A recent breakthrough has moved the concept of a "warp drive" another step along its path from a fictional SF prop-idea to a well founded physics concept that might one day be realized. This improvement on the Alcubierre warp drive was devised by general relativity theorist Chris Van Den Broeck of the Catholic University of Leuven in Belgium. He has eliminated seemingly insurmountable problems with the Alcubierre warp-drive scheme. His improvement employs topological gymnastics to keep the interior of the warp bubble large while making its external surface very small. But before describing Van Den Broeck's work, I'll summarize the Alcubierre warp drive concept itself, first featured in my column (#81) in the November-1996 Analog.
Until 1994 a "warp drive" was one of the myths of science fiction, a rubber-science concept used principally to permit space-opera heroes to flit from one star system to another at faster-than-light speeds, moving the plot forward in the process. Those familiar with the laws of physics saw the warp drive as a flagrant violation of the principles of special relativity, conservation of energy, and physics-as-we-know-it. It was tolerated as an excessive but perhaps necessary use of literary license by SF authors.
The status of the warp drive changed dramatically in 1994, when Dr. Miguel Alcubierre published a paper entitled "The Warp Drive: hyper-fast travel within general relativity" in the journal Classical and Quantum Gravity. Alcubierre is a theoretical physicist from Mexico who in 1994 was working at the University of Wales and is now at the Albert Einstein Institute in Potsdam, Germany. Also a fan of SF, he was steeped in the SF tradition and turned his physics expertise to the task of considering how a warp drive might be constructed within the restrictions of general relativity, our present "standard model" of gravity. Alcubierre constructed a "metric", a mathematical specification of the curvature of space-time that had all the characteristics of a SF warp-drive including the capability for faster-than-light travel. Surprisingly, Alcubierre's warp-drive metric is a solution of Einstein's equations of general relativity and is completely consistent with them. Science fiction's warp drive had been given a consistent theoretical and mathematical basis.
When theoretical physicist use general relativity, their normal procedure is to start with some distribution of massive objects and to calculate the metric describing space-time curvature that such a distribution would produce. Alcubierre reversed this procedure. Without worrying about how it might be formed, he constructed a metric that could transport volume of flat space, perhaps containing a spaceship, at superluminal speed. This was accomplished by placing the volume of flat space inside a "bubble" of highly curved space, then destroying space in front of the bubble while creating new space behind it. Effectively, the warp bubble is driven forward by creating and annihilating space as if a local Big Bang were occurring at the rear of the space ship while a local Big Crunch was occurring in front of it.
How does Alcubierre's metric manage to move an object faster than the speed of light? Isn't that in direct contradiction to Einstein's special theory of relativity? Actually, no. General relativity treats special relativity as a restricted sub-theory that applies locally to any region of space that is sufficiently small that its curvature can be neglected. General relativity does not forbid faster-than-light travel or communication, but it does require that the local restrictions of special relativity must apply. In other words, light speed is the local speed limit, but the broader context of general relativity may provide ways of circumventing this local statute. One example of this is a wormhole (see my AV columns, Analog 6/89 and 5/90) connecting two widely separated locations in space, say five light-years apart. An object might take a few minutes to move with at low speed through the neck of a wormhole, observing the local speed-limit laws all the way. However, by transiting the wormhole the object has traveled five light years in a few minutes, producing an effective speed of a million times the velocity of light.
Another example of a faster than light phenomenon is the expansion of the universe itself. As the universe expands, new space is created between any two separated objects. The objects may each be at rest in their local space-time, but nevertheless the distance between them may grow at a rate that is much greater than the speed of light. According to the current standard model of cosmology, most of the universe is receding from us at FTL speeds and therefore is completely isolated from us.
Alcubierre's metric uses an analogous expansion of space to drive the warp bubble forward. However, since the ship within the bubble is at rest in its local space, the occupants will feel no acceleration forces when the forward speed of the bubble changes, nor will they experience the "usual" relativistic effects of mass increase and time dilation. If an Alcubierre warp-drive ship travels 100 light years at 100 times the velocity of light, to both the occupants and outside observers the trip takes one year, no more and no less.
Alcubierre's publication stimulated a flurry of activity among general relativity theorists, who investigated the implications of the new idea, It was found, by himself and others, that Alcubierre's original warp-drive idea had a number of serious problems. It violated the strong, dominant, and weak energy conditions of general relativity. The net energy of the warp bubble, as it turned out, was extremely large and negative. For example, a warp bubble 100 meters in radius that might contain a space ship of reasonable size would have a net negative energy that was roughly ten times larger in magnitude than the entire (positive) energy of the visible universe. Another problem was that the walls of the bubble would have to be so thin that they could not be constructed with matter, even "collapsed matter" of nuclear density. It was also found that most of the warp bubble is disconnected from a sizable part of the external negative energy region. Therefore, the surface part of the bubble could not be carried along and would have to be continuously generated externally. The drive could not be self-contained or self-operated. These problems have seemed so overwhelming that recent attention has been focused on alternatives like the Krasnikov Tube (see my column #86 in the September-'97 Analog) that might present fewer problems of implementation and control.
