Bertrand Spaces and Projected Closed Orbits in General Relativity

TL;DR

This paper identifies specific spacetime metrics in general relativity that produce projected closed orbits by comparing relativistic and classical orbit equations, exploring their properties and relation to Bertrand spacetimes.

Contribution

It introduces a method to find metrics with projected closed orbits and analyzes their physical and geometrical characteristics, linking some to Bertrand spacetimes.

Findings

Metrics leading to projected closed orbits are identified.

Some of these metrics are classified as Bertrand spacetimes.

The physical and geometrical properties of these spacetimes are discussed.

Abstract

In the present work, metrics which lead to projected closed orbits are found by comparing the relativistic differential equation of orbits with the corresponding classical differential equation. Physical and geometrical properties of these peculiar spacetimes are derived and discussed. It is also shown that some of these spacetimes belong to the broader class of the Bertrand spacetimes.