Would-be
wormhole engineers face the challenging problem of how to stabilize a wormhole,
a topological shortcut from one part of the universe to another. Wormholes have a strong tendency to pinch off
and disappear. It is well established
that a sizable quantity of negative mass-energy is needed to overcome this
tendency. That requirement has been
perceived as a "show-stopper" because our universe does not seem to
contain "exotic matter" objects with negative masses. The only known way of producing negative
mass-energy in the laboratory, the Casimir Effect, can only make it in exceedingly
tiny quantities.
However, Sergei V. Krasnikov of the Central Astronomical Observatory in Pulkovo, Russia, using Einstein's general theory of relativity, may have found a way around the negative energy problem. (For a discussion of earlier work of Krasnikov, see my column in the September-`97 Analog). Krasnikov has devised a new class of wormholes that have the unusual characteristic of "puckered" throats. He has shown that for these wormholes, the negative mass-energy needed for stabilization is supplied solely by quantum fluctuations of fields in the vacuum. Thus, these wormholes need no "exotic matter" for their construction. His calculations indicate that it may be possible to construct a large transversable wormhole using only normal positive-mass matter and fields.
To
understand what Krasnikov has accomplished, let us start with a review of the
history of wormholes. For more than 80
years, Einstein's general theory of relativity has remained our "standard
model" for gravity. In 1935
Einstein and his colleague Nathan Rosen discovered that implicit in general
relativity is a tunnel-like structure in the topology of space-time, which we
now call a wormhole. The mathematical
equation (or "metric") of a wormhole describes a curved-space object
that is a shortcut through space-time itself. A wormhole may connect two regions of space-time in the same universe
(or can even connect two separate universes).
In
principle, a wormhole can also make a "timelike" connection between
one time and another in the same region of space, so that it becomes in effect
a time machine, allowing communication and travel between the past and the
future. However, some wormhole
theorists, including Steven Hawking, have suggested that our universe may
enforce a "chronology protection", with increasing vacuum
fluctuations destroying any wormhole that is on the verge of become a time
machine. These characteristics of
wormholes were used in my hard SF novel, Einstein's Bridge, in which
alien-constructed wormholes were used for inter-universe communication,
transport, and ultimately for destroying and "rewinding" our own
universe back to 1987.
The
mathematical possibility that wormholes may exist, implicit in Einstein's
equations, raises a number of interesting questions:
Are there classes of wormholes that are
"stable", that continue to exist for extended time periods?
Are there classes of wormholes that are
"transversable", in the sense that light or matter or even people
could pass through them?
Do natural wormholes exist, and if so how
can we find them?
Could artificial wormholes be produced,
and if so how can we make them?
Could wormholes be used for
faster-than-light travel and/or for time travel?
Not all of these
questions can be considered here, but all of them have been at least partially
addressed in several of my previous columns (see my Alternate View columns numbers 33, 39, 53, and 69, published
in Analog
in the June-`89, May-`90, July-`92 and
mid-December-`94 issues).
If
wormholes are to be used for FTL communication or transportation, the issue of
stability is an extremely important one. In 1962, Wheeler and Fuller demonstrated that Einstein-Rosen
wormholes (or "bridges") are extremely unstable, so that if such a
wormhole should happen to appear spontaneously it would pinch off so rapidly
that not even one photon of light could pass through it before it closed.
In
1988 Michael Morris and Kip Thorne of Cal Tech showed that stable wormholes are
possible after all, and they described how a stable wormhole might be
constructed by an "advanced civilization" (i.e., not us.) They found that to stabilize a wormhole, a
region of negative mass-energy was needed in the wormhole's
"throat". They suggested
creating this negative energy region by using the Casimir effect, a quantum
effect in which long-wavelength vacuum fluctuations are suppressed in a region
between conducting surfaces. Morris and
Thorne suggested creating the required region of negative energy by placing two
electrically charged spherical capacitor plates in the curved space of the wormhole
throat (with each sphere geometrically inside
the other, and with the spheres spaced about a proton diameter apart!). Subsequent analyses showed that a
Morris-Throne wormhole would have to be of planetary dimensions, would require
planet-mass quantities of negative mass-energy, and that the tidal forces
created by the space curvature of the wormhole throat would be likely to
destroy atoms (or people) attempting passage through it. Therefore, Morris-Thorne wormholes, while perhaps
stable, cannot be considered to be
transversable.
A few years later, Matt Visser of Washington University in St. Louis suggested a more user-friendly class of transversable wormhole. He describes his flat-space wormholes as produced by cutting holes in two separated regions of space time and then sewing the edges of the holes together with cosmic string. In other words, two joined regions of flat space are framed by a loop of "cosmic string" of negative mass and string tension. The cosmic string (another exotic artifact of general relativity) provides the needed negative energy. However, it is questionable (a) whether cosmic strings actually exist in our universe, (b) if they do, whether they can have negative mass and string tension, and (c) whether the tendencies of the wormhole to close up and of the negative-tension cosmic string loop to expand could be precisely balanced to produce a stable Visser wormhole. Therefore, neither Einstein-Rosen, Morris-Thorne, nor Visser wormholes appeared feasible for FTL transport in our universe.
At
this point let us inquire just what theorists like Einstein, Thorne, Visser,
and Krasnikov are doing when they use mathematics to design a wormhole. General relativity provides us with a
procedure for designing a wormhole (or any other space warp) by following these
three steps:
1. Describe the kind of space-curvature that
is desired by using a "metric", a symmetric 4 × 4 matrix that is a mathematical description of curved space-time.
