☸ In 1988, Morris and Throne proposed a static spherically symmetric wormhole. In order to kept this wormhole open and to make interstellar travel possible one needs ‘exotic matter’. But, this exotic matter present on the throat can cause harm to humans. Thereby making interstellar travel impossible.
☸ Later in 1998, Teo proposed a ‘rotating wormhole’, the spacetimes of his wormholes is stationary and axially symmetric. He said that, “it is possible to move the exotic matter around the throat of the wormhole so that some observers falling through the wormhole would not encounter it.”
☸ According to him, we cannot avoid the use of exotic matter, but we can minimize it. This could be achieved by confining the exotic matter to a small region around the throat, and surrounding it with ordinary matter.
☸ He also proposed that ergoregion is present in rotating wormholes. When an ergoregion is present, it is possible to extract (rotational) energy from the wormhole by the Penrose process. The ergoregion does not completely surround the throat, but forms a ‘tube’ around the equatorial region.
☸ He showed that time-like and null geodesics are able to traverse the wormhole without encountering any exotic matter having negative energy density.
☸ Later in 2007, Kashargin and Sushkov carried out the analysis of motion of test particles in the rotating wormhole spacetime which reveals an interesting feature.
Friends, in 1988, Morris and Thorne put forth the concept of traversable wormholes. Unlike, Einstein–Rosen bridge or the microscopic charge-carrying wormholes of Wheeler, traversable wormholes by definition permit the two-way travel of objects like human beings. Despite the dubious possibility of ever creating or finding such a wormhole, their study has opened up remarkably fruitful avenues of research. These include the fundamental properties of such wormholes, their use as time-machines and the associated problems of causality violation, as well as the structure of quantum or Planck-scale wormholes
Perhaps the key feature in the analysis of Morris and Thorne is that, they first list the conditions that a traversable wormhole must satisfy, and then use the Einstein equations to deduce the form of the matter required to maintain the wormhole. This is opposite to the usual procedure of postulating the matter content, and then solving the Einstein equations to obtain the space-time geometry (a step which is often very difficult, if not impossible). The paradigm shift enables a surprising amount of information to be deduced about such wormholes.
They considered a static, spherically symmetric and asymptotically flat space-time with the metric
The former determines the gravitational redshift of an infalling object, while the latter characterizes the shape of the wormhole as seen using an embedding diagram; hence their names. It was shown by them that for this metric to describe a wormhole, b must satisfy a certain flare-out condition, in which above case describes two identical asymptotic universes joined together at the ‘throat’ r = b. The condition that the wormhole be traversable, in particular, means that there are no event horizons or curvature singularities. This translates to the requirement that Φ be finite everywhere.
Morris and Thorne then went on to prove that the metric, together with the wormhole-shaping and traversality conditions on b and Φ, imply that the corresponding stress-energy tensor necessarily violates the null (and therefore also the weak) energy condition. They called this form of matter ‘exotic’, an acknowledgement of the fact that there is an astronomical, perhaps impossible, price to be paid for interstellar travel using wormholes.
This has not prevented some authors from studying other classes of traversable wormholes, in the hope of minimizing the violation of the energy conditions. It was realized early on that by giving up spherical symmetry, it is possible to move the exotic matter around in space so that some observers falling through the wormhole would not encounter it. Edward Teo is one of them.
He proposed a new class of wormholes called rotating wormholes. Unlike Morris Thorne wormhole (static, spherically symmetric ones), the spacetimes of his wormholes is stationary and axially symmetric. The former means the space-time possesses a time-like Killing vector field ξ^a ≡ (∂/∂t)^a generating invariant time translations, while the latter means it has a space-like Killing vector field ψ^a ≡ (∂/∂ϕ)^a generating invariant rotations with respect to the angular co-ordinate ϕ.
