Planes, trains and wormholes -
For Einstein, they were just a sideshow, but for frontier physicists proving
their existence, let alone building them, is the stuff of dreams.
IMAGINE a tunnel through the fabric of space-time which you could enter near the Earth and, after travelling for just a few hours, emerge near Alpha Centauri more than 4 light years away. If the ends of the tunnel were moving relative to each other you could even use it to travel backwards or forwards in time.
As yet, no one has come even close to constructing such a "wormhole", but Einstein's general theory of relativity clearly shows that such shortcuts could exist. What is more, last year an Italian physicist suggested that one day there might be a way to construct a "microwormhole" in the laboratory. And a group of American physicists believes that if there are wormholes out in space, either surviving from the big bang or constructed by advanced extraterrestrials as part of a pan-Galactic transport system, then we might even be able to detect them.
Space travel
The fact that general relativity allows for the existence of wormholes
was discovered very soon after the publication of Einstein's theory in 1915.
But would it be possible in principle for light or a spaceship to travel
through them? This question was not answered until 1988, when Kip Thorne and
Michael Morris of the California Institute of Technology in Pasadena were
asked to concoct a plausible faster-than-light method of travelling between
stars for Carl Sagan's science fiction novel
Thorne and Morris recognised that such a wormhole would have to permit travel both in one end and out the other and therefore cannot have a one-way "horizon" (black holes are said to have a one-way horizon because nothing, even light, ever escapes).
Furthermore, travellers must be subjected to only modest accelerations and must not be torn apart by tidal forces—if you were falling feet-first towards a black hole, the gravitational field would be so huge that the difference between the pull on your feet and your head would rip you apart. When Thorne and Morris imposed these constraints on the equations of general relativity, they discovered a set of general solutions each corresponding to a traversable wormhole.
According to Thorne, anyone trying to create a traversable wormhole could try one of two strategies. The first would be to conjure one literally out of nothing. Quantum theory states that if we could zoom in on a small enough scrap of space we would discover that the amount by which it is "warped"—another way of describing the size of its gravitational field—fluctuates randomly from place to place like a choppy sea as adjacent regions steal energy from each other in an eternal game of give and take.
On scales smaller than the so-called Planck-Wheeler length—a mere 1.62 ×
10
Some physicists have speculated that it might be possible to enlarge these. One possibility, raised three years ago by Thomas Roman of Central Connecticut State University, New Britain, in 1993, is that such an enlargement may even have occurred spontaneously during the first split-second of the Universe when, according to the theory of inflation, space expanded at a truly phenomenal rate.
Another possibility is that a sufficiently advanced civilisation might be able to reach down into the quantum foam and artificially enlarge a wormhole to a usable size. The problem is that, at present, theorists have no quantum theory of gravity to tell them about the properties of the quantum foam, or even whether it exists.
The second strategy for creating wormholes proposed by Thorne involves
starting from scratch and warping and twisting macroscopic space. However,
this means doing some pretty drastic things to space. To see why, imagine
intelligent ants that live on the surface of a sheet of paper and wish to
travel between two points. If the paper is gently folded in two so that the
points are brought closer together, the ants can either take the long route
over the two-dimensional surface or connect the two points by a paper tube
which provides a shortcut through the third dimension (see
Space riddled with wormholes
Such a tube is analogous to a wormhole driven through higher dimensional space. Just as the intelligent ants will be unable to connect their tube without tearing two holes in the paper, no physical process can make a wormhole without ripping space itself.
A rip in space is a place where space-time comes to an abrupt end, as it does at the enormously dense "singularity" at the heart of a black hole. However, singularities are described by quantum gravity. So, once again, we will not know whether the strategy is possible until theorists have developed a theory of quantum gravity.
Earth to Alpha Centauri by wormhole
But there could be a third way to make a wormhole. Last year, space scientist Claudio Maccone from Turin suggested using a magnetic field to warp space.
It might appear strange that a magnetic field can have a gravitational effect, but general relativity states that anything that contains energy, including a magnetic field, warps space. The proof of this was published within two years of the publication of the general relativity theory by the Italian physicist Tullio Levi-Civita, who had discovered an exact solution of Einstein's field equations corresponding to "magnetic gravity".
Deflection of light by a wormhole
Massive magnets
Levi-Civita's solution revealed that a static uniform magnetic field, created along the axis of a long solenoid, would create a gravitational field inside the solenoid. The only snag was that the artificial gravity would be measurable only if the magnetic field were about a billion billion times greater than was then possible. Not surprisingly, magnetic gravity was set aside as a mere curiosity.
But last year Maccone pointed out that Levi-Civita's "magnetic gravity" solution to Einstein's equations bears a remarkable resemblance to the general class of traversable wormholes suggested by Thorne and Morris for Sagan's novel. "What Levi-Civita's solution in fact describes is a magnetic wormhole," says Maccone. However, any wormhole that could conceivably be created by a magnetic field in a laboratory would be so large that only a small portion would be inside the laboratory, which is why Maccone refers to this part of it as a "microwormhole".
