Ring wormholes via duality rotations

Understand

We apply duality rotations and complex transformations to the Schwarzschild metric to obtain wormhole geometries with two asymptotically flat regions connected by a throat.

• In the simplest case these are the well-known wormholes supported by phantom scalar field.
• Further duality rotations remove the scalar field to yield less well known vacuum metrics of the oblate Zipoy-Voorhees-Weyl class, which describe ring wormholes.
• The ring encircles the wormhole throat and can have any radius, whereas its tension is always negative and should be less than $-c^4/4G$.