# Tolman's Luminosity-Distance, Poincare's Light-Distance and Cayley-Klein's Hyperbolic Distance

## TL;DR

This paper derives a new cosmological distance formula using Poincare's light-distance and hyperbolic geometry, transforming special relativity into a hyperbolic cosmological framework with implications for Hubble's law and acceleration.

## Contribution

It introduces a hyperbolic geometric approach to cosmological distances, linking Poincare's light-distance with hyperbolic geometry and reformulating special relativity into a hyperbolic cosmological relativity.

## Key findings

- Derivation of Tolman's luminosity-distance from Poincare's light-distance.
- Identification of hyperbolic distance as the key to cosmological measurements.
- Connection between Hubble constant and hyperbolic angular velocity.

## Abstract

We deduce Tolman's formula of luminosity-distance in Cosmology from Poincare's definition of light-distance with Lorentz Transformation (LT).In Minkowskian metric, if distance is proper time (as it is often argued) then light-distance must be also the shortest distance, like proper duration (unlike Einstein's longest length within rest system). By introducing Poincare's proper light-distance in Einstein's basic synchronization we deduce a dilated distance between observer and receding mirror (with relativistic Doppler factor). Such a distance corresponds not to an Euclidean distance (Einstein's rigid rod) but to an Hyperbolic distance (Cayley-Klein) with a Lobatchevskian Horizon. From a basic proportionality hyperbolic distance-velocity, we deduce the law of Hubble. By following Penrose's Lobatchevskian representation of LT, we transform Special Relativity (SR) into an Hyperbolic Cosmological Relativity (HCR). by using only the LT but the whole LT. In Hyperbolic Rotation motion (basic active LT or Einstein's boost), Hubble constant (hyperbolic angular velocity) is coupled with a minimal Milgrom's basic proper acceleration (hyperbolic centrifugal acceleration).