○ Ewha Womans University researcher Sung-Won Kim described the effects of extra matter and scalar field on the stability and geometry of wormhole in his 1999 paper.
○ There are two effects of extra matter fields on the Lorentzian traversable wormhole. The “primary effect” says that the extra matter can afford to be a part of source or whole source of the wormhole when the wormhole is being formed. Thus the matter does not affect the stability of wormhole and the wormhole is still safe.
○ The “auxiliary effect” is that the extra matter plays the role of the additional matter to the stably-existed wormhole by the other exotic matter. This additional matter will change the geometry of wormhole enough to prevent from forming the wormhole by backreaction.
One of the most important issues in making a practically usable Lorentzian wormhole is just the traversability. If it is traversable, there is a good usability, such as short-cut space travels, the time machine, and inspection of the interior of a black hole.
To make a Lorentzian wormhole traversable, one has conventionally used an exotic matter which violates the well-known energy conditions. For instance, a wormhole in an inﬂating cosmological model still requires exotic matter to be traversable and to maintain its shape. It is known that the vacuum energy of the inﬂating wormhole does not change the sign of the exoticity function. A traversable wormhole in the Friedmann-Robertson-Walker(FRW) cosmological model, however, does not necessarily require exotic matter at very early times. The result means that there was an exotic period in the early Universe.
The problem about maintaining a wormhole by other ﬁelds relating with the exotic property has also been of interest to us. There are two ways to generalize or modify the Lorentzian traversable wormhole space-time. (From now on the “wormhole” will be simply used in the sense “Lorentzian traversable wormhole”, unless there is a confusion.) One way is generalization of the wormhole in an alternative theory, for example, Brans-Dicke theory, Einstein-Cartan theory, etc. The other is the generalization by adding the extra matter.
In the case of the latter generalization, the added matter will play two kinds of roles in affecting the wormhole spacetime. The ﬁrst role of such matter (e.g.,scalar ﬁelds, charge, spin, etc.) is the “primary effect”, which means that the added matter is a partial or total source of the wormhole. The matter gets involved in the constructing stage of the wormhole.
The wormhole is then safe under the addition of matter, which is a part of the sources for wormhole constructing. This means that if this added extra matter ﬁeld has exotic properties, it shares exoticity with other matter, or, if the latter is not exotic, the added matter will monopolize the exotic property. In this case the wormhole cannot exist without this extra matter. Its example is the case of the wormhole solution with a scalar ﬁelds.
The second role is the “auxiliary effect”. In this effect, the added matter is not a source, but causes an extra effect on an existing wormhole. Therefore, it is not involved on the wormhole construction stage but affects it afterwards. Since this matter makes extra geometry, the wormhole is not safe when this effect dominates the exotic constructing matter. This is back reaction to the wormhole from an additional ﬁeld.
What Kim found is that a self-consistent solution for a worm-hole with a classical, minimally-coupled, massless scalar ﬁeld. He also found the back reaction of the scalar ﬁeld on the wormhole spacetime, to see the stabilities of the wormhole. The result is that the scalar field effect can break the wormhole structure when the field and the variation of the field is large.
He obtained similar consequences in charged wormhole case, in which there is the interaction term in geometry, even though no interaction term in matter. It is natural that the addition of the nonexotic matter will break wormhole if the “auxiliary effect” is large.
In this paper, he neglected the interaction between the extra field and the original matter. If the interaction exists and it is large, it can change the whole geometry drastically. But, if it is very small, it does not change the main structure of the wormhole.
Reference: Sung-Won Kim, “Backreaction to wormhole by classical scalar field: Will classical scalar field destroy wormhole?”, Astronomical Journal, pp. 1-9, 1999. https://arxiv.org/abs/gr-qc/9911099
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