Every timelike geodesic in anti--de Sitter spacetime is a circle of the same radius

TL;DR

This paper analytically proves that all timelike geodesics in anti--de Sitter space are circles of the same radius in the ambient space, refining a previous proposition and providing an alternative proof for the four-dimensional case.

Contribution

It provides a rigorous analytical proof that all timelike geodesics in anti--de Sitter space are circles of identical radius, clarifying their geometric shape in the ambient flat space.

Findings

All timelike geodesics form circles of the same radius in the ambient space.

The radius of these circles is determined by the cosmological constant .

An alternative proof is outlined specifically for .

Abstract

We refine and analytically prove an old proposition due to Calabi and Markus on the shape of timelike geodesics of anti--de Sitter space in the ambient flat space. We prove that each timelike geodesic forms in the ambient space a circle of the radius determined by $\Lambda$, lying on a Euclidean two--plane. Then we outline an alternative proof for $AdS_4$. We also make a comment on the shape of timelike geodesics in de Sitter space.