velocity space

Part III, Observers of Observers · defined in Chapter XIII, The Level-Seven Shadow

The three-dimensional hyperbolic space of rest frames, on which the Lorentz group acts.

Plate W.26Records as points of velocity space, each carrying its form: a step of a history carries the record

T

gTg^\dagger

and its form

\xi^\dagger T^{-1}\eta

onto the next record’s, and at the light directions the report states turn by phases.

The lift is its quotient by the integral Lorentz transformations of the report counts that become the identity modulo a prime above seven. The canonical transport between observers is velocity space’s parallel transport, reduced modulo $\mathfrak p$ , and its holonomy is the Thomas–Wigner rotation: the curvature of comparing charts on the finite sky is the shadow modulo $\mathfrak p$ of the curvature of velocity space. Each promotion is carried by a unique null rotation fixing the shared light direction.

Each letter is a point of velocity space, a unit rest frame, and the records of a history are a path of boosts through it. Its sphere at infinity is the celestial sphere, of which the lift’s eight light directions are eight classes of points, and the same sphere is the Bloch sphere of the observer’s report qubit.

As mathematics

$\mathbb H^3=\SL(2,\C)/\mathrm{SU}(2)$ , the future unit timelike vectors of Minkowski space written as Hermitian $2\times2$ matrices with $\SL(2,\C)$ acting by $X\mapsto AXA^\dagger$ : equivalently the positive-definite Hermitian matrices of determinant one. Its sphere at infinity is $\Proj^1(\C)$ , and $[\psi]\mapsto\psi\psi^\dagger$ is an equivariant bijection from $\Proj^1(\C)$ onto the future null rays, under which rotations act as on the Bloch sphere and boosts by Möbius maps that are not isometries.

The Bianchi group $\PSL(2,\Z[\omega])$ acts on it discretely with finite covolume, with cusps at the points of $\Proj^1(\Q(\sqrt{-3}))$ , and the tessellation by regular ideal tetrahedra with one face at $\infty$ , 0, 1 has orientation-preserving symmetry group $\PGL(2,\Z[\omega])$ .

Its name in another field	Bridge
the positive-definite Hermitian $2\times2$ matrices of determinant one	built
$\SL(2,\C)/\mathrm{SU}(2)$ , the future unit timelike vectors	classical
the upper half-space, bounded by the celestial sphere $\Proj^1(\C)$	classical
the space of rest frames	a reading

Builds: lift
In the dictionary: continuum
In the volume: IVWorld, Kernel, ObserverIXWhat Space ForgetsXIRulial RelativityXIIIThe Level-Seven ShadowXVSpin from the Double Cover