The relativity of hyperbolic space

TL;DR

This paper explores the geometric and relativistic properties of hyperbolic space, linking Doppler shifts, accelerated frames, and cosmological models without approximations, revealing new insights into the structure of spacetime.

Contribution

It introduces a novel time-velocity space metric based on hyperbolic geometry, connecting Doppler shifts to cosmological phenomena without simplifying assumptions.

Findings

Hyperbolic velocities relate to Lambert quadrilaterals.

The derived metric aligns with Friedmann's cosmological models.

Connections to Hubble's law and exponential redshift are established.

Abstract

The longitudinal Doppler shift is a measure of hyperbolic distance. Transformations of uniform motion are determined by the Doppler shift, while its square root transforms to a uniformly accelerated frame. A time-velocity space metric is derived, by magnifying the Beltrami coordinates with the geometric time, which is similar to the one obtained by Friedmann using Einstein's equations in which the mass tensor describes a universe of dust at zero pressure. No such assumption nor any approximation in which the coordinates increase with time (i.e., constant velocities) need be made. The hyperbolic velocities are related to the sides of a Lambert quadrilateral. In the limit when the acute angle becomes an ideal point, the case of uniform acceleration arises. The relations to Hubble's law, and to the exponential red shift, are discussed.