cf. my Phys.SE answer here.

3. A point $\sigma\in u(2)$ on the past (future) light-cone ${\cal V}^{\mp}$ is uniquely of the form $$\sigma~=~\mp\lambda\lambda^{\dagger}, \qquad \lambda~\in~ (\mathbb{C}^2)^{\times}~:=~\mathbb{C}^2\backslash\{(0,0)\},$$ modulo a little group $U(1)$ phase ambiguity, respectively, cf. my Phys.SE answer here. Twice the time is $$2x^0~=~{\rm tr}(\sigma)~=~\mp\sum_{\alpha=1}^2|\lambda^{\alpha}|^2~=:~\mp||\lambda||^2,$$ respectively.

   One may hence identify the celestial (anti-celestial) sphere $$\mathbb{S}^2_{\mp}~\cong~\mathbb{P}^1(\mathbb{C})$$ with the complex projective line, i.e. the Riemann sphere, cf. OP's title question.

4. The double cover $SL(2,\mathbb{C})$ acts transitively $\rho(g)\lambda=g\lambda$ on the left Weyl spinor space $(\mathbb{C}^2)^{\times}$ via regular $2\times 2$ matrices $g$.

References:

1. R. Penrose & W. Rindler, Spinors and space-time, vol. 2; p. 1.