Hyperbolic motion generated by inversion

TL;DR

This paper explores how inversion transformations create a conformally flat geometry where static observers experience uniform acceleration, revealing a novel relationship between acceleration and spatial coordinates.

Contribution

It introduces a method to generate hyperbolic motion via inversion, linking static and inertial observers through a specific acceleration transformation.

Findings

Static observers in the new geometry are uniformly accelerating.

Acceleration transforms as g → 1/b² g under inversion.

Scalar expansion of a static observer grows linearly with time.

Abstract

An inversion transformation applied to an inertial observer is used to generate a nonstatic conformally flat geometry in spherical coordinates. A static observer in the new geometry is uniformly accelerating with respect to the inertial one and vice versa, but its acceleration $g$ undergoes the transformation $g \rightarrow 1/b^{2}g$, where $b$ is a constant. A nongeodesic congruence of a static observer has a scalar expansion which grows linearly with time but the acceleration is proportional to $r$, as for the classical rotation.