According to Maccone, if such a wormhole were created with a laboratory
magnetic field of 2.5 tesla, the radius of curvature of the space inside, a
measure of the size of the wormhole, would be about seventeen times the
distance between the Sun and Sirius, the brightest star in the sky about 8.7
light years away from Earth. It would take an almost unimaginable magnetic
field of a billion billion tesla to create a radius of curvature of 1 metre
to squeeze the wormhole into a typical laboratory (
Maccone admits that fields large enough to create a detectable wormhole are way beyond our present capabilities—the largest field obtainable in a laboratory is around 10 tesla. But he points out that the magnetic field on the surface of compact neutron stars—stars that are the remnants of supernovae and are made entirely out of densely packed neutrons—can be close to a billion tesla and might spontaneously create a magnetic wormhole.
If it became possible to create a field this large on Earth, he imagines that a superlong solenoid could create a measurable wormhole. "I'm thinking of a solenoid as big as the 3 kilometre-long Stanford Linear Accelerator in California," says Maccone.
Light test
Maccone says that over such a distance the curvature of space caused by the artificial, magnetic gravity would be sufficient to have an effect on a beam of light. This effect could provide a test for the existence of a wormhole. "Light is slowed down in a gravitational field," he says. "So if a light beam were sent through the solenoid from end to end, its speed would be less than it would be in a vacuum. Measuring this dip in the speed of light would prove the existence of a wormhole in the laboratory."
For the moment, Maccone says he has simply demonstrated that with a large enough static magnetic field, a solution to Einstein's equations exists that would make a wormhole possible. But critics point out that it is very difficult to see how a magnetic field could create a wormhole in the first place. "It's hard to see how a magnetic field could change the topology of space," says Ian Moss of the University of Newcastle.
However, Maccone insists that a magnetic field can do just this. "No cutting and pasting of space is necessary," he says. He admits that he does not yet understand how the magnetic field changes the topology of space. However, he intends to investigate the effect by seeing what happens as his magnetic field grows from zero, relaxing the condition that the field be static and uniform.
"If what Maccone is saying is possible then it would be wonderful," says John Cramer of Washington State University in Pullman. "But I have my doubts."
Oddly enough, a mechanism for creating a wormhole from scratch without ripping space has already been proposed. In 1966, Robert Geroch of Princeton University in New Jersey found a way to smoothly warp and twist space into a wormhole. But, the price turned out to be very high. Time itself would become twisted so strongly that, during the construction, the machinery making the wormhole would briefly act as a time machine, carrying things back from late to early moments in the construction.
Although theorists initially reacted to Geroch's discovery with derision,
it is now known that general relativity permits the existence of wormhole
time machines and the subject has become the focus of intense theoretical
interest ("Wormholes, time travel and quantum gravity",
There is, however, yet another snag. It is not enough simply to find a way to make a wormhole, you also need a mechanism to keep it open. The space in a wormhole is warped so unnaturally that without something holding it open it would snap shut in the blink of an eye. In 1988, Thorne and Morris discovered that their general solutions for traversable wormholes all needed to be threaded with strange material called "exotic matter". This maintained the unnatural warpage of space by pushing outwards on the walls of the wormholes with an enormous "repulsive gravity".
Repulsive gravity is not as crazy as it sounds. According to Einstein, two separate properties of matter contribute to its gravity. One is the amount of energy contained in a unit volume of the matter — its so-called "energy density", equal to its density times the square of the speed of light—and is always positive.
Pressure point
The other contribution comes from the pressure exerted by the matter on its surroundings, just as gas exerts a pressure by hammering on the walls of its container, and in principle the pressure can be positive or negative.
Usually, this pressure is very small compared to the vast amount of energy locked up in the matter. But Thorne and Morris envisaged a kind of matter with vast negative pressure. Matter with negative pressure has a tension, rather like a stretched spring. Exotic material possesses this property in spades. In fact, its negative pressure is so enormous that it actually exceeds its energy density. This changes the "sign" of the warpage of space, making gravity repulsive rather than attractive.
Exotic matter is weird stuff. Even so, some theorists have willingly
entertained its existence in the first split-second of creation. According
to the theory of "inflation", the force that drove the enormous expansion of
the Universe immediately after its creation was none other than the
repulsive gravity of an "exotic" vacuum with a huge negative pressure ("Nothing
like a vacuum",
Inflation might also provide the means of holding open a macroscopic wormhole that started life as a microscopic wormhole in the first moments of the Universe. Richard Gott, a theorist at Princeton University, and other physicists think that inflation may have created loops of "cosmic string", one-dimensional faults in space-time in which the "exotic" conditions of the inflationary vacuum are permanently preserved.
And Matt Visser of Washington University in St Louis, Missouri, has suggested that stable macroscopic wormholes could have been created by the simultaneous inflation of quantum wormholes and loops of cosmic string. The loops of string could have held the wormhole open as it rapidly expanded. Alternatively, Visser believes that an artificial wormhole, created by an advanced civilisation, could be stabilised with "struts" of exotic material, which would behave like a loop of cosmic string.
