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Traversable Kaluza-Klein wormholes?

Christopher Simmonds, Matt Visser

TL;DR

The paper addresses whether a traditional 4+1 Kaluza–Klein extension can mitigate the null convergence condition (NCC) and related curvature-condition violations in traversable wormholes. By analyzing both area and proper-distance coordinate formulations, it shows that eliminating NCC violations would require the 5th dimension to become macroscopically large near the throat, effectively decompactifying, which is incompatible with the KK ansatz. Consequently, extra dimensions can at best relocate curvature-condition violations rather than remove them at a reasonable physical cost. The work clarifies the limitations of KK approaches for traversable wormholes and points to the need for alternative higher-dimensional or membrane-based frameworks to circumvent NCC/NEC violations.

Abstract

Various authors have suggested that Kaluza--Klein variants of traversable wormholes might to some extent ameliorate the defocussing properties (the curvature condition violations, and implied energy condition violations) inherent in positing the existence of a traversable wormhole throat. Unfortunately such a hope is ill-founded. We shall show that in a traditional Kaluza--Klein context the price paid for completely eliminating the defocussing properties of the wormhole throat is extremely high -- to completely eliminate curvature condition violations the 5th dimension has to become truly enormous (formally infinite) in the vicinity of the wormhole throat, in a manner that is fundamentally incompatible with the traditional Kaluza--Klein ansatz. At best, the extra dimensions allow one to move the curvature condition violations around, they cannot be eliminated except at prohibitive cost. While traversable Kaluza--Klein wormholes might be interesting for other reasons, it must be emphasized that adding a 5th dimension is not particularly useful in terms of ameliorating violations of the curvature conditions.

Traversable Kaluza-Klein wormholes?

TL;DR

The paper addresses whether a traditional 4+1 Kaluza–Klein extension can mitigate the null convergence condition (NCC) and related curvature-condition violations in traversable wormholes. By analyzing both area and proper-distance coordinate formulations, it shows that eliminating NCC violations would require the 5th dimension to become macroscopically large near the throat, effectively decompactifying, which is incompatible with the KK ansatz. Consequently, extra dimensions can at best relocate curvature-condition violations rather than remove them at a reasonable physical cost. The work clarifies the limitations of KK approaches for traversable wormholes and points to the need for alternative higher-dimensional or membrane-based frameworks to circumvent NCC/NEC violations.

Abstract

Various authors have suggested that Kaluza--Klein variants of traversable wormholes might to some extent ameliorate the defocussing properties (the curvature condition violations, and implied energy condition violations) inherent in positing the existence of a traversable wormhole throat. Unfortunately such a hope is ill-founded. We shall show that in a traditional Kaluza--Klein context the price paid for completely eliminating the defocussing properties of the wormhole throat is extremely high -- to completely eliminate curvature condition violations the 5th dimension has to become truly enormous (formally infinite) in the vicinity of the wormhole throat, in a manner that is fundamentally incompatible with the traditional Kaluza--Klein ansatz. At best, the extra dimensions allow one to move the curvature condition violations around, they cannot be eliminated except at prohibitive cost. While traversable Kaluza--Klein wormholes might be interesting for other reasons, it must be emphasized that adding a 5th dimension is not particularly useful in terms of ameliorating violations of the curvature conditions.

Paper Structure

This paper contains 12 sections, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Area coordinates versus proper distance coordinates
  3. Null geodesic de-focussing at the throat
  4. Curvature conditions versus energy conditions
  5. Traditional Kaluza--Klein ansatz
  6. Traversable Kaluza--Klein wormholes
  7. Area coordinates
  8. Proper distance coordinates
  9. Spacetime curvature
  10. 3+1 traversable wormhole in proper distance coordinates
  11. 4+1 Kaluza--Klein wormhole in proper distance coordinates
  12. Discussion

Figures (1)

  • Figure 1: Idealized example of an inter-universe traversable wormhole, connecting two universes "A" and "B". Note that the wormhole throat acts to de-focus initially incoming converging bundles of light rays, converting them into outgoing diverging bundles of light rays. This de-focussing is a key unavoidable characteristic of any geometry one might plausibly wish to call a traversable wormhole.