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AV Columns On-line: Electronic reprints of 95 or so past "The Alternate View" columns
by John G. Cramer, previously published in Analog, are available on-line at the URL: http://www.npl.washington.edu/av. The preprint referenced below can be obtained at: http://xxx.lanl.gov.
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. Seemingly insurmountable problems with the
Alcubierre warp-drive scheme have been eliminated by Van Den Broeck.
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 ’96 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 an 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 physicists 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 a 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 spaceship 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 subtheory 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 June 1989 and May 1990) connecting two widely separated locations
in space, say five light years apart. An object might take a few
minutes to move 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 spaceship 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 focussed on alternatives like the Krasnikov
Tube (see my column #86 in the September 1997 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 were 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 4/3(pi)R^3.
The new metric of the Van Den Broeck /Alcubierre warp bubble is
like a bull’s-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/(alpha). 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 ± to 10^34,
so that Region 4 is only about 3 x 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 x 10^ -32 meters in radius, much smaller
than a proton and approaching the Planck length (1.62 x 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 x 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 many 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 (December 1999 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.
* * *
References:
General Relativity: 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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