Previously on “Is it a wormhole which is expanding?: PART 1”, we saw that in 1999, Sean hayward proposed a concept that if wormhole negative energy source fails it may collapse into a black hole. He used 3-step penrose diagrams in order to explain this process. Along with all these concepts, he also proposed that if you want to transport matter to another universe through black hole, you need additional negative energy and suggested that:
It is useful for theoretical purposes to have a simple matter model which allows negative energy.— Said Hayward.
One such suggestion given by Hayward is a Klein-Gordon field whose gravitational coupling takes the opposite sign to normal. That is, the Lagrangian and therefore energy tensor take the new sign, while the Klein-Gordon equation itself is therefore preserved. This may be thought of as a simple model of negative-energy matter or radiation, respectively in the massive and massless cases.
And that’s what Shinkai and Hayward studied in their next paper. They studied dynamical perturbations of static wormhole, using the spherically symmetric Einstein system with the exotic matter model, the massless ghost Klein-Gordon field. They developed a numerical code based on a dual-null coordinate system, in order to follow the horizon dynamics and radiation propagation clearly. Their main experiment involves, adding or subtracting Gaussian pulses in the ghost field, i.e. respectively with negative or positive energy. They also considered Gaussian pulses in a normal Klein-Gordon field, to see the effect on the wormhole of normal matter, like a human being traversing the wormhole.
They found that the dynamical structure will be characterized by the total mass or energy of the system, the Bondi energy. For the static wormhole, the Bondi energy is everywhere positive, maximal at the throat and zero at infinity, i.e. the Bondi energy is zero. Generally, the Bondi energy-loss property, that it should be non-increasing for matter satisfying the null energy condition, is reversed for the ghost field.
As a physical measure of the size of the perturbation, they took the initial Bondi energy, scaled by the initial throat radius ‘a’. In practice, they took E(20, 0) and subtract the corresponding value for the static wormhole. Some data are presented in Table I. They found that positive Bondi energy, E0 will cause collapse to a black hole, while negative Bondi energy E0 will cause explosion to an inflationary universe. Also, the speed of the horizon bifurcation increases with the energy. One could also scale the total energy by the maximal initial energy E, but this occurs at the throat and is a/2. In either case, the perturbations are small, down to 1% in energy, yet the final structure is dramatically different. Thus they concluded that the static wormhole is unstable.
They plot the energy E(x+, x–) as a function of x–, for x+ = 12, 16, 20 for the three cases. In each case, the mass increases rapidly in x– as the horizon ϑ± = 0 forms, but rapidly approaches a constant value in x+ at the horizon. This value M = E(∞, xH–) is the final mass of the black hole. The values are shown in Table I. The graphs indicated that an observer at infinity will see a burst of radiation as the wormhole collapses or explodes. For collapse, a certain fraction of the field energy radiates away, the rest being captured by the black hole, constituting its mass. For explosion, the radiated energy continues to rise in an apparently self-supporting way as the universe inflates.
Similarly, they next considered adding a small amount of conventional scalar field to the static wormhole solution. Just refer fig. 3 above you will get the idea. To extend the life of wormhole, they used additional pulse which they sent with scalar pulse. With this maintainance pulse, you will get inflation i.e. Expansion of universe. But, if you avoid to add this ghost pulse you will get a black hole.
In simple terms, just assume normal field pulse as an actual traveller of the traversible wormhole, then he, she or it may go through the wormhole and exit safely into the other universe if the speed is high enough, as can be seen from the Penrose diagram obtained from Fig. 4. However the wormhole itself will collapse to a black hole, ending its usefulness for travellers by killing them.
So, Hayward and Shinkai demonstrated a kind of temporary maintenance of the wormhole, by sending in an additional ghost pulse just after the passing of the traveller. In Fig. 3 they showed the results for pulse parameters (ca’, cb’, cc’) = (0.1, 6.0, 2.0) for the normal field representing the traveller, combined with a balancing pulse, (ca’, cb’, cc’) = (0.02390, 6.0, 3.0), case B, or (ca’, cb’, cc’) = (0.02385, 6.0, 3.0), case C. If they do not send the balancing pulse, the wormhole collapses to a black hole with horizons given by case A in the plot. The case B ends up with an inflationary expansion, while the case C keeps the wormhole structure at least until x’ = 10. Since the final fate of the wormhole is either a black hole or an inflationary expansion, to keep the throat as it was requires a fine-tuning of the parameters, and may not be realistic. However, it shows how the wormhole life may be extended.
This indicates that the wormhole might be maintained by continual adjustments to the radiation level, though it would be a never-ending project.
You might think Hayward’s idea is weird but he let us think alot deeper than anyone ever have been. If proved someday, this will change the way we look at our universe today.
Reference: Hisa-aki Shinkai and Sean A. Hayward, “Fate of the first traversible wormhole: Black-hole collapse or inflationary expansion”, Phys. Rev. D 66, 044005 – Published 16 August 2002. DOI: https://doi.org/10.1103/PhysRevD.66.044005 https://journals.aps.org/prd/abstract/10.1103/PhysRevD.66.044005
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