H3 frames and the Lorentz action
A Poincare ball with the measured objects inside and a celestial sphere outside. Apply a boost and the sky aberrates.
The relation on screen
Chart changes between observers act on the celestial sphere as Mobius maps, the aberration formula.
Symbols
- rapidity between two observers
- angle on the sky
what to look for · Boosts crowd the sky forward and thin it behind. The hyperbolic structure is the point.
Consensus to physics
- 01the observer's direction set is a two-sphere
- 02modular flow acts on that sphere
- 03the action is by Mobius transformations, which is the Lorentz group on the celestial sphere
- 04the space of observer frames is the hyperboloid H3
In standard physics
Relativistic aberration and the celestial sphere
Boosts act on the celestial sphere by Mobius maps and the set of unit timelike directions is H3, so an atlas of H3 charts with a modular clock is the raw material of a 3+1 spacetime. The chart holds 136 localised objects against a shuffled control of 7, a nineteen-fold separation.
boundary · The boost animation illustrates how modular flow acts on the direction sphere. The object coordinates are the measured part.