# On wormholes with long throats and the stability problem

## TL;DR

This paper constructs explicit, regular wormhole and black hole solutions with phantom scalar fields, introduces the concept of 'invisible ghosts' to explain their observational invisibility, and discusses the instability of long-throat wormholes.

## Contribution

It provides new explicit models of traversable wormholes with long throats using phantom fields and analyzes their stability, revealing inherent instabilities in long-throat configurations.

## Key findings

- Long-throat wormholes are generally unstable.
- Maximally long throats correspond to unstable Nariai-like solutions.
- 'Invisible ghosts' concept explains the lack of observational evidence for phantom fields.

## Abstract

We construct explicit examples of globally regular static, spherically symmetric solutions of general relativity with a phantom scalar field as the source of gravity, describing traversable wormholes with flat asymptotic regions on both sides of the throat as well as regular black holes, in particular, those called black universes. To explain why such phantom fields are not observed under usual conditions, we invoke the concept of "invisible ghosts," which means that the phantom field decays quickly enough at infinity and is there too weak to be observed. This approach leads to wormhole models in which the spherical radius is almost constant in some range of the radial coordinate near the throat, forming a "long throat". We discuss the peculiar features and difficulties of the stability problem for such configurations. It is shown that the limiting case of a "maximally long throat" has the form of an unstable model with the Nariai metric. This allows us to conjecture that a long throat does not stabilize wormholes with a scalar source.