2. Solve Einstein's equations for the
"stress-energy tensor" (a mathematical description of how mass-energy
from matter and fields is distributed in space), such that the stress-energy
tensor will produce the desired metric.
3. By successive approximations, find a
configuration of matter and fields that will produce the required stress-energy
tensor.
That's
all there is to it. However, while many
wormhole theorists have been able to carry out steps 1 and 2, the problem lies
in accomplishing step 3. Einstein's
equations tell us that the stress-energy tensor needed to produce the metric
for wormholes (and other space warps like "warp-drives" that are of
interest to SF readers and writers) requires a large quantity of negative mass-energy that must be
concentrated in a very small region of space. This violates what theorists call the "Weak Energy Condition"
and has been viewed as requiring the existence of "exotic matter" having
negative mass-energy. Unfortunately,
all the matter and fields of our acquaintance have positive mass-energy. There has been a growing consensus among
physicists that the requirement of negative mass-energy makes it impossible to
construct a wormhole with normal matter and that some "exotic"
material like Visser's negative-tension cosmic would be required.
However,
this consensus may be wrong. Krasnikov
has shown a third way of obtaining the negative energy needed to form a stable
wormhole. He demonstrates that the
fluctuating energy of the vacuum itself can be used as the source of negative
mass energy, so that the wormhole that can be constructed with only normal
matter and fields.
Empty
space, according to quantum mechanics, is not static and unchanging, as one
would naively expect. As the quantum
vacuum is examined microscopically at smaller and smaller distances, it is
found that virtual particles with both positive and negative energies spontaneously
appear and then disappear, their brief period of existence governed by
Heisenberg's uncertainty principle. Krasnikov's calculations indicate that the negative energy part of this
process is useful for wormhole engineering.
Krasnikov
separated the stress-energy tensor, developed in step 2 above, into two parts,
one part from the mass-energy of quantum vacuum fluctuations and the other part
from the matter and fields that form the construction materials of the
wormhole. He performed "heroic" calculations with a fast computer using
the large numerical relativity program GrtensorII. From these calculations he demonstrated that
for his particular kind of puckered wormhole, the second part of the
stress-energy tensor (the non-quantum-mechanical part) has positive energy and
therefore can be produced, at least in principle, using only ordinary matter
and fields. Moreover, he has shown that
there is no particular size limitation to the new class of wormholes, and that
they could be made as large as is needed.
It
is not clear from his publications whether Krasnikov has actually evaluated the
quantum mechanical part of the stress-energy tensor. However, it appears that for the new class of wormholes the
requirement for exotic matter seems to have been ,with the quantum vacuum
itself provides the negative energy contribution from quantum fluctuations of
the electromagnetic field, the neutrino field, and the massless scalar fields (as
predicted by some theories).
The
metric of Krasnikov's new wormhole has a very simple mathematical form, a
simple four-term equation. However, the
curvature of space in the throat of the wormhole is peculiar. It is wrinkled or puckered like crepe paper,
folded into sine-wave rings from the center to the edge to make a sinusoidaly varying
space warp. Krasnikov has not reported
the results of step 3 of the procedure above, so we do not know what configuration
of matter and fields might be needed to produce this metric. In particular, we do not know whether the needed
matter densities and field strengths lie within the ranges of what is currently
feasible.
These
"engineering details" need to be worked out. Even if only normal positive-mass matter is
required, it may well turn out that Krasnikov wormhole can only be built out of
"unobtanium". Also, the tidal
forces from the space curvature and gravitational field gradients in wormhole throats
need to be examined, to determine whether a space traveler passing through a wormhole
throat of reasonable size would have a chance of survival. Moreover, Krasnikov's calculations and
assumptions need to be verified by other relativity theorists. In particular, it is well known that
calculations of the mass-energy of quantum vacuum fluctuations are tricky, and that
results can be wrong by many orders of magnitude. It is important to verify that Krasnikov has avoided this pitfall.
Nevertheless,
an important step has been made in the theory of wormholes. It has been demonstrated that stable
transversable wormholes constructed from normal matter are possible, at least
in principle. The details remain to be
worked out, but we have the beginnings of the new art of wormhole engineering.
References:
Krasnikov Wormholes:
S. V. Krasnikov, "Toward a Transversable Wormhole", LANL preprint gr-qc/0003093, Proc. of the STAIF-2000 Conference, Albuquerque, NM, February 1-4, 2000.
S. V. Krasnikov, "A Transversable Wormhole", LANL preprint gr-qc/9909016 (September 26, 1999), (to be published).
Morris-Thorne Wormholes:
Michael S. Morris and Kip S. Thorne, American Journal of Physics 56, 395 (1988).
Visser Wormholes:
Matt Visser, Lorentzian Wormholes - from Einstein to Hawking, AIP Press, New York (1995).
Einstein-Rosen Wormholes:
A. Einstein and N. Rosen, Physical Review 48, 73-77 (1935).
R. W. Fuller and J. A. Wheeler, Physical Review 128, 919-929 (1962).
John
Cramer's new book: a non-fiction work describing his Transactional
Interpretation of quantum mechanics, The Quantum Handshake - Entanglement,
Nonlocality, and Transactions, (Springer, January-2016)
is available for purchase online as a printed or eBook at: http://www.springer.com/gp/book/9783319246406
.
SF Novels by John Cramer: my two hard SF novels, Twistor and Einstein's Bridge, are newly released as eBooks by Book View Cafe and are available at : http://bookviewcafe.com/bookstore/?s=Cramer .
AV
Columns Online: Electronic
reprints of about 194 "The Alternate View" columns by John G.
Cramer, previously published in