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Lawrence H. Ford and Thomas A. Roman Scientific American, January 2000 The construction of worm holes and warp
drive would If a wormhole could exist, it would appear as a spherical opening to an otherwise distant
part of the cosmos. In this doctored photograph of Times Square, the wormhole allows New Yorkers to walk to the Sahara with
a single step, rather than spending hours on the plane to Tamanrasset. although such a wormhole does not break any known laws
of physics, it would require the production of unrealistic amounts of negative energy. Can a region of space contain less than nothing? Common sense would say no; the most one could do is remove all matter
and radiation and be left with vacuum. But quantum physics has a proven ability to confound intuition, and this case is no
exception. A region of space, it turns out, can contain less than nothing. Its energy per unit volume–the energy density–can
be less than zero. Needless to say, the implications are bizarre. According to Einstein's theory of gravity, general relativity, the presence
of matter and energy warps the geometric fabric of space and time. What we perceive as gravity is the space-time distortion
produced by normal, positive energy or mass. But when negative energy or mass–so-called exotic matter–bends space-time,
all sorts of amazing phenomena might become possible: traversable wormholes, which could act as tunnels to otherwise distant
parts of the universe; warp drive, which would allow for faster-than-light travel; and time machines, which might permit journeys
into the past. Negative energy could even be used to make perpetual-motion machines or to destroy black holes. A Star Trek episode could not ask for more. For physicists, these ramifications set off alarm bells. The potential paradoxes of backward time travel–such
as killing your grandfather before your father is conceived–have long been explored in science fiction, and the other
consequences of exotic matter are also problematic. They raise a question of fundamental importance: Do the laws of physics
that permit negative energy place any limits on its behavior? We and others have discovered that nature imposes stringent
constraints on the magnitude and duration of negative energy, which (unfortunately, some would say) appear to render the construction
of wormholes and warp drives very unlikely. Double Negative Before proceeding further,
we should draw the reader's attention to what negative energy is not. It should not be confused with antimatter, which has
positive energy. When an electron and its antiparticle, a positron, collide, they annihilate. The end products are gamma rays,
which carry positive energy. If antiparticles were composed of negative energy, such an interaction would result in a final
energy of zero. One should also not confuse negative energy with the energy associated with the cosmological constant, postulated
in inflationary models of the universe [see "Cosmological Antigravity, by Lawrence M. Krauss; SCIENTIFIC AMERICAN, January
1999]. Such a constant represents negative pressure but positive energy. (Some authors call this exotic matter; we reserve
the term for negative energy densities.) The concept of negative energy is not pure fantasy; some of its effects have even been produced in the laboratory.
They arise from Heisenberg's uncertainty principle, which requires that the energy density of any electric, magnetic or other
field fluctuate randomly. Even when the energy density is zero on average, as in a vacuum, it fluctuates. Thus, the quantum
vacuum can never remain empty in the classical sense of the term; it is a roiling sea of "virtual" particles spontaneously
popping in and out of existence [see "Exploiting Zero-Point Energy," by Philip Yam; SCIENTIFIC AMERICAN, December 1997]. In
quantum theory, the usual notion of zero energy corresponds to the vacuum with all these fluctuations. So if one can somehow
contrive to dampen the undulations, the vacuum will have less energy than it normally does–that is, less than zero energy. Waves of light ordinarily have a positive or zero energy density at different points in
space (top). But in a so-called squeezed state, the energy density at a particular
instant in time can become negative at some locations (bottom). To compensate,
the peak positive density must increase. Not Separate and Not Equal Fortunately (or not, depending on your point of view), although quantum theory allows the existence of negative energy,
it also appears to place strong restrictions - known as quantum inequalities - on its magnitude and duration. These inequalities
were first suggested by Ford in 1978. Over the past decade they have been proved and refined by us and others, including Eanna
E. Flanagan of Cornell University, Michael J. Pfenning, then at Tufts, Christopher J. Fewster and Simon P. Eveson of the University
of York, and Edward Teo of the National University of Singapore. The inequalities bear some resemblance to the uncertainty principle. They say that a beam of negative energy cannot
be arbitrarily intense for an arbitrarily long time. The permissible magnitude of the negative energy is inversely related
to its temporal or spatial extent. An intense pulse of negative energy can last for a short time; a weak pulse can last longer.
