[Paper Review] Viable models of traversable wormholes supported by small amounts of exotic matter

Peter K. F. Kuhfittig|ArXiv.org|Jun 6, 2008

Black Holes and Theoretical Physics6 references3 citations

TL;DR

This paper presents a class of traversable wormhole models that minimize exotic matter content and fine-tuning while satisfying quantum inequalities and traversability criteria. By generalizing Ford-Roman quantum inequalities to the entire exotic region and optimizing metric functions, the model achieves a proper thickness of the exotic region as low as ~0.1 mm, with fine-tuning being a generic but manageable feature across all solutions.

ABSTRACT

Wormholes allowed by the general theory of relativity that are simultaneously traversable by humanoid travelers are subject to severe constraints from quantum field theory, particularly the so-called quantum inequalities, here slightly extended. Moreover, self-collapse of such wormholes can only be prevented by the use of "exotic matter," which, being rather problematical, should be used in only minimal quantities. However, making the layer of exotic matter arbitrarily thin leads to other problems, such as the need for extreme fine-tuning. This paper discusses a class of wormhole geometries that strike a balance between reducing the proper distance across the exotic region and the degree of fine-tuning required to achieve this reduction. Surprisingly, the degree of fine-tuning appears to be a generic feature of the type of wormhole discussed. No particular restriction is placed on the throat size, even though the proper thickness of the exotic region can indeed be quite small. Various traversability criteria are shown to be met.

Motivation & Objective

• To develop viable traversable wormhole models that minimize exotic matter use while avoiding extreme fine-tuning.
• To satisfy quantum inequalities across the entire exotic region, not just at the throat.
• To reduce the proper thickness of the exotic matter layer to macroscopically feasible levels, such as ~0.1 mm.
• To maintain traversability for humanoid travelers by ensuring finite redshift and geodesic completeness.
• To show that fine-tuning is a generic, not pathological, feature of such wormhole geometries.

Proposed method

• Generalizes Ford-Roman quantum inequalities to apply across the entire exotic region, not just at the throat.
• Uses a spherically symmetric, static line element with metric functions α(r) and γ(r) to describe the wormhole geometry.
• Imposes constraints on α′(r) < 0, γ′(r) > 0, α′′(r) > 0, and γ′′(r) ≤ 0 to ensure physical consistency.
• Derives stress-energy tensor components from Einstein field equations to identify regions violating the weak energy condition.
• Applies Lorentz transformation to the energy density in a boosted frame to derive a quantum inequality bound involving proper time sampling.
• Uses a sampling time τ₀ = f rₘ / γ with f ≪ 1 to derive a constraint on the metric coefficients, enabling fine-tuning control.

Experimental results

Research questions

• RQ1 Can traversable wormholes be constructed with minimal exotic matter while satisfying quantum inequalities beyond the throat?
• RQ2 What is the minimal proper thickness of the exotic matter region that still allows macroscopic traversability?
• RQ3 Is fine-tuning of metric parameters an inherent feature of such wormhole models, or can it be avoided?
• RQ4 How do generalized quantum inequalities constrain the geometry and stress-energy distribution in the exotic region?
• RQ5 Can viable wormhole solutions be constructed with minimal assumptions on the metric functions?

Key findings

• The proper thickness of the exotic matter region can be reduced to approximately 0.1 mm, corresponding to a distance of about 100,000 km to a space station.
• The degree of fine-tuning required for such thin exotic layers is a generic feature of the wormhole class, not an artifact of specific parameter choices.
• Quantum inequalities are generalized to apply across the entire exotic region, ensuring consistency with quantum field theory constraints.
• The models satisfy all standard traversability criteria, including finite redshift and geodesic completeness.
• Solutions exist for a wide range of parameters, with conservative choices yielding stable, macroscopic wormholes.
• Further reduction of the exotic region thickness is theoretically possible but would require increasingly extreme fine-tuning, eventually exceeding practical limits.