Previously on “How could we produce exotic matter required to hold a wormhole open, classically?”, we saw that in 1997, Dan Vollick, a physicist at University of Victoria, published a paper, in which he showed that it is possible to produce the exotic matter required to hold a wormhole open. This was accomplished by coupling a scalar field to matter that satisfies the weak energy condition. But then, an year later, Dan Vollick came up with a wormhole which was constructed by cutting and joining two spacetimes satisfying the low energy string equations with a dilaton field. But why?
Well friends, to keep a wormhole open it is necessary to thread its throat with matter that violates the averaged null energy condition (also called the averaged weak energy condition). That is, there must exist null geodesics with tangent vectors k^µ = dx^µ/ds that satisfy:
Most discussions of such exotic matter involve quantum field theory effects, such as the Casimir effect. But, after Dan Vollick came across the researches done by Ford et al. and Taylor et al. it changed his view. In 1995, Ford and Roman have showed that the quantum inequalities satisfied by the negative energy densities in scalar and vector quantum field theories tightly constrain the geometry of wormholes. Their analysis showed that the wormhole can either be on the order of ≃ Planck size or that the negative energy density is concentrated in a “thin-shell” at the throat of the wormhole. In another paper Taylor, Hiscock, and Anderson have examined the energy-momentum tensor of a “test” quantized scalar field in a fixed background wormhole spacetime. They found that for five different wormhole geometries the energy-momentum tensor of a Minimally or conformally coupled scalar field does not even come close to having the properties required to support the wormhole. Thus, Vollick realized that the prospect of maintaining a Wormhole through quantum field theory effects is not very promising.
So, an year later, he came up with a wormhole which was constructed by cutting and joining two spacetimes satisfying the low energy string equations with a dilaton field. In spacetimes described by the “string metric” the dilaton energy-momentum tensor need not satisfy the weak or dominant energy conditions (as in case of scalar field+ matter, in last paper). In the cases considered in paper, he showed that the dilaton field violates these energy conditions and is the source of the exotic matter required to maintain the wormhole.
There is also a surface stress-energy, that must be produced by additional matter, where the spacetimes are joined. He showed that wormholes can be constructed for which this additional matter satisfies the weak and dominant energy conditions, so that it could be a form of “normal” matter.
He also briefly discussed creating charged dilaton wormholes in which the coupling of the dilaton to the electromagnetic field is more general than in string theory. I summarized the method below:
Dan Vollick showed that wormholes cannot be constructed by using Field Equations & Spherically Symmetric Solutions method. Since it is necessary that a > 1 for a wormhole throat to exist. So, he suggested one alternative method, where he considered taking A = 0 (where A is an independent parameter) in field equations (5)-(12) given in paper. This implies that Q = 0 and Fµν = 0. The action for gravity coupled to a dilaton and Maxwell field equation becomes independent of ‘a’ and the static spherically symmetric solutions is still the static, spherically symmetric solution with a now being a free parameter. It is well-known that the static, spherically symmetric solution to the usual Einstein-scalar field equations contains an additional free parameter related to the scalar charge. Thus, in string theory a wormhole throat can be created by taking A = 0 and a > 1.
He showed that charged dilaton wormholes with a > 1 can also be produced by similar methods. In this case, the charge associated with the two asymptotically flat regions will be different & should be chosen so as to make the electromagnetic field continuous across the jump. This implies that A and B will generally have different values on M+ and M–.
Note: Where, M+ and M– are two different spacetimes, while, A and B and independent parameters.
Reference: Dan Vollick, “Wormholes in string theory”, Classical and Quantum Gravity, Volume 16, Number 5, 1998. https://iopscience.iop.org/article/10.1088/0264-9381/16/5/309
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