One interesting thing that comes to mind is spinor calculus.
Spinor calculus is a bit like geometric algebra; there is a mention to them in the paper (more precisely, to "twistors"). The central idea of spinor calculus relates like this: take your 4-vector (v^0, v^1, v^2, v^3) and form a 2x2 Hermitian matrix by:
V = v^0 I + v^1 s_1 + v^2 s_2 + v^3 s_3
where s_1, x_2, and s_3 are the Pauli matrices. Then it turns out that det V is the 4-norm of v^\mu:
det V = v^0 v^0 − v^1 v^1 − v^2 v^2 − v^3 v^3.
The Lorentz transforms must preserve the 4-norm and hence det V, but they must also be linear and map Hermitian matrices to Hermitian matrices, so that given a lorentz transform t, there is a Lorentz matrix L such that:
matrix (t v) = L (matrix v) L†
(that's not 100% accurate because it can't do PT flips; I think P is something like V → V^-1 while a 4-flip is V → -V; combine them together to get T). The Lorentz transforms are just the group det L = 1 -- the Möbius transformations SL(2, C).
The elegance of this comes when you look at null vectors, where det V = 0, making V a projection -- so V = u ⊗ u† for some u. The action of a Lorentz transform on u is then just u → L u, where L is the Lorentz matrix. Moreover when you work out what the ratio of the components of u are, tracing back through the mathematics, you get varios stereographic projections (x + i y) / (R - z), depending whether it's future-pointing or past-pointing.
So all the light that is coming in towards you is a bunch of null vectors that you can paint on a celestial sphere, projected to the complex plane by a stereographic projection, with Lorentz boosts as Möbius transformations of those points.
Immediate freebies: when a marble is speeding past you it still "looks like" a marble to you; it just seems "rotated" in a strange way, because Möbius transformations map circles to circles. Yes if you try to "work backwards" in your coordinates you'll construct a warped model of the system which is Lorentz-contracted, but that's not what you'll see.
Another freebie: as you accelerate faster and faster, the stars all "tilt" in the direction that you're going, crowding around the point you're travelling to. This is in sharp contrast to all those spacey TV shows where the stars "streak away." One can imagine that for a photon's timeless life, the event of its origin is the only thing behind it; and the entire rest of the universe is in front of it.
It actually gets even better; it turns out that you get to unify the spinor equations for the massless neutrino ∇_{AA'} u^A = 0; the photon ∇_{AA'} u^{AB} = 0, and the weak-field limit for gravity is something like ∇_{AA'} u^{ABCD} = 0 for the graviton. (That may not be 100% correct; I am working from memory here.)
All of that comes from something which is basically a quaternion/geometric algebra application to spacetime.
Spinor calculus is a bit like geometric algebra; there is a mention to them in the paper (more precisely, to "twistors"). The central idea of spinor calculus relates like this: take your 4-vector (v^0, v^1, v^2, v^3) and form a 2x2 Hermitian matrix by:
where s_1, x_2, and s_3 are the Pauli matrices. Then it turns out that det V is the 4-norm of v^\mu: The Lorentz transforms must preserve the 4-norm and hence det V, but they must also be linear and map Hermitian matrices to Hermitian matrices, so that given a lorentz transform t, there is a Lorentz matrix L such that: (that's not 100% accurate because it can't do PT flips; I think P is something like V → V^-1 while a 4-flip is V → -V; combine them together to get T). The Lorentz transforms are just the group det L = 1 -- the Möbius transformations SL(2, C).The elegance of this comes when you look at null vectors, where det V = 0, making V a projection -- so V = u ⊗ u† for some u. The action of a Lorentz transform on u is then just u → L u, where L is the Lorentz matrix. Moreover when you work out what the ratio of the components of u are, tracing back through the mathematics, you get varios stereographic projections (x + i y) / (R - z), depending whether it's future-pointing or past-pointing.
So all the light that is coming in towards you is a bunch of null vectors that you can paint on a celestial sphere, projected to the complex plane by a stereographic projection, with Lorentz boosts as Möbius transformations of those points.
Immediate freebies: when a marble is speeding past you it still "looks like" a marble to you; it just seems "rotated" in a strange way, because Möbius transformations map circles to circles. Yes if you try to "work backwards" in your coordinates you'll construct a warped model of the system which is Lorentz-contracted, but that's not what you'll see.
Another freebie: as you accelerate faster and faster, the stars all "tilt" in the direction that you're going, crowding around the point you're travelling to. This is in sharp contrast to all those spacey TV shows where the stars "streak away." One can imagine that for a photon's timeless life, the event of its origin is the only thing behind it; and the entire rest of the universe is in front of it.
It actually gets even better; it turns out that you get to unify the spinor equations for the massless neutrino ∇_{AA'} u^A = 0; the photon ∇_{AA'} u^{AB} = 0, and the weak-field limit for gravity is something like ∇_{AA'} u^{ABCD} = 0 for the graviton. (That may not be 100% correct; I am working from memory here.)
All of that comes from something which is basically a quaternion/geometric algebra application to spacetime.