Traversable wormholes represent a class of exact metric solutions of the general relativistic field equations. The solutions are “exact” in the sense that no approximations requiring simplifying assumptions or specialized symmetries have to be made in order to derive the appropriate spacetime geometry. To define a stable traversable wormhole one needs to define the desirable physical requirements it is to have in order to achieve the desired FTL travel benefit. The requirements we desire are the following:
1. Travel time through the wormhole tunnel or throat should be ≤ 1 year as seen by both the travelers and outside static observers.
2. Proper time as measured by travelers should not be dilated by relativistic effects.
3. ������������������ The gravitational acceleration and tidal-gravity accelerations between different parts of a traveler’s body should be ≤ 1-g when going through the wormhole.
4. Travel speed through the tunnel/throat should be < c.
5. Travelers (made of ordinary matter) must not couple strongly to the material that generates the wormhole curvature; the wormhole must be threaded by a vacuum tube through which the travelers can move.
6. There is no event horizon at the wormhole throat.
7. There is no singularity of infinitely collapsed matter residing at the wormhole throat.
The reader should consult Ref. 3 for the full technical details.
The figure shows two diagrams representing the embedded space (Flamm) representation of a traversable wormhole, which depicts the geometry of an equatorial (Θ = π/2) slice through space at a specific moment of time (t = const). The top of the figure shows the embedding diagram for a traversable wormhole that connects two different universes (i.e., an inter-universe wormhole). The bottom diagram in the figure is an intra-universe wormhole with a throat that connects two distant regions of our own universe. These diagrams serve to aide in visualizing traversable wormhole geometry and are merely a geometrical exaggeration.
There was originally one other criterion for defining a traversable wormhole, which was that it must be embedded within the surrounding (asymptotically) flat spacetime. However, Hochberg and Visser [4] proved that it is only the behavior near the wormhole throat that is critical to understanding the physics, and that a generic throat can be defined without having to make all the symmetry assumptions and without assuming the existence of an asymptotically flat spacetime in which to embed the wormhole. Therefore, one only needs to know the generic features of the geometry near the throat in order to guarantee violations of theNull Energy Condition for certain open regions near the throat. So we are free to place our wormhole anywhere in spacetime we want because it is only the geometry and physics near the throat that matters for any analysis. This fact led to the development of a number of different traversable wormhole throat designs that are cubic shaped, polyhedral shaped, flat-face shaped, generic shaped, etc. The reader should consult Ref. 3 for a complete technical review of the various types (and shapes) of traversable wormhole solutions found in general relativity theory.
Research Program
Our research program explores the feasibility of producing a traversable wormhole in the lab which includes, but is not limited to, studying accompanying issues such as momentum conservation in traversable wormhole spacetimes, quantum back-reaction effects due to vacuum polarization, causality, etc. We are also exploring the use of computer model simulations and transformational optics and metamaterials to produce an electromagnetic analog of traversable wormholes in order to discover useful new insights.
A Few Key References
[1] Morris, M. S., and Thorne, K. S. (1988), “Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity,” Am. J. Phys., Vol. 56, pp. 395-412.
[2] Morris, M. S., Thorne, K. S., and Yurtsever, U. (1988), “Wormholes, time machines, and the weak energy conditions,” Phys. Rev. Lett., Vol. 61, pp. 1446-1449.
[3] Visser, M. (1995), Lorentzian Wormholes: From Einstein to Hawking, AIP Press, New York.
[4] Hochberg, D., and Visser, M. (1997), “Geometric Structure of the Generic Static Traversable Wormhole Throat,”Phys. Rev. D, Vol. 56, pp. 4745-4755.