Now, however, Dr. Van Den Broeck has proposed an improvement on Alcubierre's scheme that appears to solve many of its problems. Van Den Broeck observed that most of the undesirable effects of Alcubierre's drive scale with the volume or surface area of the warp bubble. Therefore, his simple solution is to make the radius of the warp bubble so small that the problems go away. In doing this, he makes use of another trick from general relativity. The interior volume of a region of space bounded by a closed surface, because of space curvature, can be made much larger than the flat-space volume bounded by its surface. In curved space the inside volume of a sphere of radius R can be much greater than 4/3pR3.
The new metric of the Van Den Broeck/Alcubierre warp bubble is like a bulls-eye target with a center (Region 1) surrounded by three concentric rings (Regions 2-4). The central sphere in Region 1 is flat space large enough to hold a spaceship. Region 2 is a spherical shell containing distorted space that connects the large interior volume of Region 1 to an exterior region that is smaller in radius by a factor of 1/a. Region 3 is a transition region of flat space, a spherical shell with a volume much less than that of Region 1. Region 4 is a spherical shell that is Alcubierre's warp bubble, but now with a very small radius. Van Den Broeck makes the radius of Region 1 about 100 meters, and sets a to 1034, so that Region 4 is only about 3 × 10-32 meters in radius. With such a small radius, if the warp bubble travels at 10 times the velocity of light the amount of negative mass-energy it would require is only about −0.06 grams. Even if it travels at 100 times the velocity of light, it would require is only about −56 kilograms of negative mass-energy. Region 2, where the volume of space is compressed from inside to outside also requires a quantity of negative mass-energy, but Van Den Broeck calculates that it is only about −4 grams. These small quantities of negative energy eliminate many of the problems of Alcubierre's original concept.
However, even with these improvements, there would still be very severe "engineering problems" with any implementation of the scheme. First, although the interior of the warp bubble may be quite spacious, its exterior is only 3 × 10-32 meters in radius, mush smaller than a proton and approaching the Planck length (1.62 × 10-35 meters) in size. This is close enough to the minimum length-scale of the universe that such a size reduction is doubtful due to quantum effects. Moreover, since the diameter of the warp bubble is many orders of magnitude smaller than a wavelength of visible light (about 4 × 10-7 meters) there would be no possibility of seeing out from inside the bubble. Any trip would be a blind one, with no possibility of seeing or steering. Moreover, while the magnitude of energy required to form a warp bubble becomes more reasonable in Van Den Broeck's warp drive, the energy density requirement remains unphysically large.
And how could our space travelers enter the bubble or exit again at the end of their trip? Van Den Broeck's calculations indicate that slowing the bubble to a near stop might permit it to be expanded to any desired size. However, such an expansion would decrease the wall thickness, and it is not clear what would happen if the wall thickness became smaller than the Planck length. Van Den Broeck ends his paper by commenting that while the first warp-drive space flight remains a long way off, perhaps it has become slightly less improbable with the new scenario for a warp bubble.
From the point of view of science fiction, even the application of general relativity to create a volume of space that is larger on the inside than on the outside is very appealing. It would, for example, solve the book storage space problem for may of us. Further, I cannot wait until this principle is applied to airplane seats!
Van Den Broeck's warp drive is a large volume of flat space that is connected to normal space by a tiny "neck". It therefore resembles the more familiar general relativity topologies of wormholes or "baby universes" and perhaps has a similar behavior. This raises the issue of how the neck is prevented from pinching off altogether, isolating our space travelers in a new universe of their own rather than transporting them to a new part of the old one.
I should also comment that these calculations were performed without a proper understanding of the unknown theory-to-be that we call "quantum gravity". A warp bubble with a diameter near the Planck scale will be affected by quantum gravity effects and corrections. In particular, my previous column (12/99 Analog) described the possibility that extra space dimensions affecting gravity may be rolled up into loops about a millimeter in diameter. If this were the case, it would modify general relativity at the millimeter scale and would almost certainly render Van Den Broeck's metric unachievable.
Thus extra space dimensions might block the path to faster-than-light travel. Ours is certainly an interesting universe, and it grows more interesting as we understand it more fully.
C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W.H. Freeman (1973).
The Alcubierre Warp Drive:
Miguel Alcubierre, Classical and Quantum Gravity, v. 11, L73-L77, (1994).
The Micro-Warp Drive:
C. Van Den Broeck, preprint hep-ph/9805217 , LANL Archive, (April 2, 1999).
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