Furthermore, an initial negative energy pulse must be followed by a larger pulse of positive energy [see illustration below]. The larger the magnitude of the negative energy,
the nearer must be its positive energy counterpart. These restrictions are independent of the details of how the negative
energy is produced. One can think of negative energy as an energy loan. Just as a debt is negative money that has to be repaid,
negative energy is an energy deficit. As we will discuss below, the analogy goes even further. Pulses of negative energy are permitted by quantum theory but only under three conditions.
First, the longer the pulse lasts, the weaker it must be (a, b). Second, a pulse of positive energy must follow. The magnitude
of the positive pulse must exceed that of the initial negative one. Third, the longer the time interval between the two pulses,
the larger the positive one must be - an effect known as quantum interest (c). In the Casimir effect, the negative energy density between the plates can persist indefinitely, but large negative
energy densities require a very small plate separation. The magnitude of the negative energy density is inversely proportional
to the fourth power of the plate separation. Just as a pulse with a very negative energy density is limited in time, very
negative Casimir energy density must be confined between closely spaced plates. According to the quantum inequalities, the
energy density in the gap can be made more negative than the Casimir value, but only temporarily. In effect, the more one
tries to depress the energy density below the Casimir value, the shorter the time over which this situation can be maintained. When applied to wormholes and warp drives, the quantum inequalities typically imply that such structures must either
be limited to submicroscopic sizes, or if they are macroscopic the negative energy must be confined to incredibly thin bands.
In 1996 we showed that a submicroscopic wormhole would have a throat radius of no more than about 10-32 meter.
This is only slightly larger than the Planck length, 10-35 meter, the smallest distance that has definite meaning.
We found that it is possible to have models of wormholes of macroscopic size but only at the price of confining the negative
energy to an extremely thin band around the throat. For example, in one model a throat radius of 1 meter requires the negative
energy to be a band no thicker than 10-21 meter, a millionth the size of a proton. Visser has estimated that the
negative energy required for this size of wormhole has a magnitude equivalent to the total energy generated by 10 billion
stars in one year. The situation does not improve much for larger wormholes. For the same model, the maximum allowed thickness
of the negative energy band is proportional to the cube root of the throat radius. Even if the throat radius is increased
to a size of one light-year, the negative energy must still be confined to a region smaller than a proton radius, and the
total amount required increases linearly with the throat size. It seems that wormhole engineers face daunting problems. They must find a mechanism for confining large amounts of
negative energy to extremely thin volumes. So-called cosmic strings, hypothesized in some cosmological theories, involve very
large energy densities in long, narrow lines. But all known physically reasonable cosmic-string models have positive energy
densities. Warp drives are even more tightly constrained, as shown by Pfenning and Allen Everett of Tufts, working with us. In
Alcubierre's model, a warp bubble traveling at 10 times lightspeed (warp factor 2, in the parlance of Star Trek: The Next Generation) must have a wall thickness of no more than 10-32 meter. A bubble large enough
to enclose a starship 200 meters across would require a total amount of negative energy equal to 10 billion times the mass
of the observable universe. Similar constraints apply to Krasnikov's superluminal subway. A modification of Alcubierre's model
was recently constructed by Chris Van Den Broeck of the Catholic University of Louvain in Belgium. It requires much less negative
energy but places the starship in a curved space-time bottle whose neck is about 10-32 meter across, a difficult feat. These
results would seem to make it rather unlikely that one could construct wormholes and warp drives using negative energy generated
by quantum effects. Cosmic Flashing and Quantum Interest The quantum inequalities prevent violations of the second law. If one tries to use a pulse of negative energy to cool
a hot object, it will be quickly followed by a larger pulse of positive energy, which reheats the object. A weak pulse of
negative energy could remain separated from its positive counterpart for a longer time, but its effects would be indistinguishable
from normal thermal fluctuations. Attempts to capture or split off negative energy from positive energy also appear to fail.
