Geodesics around Weyl-Bach's Ring Solution

TL;DR

This paper investigates the gravitational properties of a ring in both Newtonian gravity and General Relativity, revealing unique relativistic effects such as particles being attracted to but not hitting the ring.

Contribution

It provides a detailed analysis of geodesic motion around a Weyl-Bach ring solution, highlighting novel relativistic behaviors not seen in Newtonian theory.

Findings

Particles are attracted to the ring's inner rim in relativistic spacetime.

Test particles never collide with the ring despite the attraction.

Comparison between Newtonian and relativistic particle trajectories shows distinct differences.

Abstract

We explore some of the gravitational features of a uniform ring both in the Newtonian potential theory and in General Relativity. We use a spacetime associated to a Weyl static solution of the vacuum Einstein's equations with ring like singularity. The Newtonian motion for a test particle in the gravitational field of the ring is studied and compared with the corresponding geodesic motion in the given spacetime. We have found a relativistic peculiar attraction: free falling particle geodesics are lead to the inner rim but never hit the ring.