The space-time (given in the canonical metric for a stationary, axisymmetric traversable wormhole) will, in general, have nonvanishing stress-energy tensor components Ttt, Ttϕ and Tϕϕ, as well as Tij . They have the usual physical interpretations; in particular, Ttϕ characterizes the rotation of the matter distribution. For a static, spherically symmetric wormhole, it turns out that the matter required to support it must have a radial tension at the throat exceeding its mass-energy density.
It can be shown, under very general conditions, that a traversable wormhole violates the averaged null energy condition in the region of the throat, by using the Raychaudhuri equation together with the fact that a wormhole throat by definition defocuses light rays. But Teo carried out a specific analysis, as in the present case. He showed that the null energy condition,
where Rab is the Ricci tensor of the space-time, is violated by a class of null vectors k^a at the throat. This would allowed him to determine the precise location of the violation, and identify the gravitational potentials responsible for it. So, what he proved was, the null energy condition is violated at some point on the throat. But, although the null energy condition is generically violated at the throat, it is possible for geodesics falling through the wormhole to avoid this energy-condition violating matter.
He also showed that, the exotic matter supporting the wormhole can be moved around the throat (in this rotating wormholes or holes case), so that some class of infalling observers would not encounter it. This is to be contrasted with the static, spherically symmetric case, where all such observers will experience violation of the energy condition. But the key point in the their result is that one can never avoid the use of exotic matter altogether.
But yes, the use of exotic matter can be minimized. This could be achieved by confining the exotic matter to a small region around the throat, and surrounding it with ordinary matter.— said Edward Teo
He also note that the ergoregion does not completely surround the throat, but forms a ‘tube’ around the equatorial region as illustrated in Fig. 1. This is characteristic of traversable wormholes: the ergoregion would necessarily intersect an event horizon at the poles, but since the latter is ruled out by definition, the ergoregion cannot extend to the poles.
When an ergoregion is present, it is possible to extract (rotational) energy from the wormhole by the Penrose process, which was originally proposed for the Kerr black hole. This process relies on the fact that time-like particles need not have positive energy in the ergoregion. Imagine an infalling particle breaking up into two inside the ergoregion. It is possible to arrange this breakup so that one of the resulting particles has negative total energy. The other particle can then travel out of the ergoregion along a geodesic, and would have more energy than what originally went in.
However, the particle with negative energy would have an orbit confined entirely within the ergoregion. It is not possible for it to escape without gaining additional energy from some other source.
When these geodesics have sufficiently high energy E ≥ 1.75, (35) is positive everywhere along the path. Hence, these time-like and null geodesics are able to traverse the wormhole without encountering any exotic matter having negative energy density.
These rotating wormholes have throats with radii of order of the Planck length, and could serve as a model for space-time foam.
Guys, Edward Teo told us about traversability of particle but not, about its trajectory or motion a particle could follow and that’s what Kashargin and Sushkov told us.
In 2007, Kashargin and Sushkov carried out the analysis of motion of test particles in the rotating wormhole spacetime which reveals an interesting feature. The particle initially propagating along the radial direction turns out to be involving into the wormhole rotation so that after passing through the throat of wormhole it continues its motion along a spiral trajectory moving away from the throat. The similar behavior has the propagation of light. The ray of light after passing through the rotating wormhole throat is propagating along the spiral.
Namely, the ray of light, propagating initially along the radial direction, becomes “twisted” by the wormhole rotation. This means that after the ray of light has passed through the wormhole throat and gone to the region r = −∞, it becomes to be propagating along a spiral trajectory.— said Sushkov
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Reference: (1) Michael S. Morris and Kip Throne, “Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity, American Journal of Physics 56, 395 (1988); https://doi.org/10.1119/1.15620 (2) Edward Teo,”Rotating traversable wormholes”, Phys. Rev. D 58, 024014 – Published 24 June 1998. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.58.024014 (3) Kashargin, P.E., Sushkov, S.V. Slowly rotating wormholes: the first-order approximation. Gravit. Cosmol. 14, 80–85 (2008). https://doi.org/10.1134/S0202289308010106
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