The Spin Chronicles (Part 21): Rene Magritte and Flammarion

 

We’re on a journey. A journey to where, you ask? Honestly, I’m not entirely sure myself. Is the ultimate goal to solve all the mysteries of the universe? (No pressure, right?) Or maybe just crack open one of them, like an intellectual piñata? Or perhaps it's something subtler, something closer to René Magritte's enigmatic wisdom: “Each thing we see hides something else we want to see.”

But Magritte, ever the paradoxical philosopher, also declared: “I think we are responsible for the universe, but that doesn’t mean we decide anything.” Really, René? Nothing? Well, I’ll take my victories where I can, starting with today’s decision to write this blog. I have no clue what cosmic ripples this choice will send out (but hey, I hear butterflies start hurricanes), yet I’ll cling to the notion that I’m somehow fulfilling my obligations to the universe by clicking away at my keyboard.

I have no clue what cosmic ripples this choice will send out 

So, where were we? Ah, yes—the journey. It turns out, we’re not exactly on the straight path. We’ve taken a detour. A scenic route, if you will. After Part 17, the plan was to dive straight into options (3) and (4) for two-sided actions of the Clifford group on Cl(V):

1. $g \cdot u = g u \tau(g)$
2. $g \cdot u = g u \nu(g)$
3. $g \cdot u = \pi(g) u \tau(g)$
4. $g\cdot u=\pi (g)u\nu (g)$

Instead, I decided to veer off-road. Why? Because sometimes you’ve got to set aside the GPS of mathematical rigor and take a little detour to explore something... different. Something fun. So, let’s play with the action of the Clifford group on the null cone in Minkowski space and the induced action on the $(x, y)$-plane.

At first blush, this might sound like one of those “because I can” exercises in pure abstraction. But hold on—this isn’t some random doodle in the margins of physics. It has a gorgeous physical interpretation! The null cone in Minkowski space represents light emanating from a single spacetime point. Picture a laboratory at time $t=0$, where light bursts forth in all directions. By $t = 1$, that light forms a sphere around the lab. Think of it as the universe’s way of saying, “Let there be geometric algebra!”

On this sphere, we can project the stars of the universe—the celestial sphere—much like Flammarion’s famous engraving, which tantalizingly invites us to peek beyond the cosmic veil:

Now, let’s spice things up a bit. What happens if we apply a Lorentz transformation to our lab? A rotation is straightforward—the sky spins, and the stars gracefully whirl like a cosmic waltz. But a Lorentz boost? That’s a whole different party. Enter relativistic aberration: the stars’ positions shift dramatically, bending and dancing under the influence of spacetime’s warping lens.

We’re going to calculate this starry choreography using the Clifford group. (And if you’re getting déjà vu, it’s because I hinted at this back in my September 4, 2024 post, “Monkeys, UFOs, and Quantum Fractals: A Mind-Bending Journey through Relativity and Imagination.”)

Here’s the twist: we’ll go beyond. We’ll project the celestial sphere onto the $(x, y)$-plane of our lab. But don’t expect a straightforward mapping. The projection originates from the South Pole of the celestial sphere—a point conspicuously absent in Flammarion’s image. (That’s because it’s “beneath” the feet of our starry-eyed astronomer. Practical, but disappointing.)

This is the plan. And in the next post, we’ll dive into the algebra to unravel what happens to the projected images of stars. Will they remain stars? Will they smear into streaks of cosmic glory? Stay tuned. Who knows—this little exploration might even shake up your view of the heavens. Or at least provoke a disagreement or two.

P.S. 04-12-24 Updated the pdf file.