*Summary:*

*○ Dan Vollick in his **paper** showed that it is possible to produce the exotic matter required to hold a wormhole open classically. This is accomplished by coupling a scalar field to matter that satisfies the weak energy condition.*

*○ The violation of the weak energy condition (for the total energy-momentum tensor) is produced by the interaction energy-momentum tensor.*

*○ In addition to satisfying the weak energy condition he showed that the matter and scalar field also satisfy the dominant energy condition. Thus a wormhole can be maintained classically by coupling a scalar field to matter that satisfies the weak and dominant energy conditions.*

*○ Finally, he showed that it is ***not possible*** for the matter energy-momentum tensor to satisfy the weak energy condition if the gravitational field is weak.*

A wormhole is a handle which connects two different space-times or two distant regions in the same space-time. To keep a wormhole open it is necessary to thread its throat with matter that violates the averaged weak energy condition. In other words, there exist null geodesics passing through the wormhole, with tangent vectors k^µ = dx^µ/dσ, which satisfy

Such matter obviously violates the weak energy condition which states that T^µν ×Uµ×Uν ≥ 0 for all non-spacelike vectors U^µ. The weak energy condition ensures that all observers will see a positive energy density. Matter that violates the weak energy condition is called exotic. Thus it takes exotic matter to hold a wormhole open.

Most discussions of exotic matter involve quantum field theory effects, such as the Casimir effect. But, Dan Vollick in his paper showed that it is possible to generate the exotic matter required to maintain a wormhole classically. This is accomplished by coupling a scalar field to matter which satisfies the weak energy condition.

I derived the energy-momentum tensor for a scalar field coupled to an ideal fluid. In addition to the energy-momentum tensor for the matter and the scalar field there exists an interaction energy-momentum tensor. The interaction energy-momentum tensor can violate the weak and dominant energy conditions even if the matter & scalar field energy-momentum tensors do not. It is the interaction energy-momentum tensor that allows the wormhole to be maintained.— said Dan Vollick, author of the study.

To create a wormhole he took two static, spherically symmetric, scalar-vac solutions of the Einstein field equations and join them together. A surface energy-momentum tensor exists on the surface where the two spacetimes are joined. This surface energy-momentum tensor violates the weak energy condition.

However, if the source of the energy-momentum tensor is taken to be a scalar field coupled to matter he showed that the energy-momentum tensor of the matter and scalar field can satisfy the weak and dominant energy conditions. The violation of the weak energy condition (for the total energy-momentum tensor) is produced by the interaction energy-momentum tensor.

In addition to satisfying the weak energy condition, he showed that the matter and scalar field also satisfy the dominant energy condition. The dominant energy condition ensures that the four velocity associated with the local flow of energy and momentum is non-spacelike. Thus a wormhole can be maintained classically by coupling a scalar field to matter that satisfies the weak and dominant energy conditions.

Finally, he showed that it is **not possible** for the matter energy-momentum tensor to satisfy the weak energy condition if the gravitational field is weak.

**Reference***: Dan N. Vollick, “Maintaining a wormhole with a scalar field”, Phys. Rev. D 56, 4724 – Published 15 October 1997. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.56.4724 DOI: https://doi.org/10.1103/PhysRevD.56.4724*

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