# Weyl metrics and wormholes

## TL;DR

This paper explores new wormhole solutions derived from Weyl metrics through dualities and complexifications, revealing configurations with conical singularities, negative tension rings, and scalar field dressings, expanding the landscape of static, axially symmetric wormholes.

## Contribution

It introduces a novel class of wormhole solutions with conical singularities and negative tension rings, extending known scalar field wormholes via duality transformations.

## Key findings

- Wormholes with conical singularities and negative tension rings are constructed.
- Flat geometry can be achieved with specific ring tension, creating holes in flat space.
- Scalar field dressings produce large classes of static, axially symmetric solutions.

## Abstract

We study solutions obtained via applying dualities and complexifications to the vacuum Weyl metrics generated by massive rods and by point masses. Rescaling them and extending to complex parameter values yields axially symmetric vacuum solutions containing conical singularities along circles that can be viewed as singular matter sources. These solutions have wormhole topology with several asymptotic regions interconnected by throats and their sources can be viewed as thin rings of negative tension encircling the throats. For a particular value of the ring tension the geometry becomes exactly flat although the topology remains non-trivial, so that the rings literally produce holes in flat space. To create a single ring wormhole of one metre radius one needs a negative energy equivalent to the mass of Jupiter. Further duality transformations dress the rings with the scalar field, either conventional or phantom. This gives rise to large classes of static, axially symmetric solutions, presumably including all previously known solutions for a gravity-coupled massless scalar field, as for example the spherically symmetric Bronnikov-Ellis wormholes with phantom scalar. The multi-wormholes contain infinite struts along the symmetry axes, apart from solutions with locally flat geometry.