One might intercept an energy beam, say, by using a box with a shutter. By closing the shutter, one might hope to trap a pulse
of negative energy before the offsetting positive energy arrives. But the very act of closing the shutter creates an energy
flux that cancels out the negative energy it was designed to trap [see illustration
below]. Attempt to circumvent the quantum laws that govern negative energy inevitably ends in disappointment.
The experimenter intends to detach a negative energy pulse from its compensating positive energy pulse. As the pulses approach
a box (a), the experimenter tries to isolate the negative one by closing the lid after it has entered (b). Yet the very act
of closing the lid creates a second positive energy pulse inside the box (c). We have shown that there are similar restrictions on violations of cosmic censorship. A pulse of negative energy injected
into a charged black hole might momentarily destroy the horizon, exposing the singularity within. But the pulse must be followed
by a pulse of positive energy, which would convert the naked singularity back into a black hole - a scenario we have dubbed
cosmic flashing. The best chance to observe cosmic flashing would be to maximize the time separation between the negative
and positive energy, allowing the naked singularity to last as long as possible. But then the magnitude of the negative energy
pulse would have to be very small, according to the quantum inequalities. The change in the mass of the black hole caused
by the negative energy pulse will get washed out by the normal quantum fluctuations in the hole's mass, which are a natural
consequence of the uncertainty principle. The view of the naked singularity would thus be blurred, so a distant observer could
not unambiguously verify that cosmic censorship had been violated. Recently we, and also Frans Pretorius, then at the University of Victoria, and Fewster and Teo, have shown that the
quantum inequalities lead to even stronger bounds on negative energy. The positive pulse that necessarily follows an initial
negative pulse must do more than compensate for the negative pulse; it must overcompensate. The amount of overcompensation
increases with the time interval between the pulses. Therefore, the negative and positive pulses can never be made to exactly
cancel each other. The positive energy must always dominate–an effect known as quantum interest. If negative energy
is thought of as an energy loan, the loan must be repaid with interest. The longer the loan period or the larger the loan
amount, the greater is the interest. Furthermore, the larger the loan, the smaller is the maximum allowed loan period. Nature
is a shrewd banker and always calls in its debts. The concept of negative energy touches on many areas of physics: gravitation, quantum theory, thermodynamics. The interweaving
of so many different parts of physics illustrates the tight logical structure of the laws of nature. On the one hand, negative
energy seems to be required to reconcile black holes with thermodynamics. On the other, quantum physics prevents unrestricted
production of negative energy, which would violate the second law of thermodynamics. Whether these restrictions are also features
of some deeper underlying theory, such as quantum gravity, remains to be seen. Nature no doubt has more surprises in store. The Authors Lawrence H. Ford and Thomas A. Roman have collaborated on negative energy issues for over a decade. Ford received his
Ph.D. from Princeton University in 1974, working under John Wheeler, one of the founders of black hole physics. He is now
a professor of physics at Tufts University and works on problems in both general relativity and quantum theory, with a special
interest in quantum fluctuations. His other pursuits include hiking in the New England woods and gathering wild mushrooms.
Roman received his Ph.D. in 1981 from Syracuse University under Peter Bergmann, who collaborated with Albert Einstein on unified
field theory. Roman has been a frequent visitor at the Tufts Institute of Cosmology during the past 10 years and is currently
a professor of physics at Central Connecticut State University. His interests include the implications of negative energy
for a quantum theory of gravity. He tends to avoid wild mushrooms. Further Information ·
BLACK HOLES AND TIME WARPS: EINSTEIN'S OUTRAGEOUS LEGACY. ·
LORENTZIAN WORMHOLES: FROM EINSTEIN TO HAWKING. ·
QUANTUM FIELD THEORY CONSTRAINS TRAVERSABLE WORMHOLE GEOMETRIES. ·
THE UNPHYSICAL NATURE OF WARP DRIVE. ·
PARADOX LOST. ·
TIME MACHINES: TIME TRAVEL IN PHYSICS, METAPHYSICS, AND SCIENCE FICTION. ·
THE QUANTUM INTEREST CONJECTURE. Topics |
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