These animations are companion to an explainer containing the mathematical details.
As a warmup, consider the action of SO(1,2) on R1,2
The video demonstrates the action of a boost. We're in 1+2 dimensions, with z being the timelike direction. We have a yellow-green checkered light-cone, and an orange spacelike slice of constant time 1. These intersect at a red 'celestial circle', with some stars added for visualisation.
During the first step, these all transform under a matrix Λ ∈ SO(1,2). The light-cone is preserved, as it must be, but the circle is distorted. We need to rescale it, and in doing so the stars end up accumulating in the direction of the boost.
Going to 1+3 dimensions, we can no longer visualise all three spatial dimensions plus time. We project onto the spacelike part of the celestial sphere. But actually, if our 4-vectors are null, and positive timelike directed, then the time component is simply the norm of the spatial part.
Let's now look at an action of SO(1,3) on a celestial sphere.
The two stages are the same as for 1+2 dimensions: the sphere is distorted, then rescaled to t = 1. This is a boost in the z-direction. Just as in the 1+2 case, we get an accumulation of stars in the direction we boost in.
Let's see the same map, but now as a Möbius transformation.
There are three steps: stereographically project, apply the dilatation, regarding the projection as the complex plane, then undo the stereographic projection. The hole opening up at the top is a side effect of cutting out the north pole, which maps to infinity under stereographic projection.
Here's an example of a map which is simple when regarded as a Lorentz transformation, but is interesting as a Möbius map. Pure, 3D rotations just do regular 3D rotation on the celestial sphere. You can visualise a rotation by a quarter turn about the x-axis. But as a Möbius map, we get
Note, as discussed in the explainer, there are two fixed points at plus and minus one on the real axis. Unfortunately, we don't see the orbits around these two fixed points too clearly; this is one of manim's quirks, that we linearly interpolate between configurations. But we can also note that 0 and infinity are now at plus or minus i, and that the grid lines intersect at right angles, since this is a conformal map.
There are also examples of maps which are simple as Möbius maps, but complicated as Lorentz transformations, in particular translation. This is z ↦ z + 1, as a Lorentz transformation.
Going through slowly, we can see a rotation in the x-axis. Then after rescaling, we see that stars have accumulated in the y-direction. This checks out with the maths in the explainer.
We've now considered Möbius maps z ↦ az in the explainer, and translations (exercise to generalise to z ↦ z + b). How about inversion? It turns out to be a rotation around the x-axis.
There's some fun to be had too with the flows which are generated by these translations. For example, loxodromic transformations correspond to a flow which spirals outwards. This corresponds to the vector field V(x,y) = (x - y, y + x)
And we can return to rotations around the x-axis. Here we can manifestly see these orbits around plus and minus one. Ignore the bits where these orbits intersect; this is just an artifact of taking too large a step computationally when integrating the flow field. These orbits should not intersect.
The vector field is V(x,y) = (2xy, 1 - x2 + y2). These vector fields can be derived by considering a transformation generated by a single real parameter, differentiating with respect to the parameter and setting the parameter to 0, then splitting into real and imaginary parts.
Let's do this for the loxodromic flow. We have z ↦ eλ(1 + i)z. Differentiating with respect to λ gives (1+i)z = (1 + i)(x + iy) = (x - y) + i(y + x). Reading off real and imaginary parts gives the result above.