Determining the Topology and Deflection Angle of Ringholes via Gauss-Bonnet Theorem

TL;DR

This paper uses the Gauss-Bonnet theorem to analyze the topology and light deflection in ringhole geometries, revealing a torus topology at the throat and deriving the deflection angle's dependence on geometric parameters.

Contribution

It applies the Gauss-Bonnet theorem to determine the topology and gravitational lensing effects of ringholes, including special cases like Ellis wormholes, in a weak field approximation.

Findings

Surface topology at the wormhole throat is a torus.

Deflection angle depends on ringhole parameters $b_0$ and $a$.

Ellis wormhole is a special case with spherical topology.

Abstract

In this letter, we use a recent wormhole solution known as a ringhole [Gonzalez-Diaz, Phys.\ Rev.\ D {\bf 54}, 6122, 1996] to determine the surface topology and the deflection angle of light in the weak limit approximation using the Gauss-Bonnet theorem (GBT). We apply the GBT and show that the surface topology at the wormhole throat is indeed a torus by computing the Euler characteristic number. As a special case of the ringhole solution, one can find the Ellis wormhole which has the surface topology of a 2-sphere at the wormhole throat. The most interesting results of this paper concerns the problem of gravitational deflection of light in the spacetime of a ringhole geometry by applying the GBT to the optical ringhole geometry. It is shown that, the deflection angle of light depends entirely on the geometric structure of the ringhole geometry encoded by the parameters $b_0$ and $a$, being the ringhole throat radius and the radius of the circumference generated by the circular axis of the torus, respectively. As special cases of our general result, the deflection angle by Ellis wormhole is obtained. Finally, we work out the problem of deflection of relativistic massive particles and show that the deflection angle remains unaltered by the speed of the particles.