The problem is that no one knows if exotic matter still exists in today's Universe, which puts rather a damper on traversable wormholes of the kind envisaged by Thorne and Morris. Maccone's answer is to dispense with exotic matter altogether. After all, he says, his magnetic wormhole is threaded by nothing but a magnetic field, and a magnetic field blatantly violates Thorne and Morris's requirement that whatever holds open a wormhole should exert a greater negative pressure than its energy density. The pressure exerted by the magnetic field because of the mutual repulsion of magnetic field lines is always less than its energy density.
But, says Maccone, Levi-Civita's solution demonstrates that a magnetic field does not have to satisfy the same conditions as matter. "It's a surprising result and I don't completely understand it yet," he admits. "But it seems that no exotic material is needed to make a magnetic wormhole."
Visser agrees with Maccone that the Levi-Civita solution does not require exotic matter. However, he disputes whether it is a wormhole at all. "It's a closed spherical universe," he says. "It doesn't take you anywhere because you're actually in it."
"I admit that a magnetic wormhole would be a very peculiar beast," says Maccone. However, he suggests that it might be possible to "cut away" small sections of the wormhole's profile just around the north and south poles to create two ends, enabling a light beam to travel through.
But Visser is still not satisfied. "To create a mouth and join the region inside to the outside Universe, space will have to fold back on itself," he says. "As far as I can see, the only way to do that is with exotic matter." If Visser is right then Maccone's wormhole does, after all, use exotic matter. It's just very well hidden.
Meanwhile, if there are any wormholes out in space, whether they are built from exotic matter or magnetic fields, some of them would have a remarkable property that could make them detectable from the Earth. The detection technique was suggested last year by a group of American physicists, including Cramer and Morris, now at Butler University in Indianapolis. It involves monitoring millions upon millions of distant stars for a period of many years. "In theory, if a wormhole mouth happens to pass between the Earth and one of the stars, its gravity will cause the star's light to fluctuate in an unusual and distinctive manner," says Morris.
Mass movements
It's all to do with the appearance and disappearance of mass. When matter enters a wormhole, the "entrance" obviously gains mass. However, according to the Russian astrophysicist Igor Novikov, at the same time the "exit" mouth loses a corresponding amount of mass. According to Novikov's theory, even after the matter has left the other side of the wormhole, it will leave behind its imprint: one side of the wormhole will keep its acquired positive mass and the other side will still behave as if it had lost mass. Ultimately, says Novikov, it might even acquire a negative mass as the effect of successive transits built up.
A negative mass object would be a remarkable thing. Not only would it
possess a repulsive gravitational force that would drive away any matter in
its vicinity, it would also wreak havoc with light. It is the effect of such
an object on light that Cramer and his colleagues have calculated (
Star bright
If a normal object with a "positive" mass happens to pass between the Earth and a star, its gravity bends and magnifies the star's light rather like a converging lens. This well-established phenomenon is known as "gravitational lensing" and it causes the star to brighten for a few days before fading again. The repulsive gravity of a negative mass object would also bend the light from a distant star, but in a different way. "Our naive expectation was that it would act as a diverging lens, temporarily dimming a distant star's light," says Cramer. "However, this was not what we found when we did the calculations."
Cramer was surprised to find that in certain circumstances a negative mass object can actually cause a distant star to brighten more than an equivalent positive mass. This counterintuitive result has its root in the peculiar nature of gravitational lensing. "When light passes through a conventional glass lens, the rays which are most severely bent are those which are farthest from the optic axis," says Morris. "However, the opposite is true for a gravitational lens since gravity, and its ability to bend light, become weaker with increasing distance from a massive body."
The implications for a negative mass object are that rays close to the
axis are repelled more strongly than those far from the axis. The rays
therefore "pile up" on either side of the lens in what physicists call a "caustic".
In effect, the rays are pushed away from the axis and focused around the
edges of the "lens" (see
There is a possibility that the repulsive gravity of exotic matter might have a similar lensing effect, increasing astronomers' chances of finding a wormhole. From the point of view of someone travelling through a wormhole, exotic matter has a negative energy density. However, the wormhole's gravitational field will always have a positive mass density so, overall, the wormhole could appear to have a positive, zero or negative mass to an external observer.
Fortunately, there is no need to launch a special search for the
signature of a negative mass object. For several years now three
international teams of astronomers have been routinely monitoring the
brightness of millions of distant stars in two places: a nearby galaxy
called the Large Magellanic Cloud and in the centre of our own Milky Way.
Their quarry has not been wormholes but dark objects which make up the
mysterious "dark matter" that shrouds the Milky Way. So far, the teams have
found the signatures of a dozen or so dark objects which have been dubbed
massive astrophysical compact halo objects, or MACHOs (
A little mischievously, Cramer and his colleagues have called their quarry gravitationally negative anomalous compact halo objects, or GNACHOs, an umbrella term for wormhole mouths and any other, as yet unimagined, negative mass objects.
So does our Galaxy contain any GNACHOs? Morris and his colleagues became very excited when they saw a double peak in lensing data recorded by the Polish Optical Gravitational Lensing Experiment team in 1994. "We were very encouraged," says Cramer. "The event was surprisingly like what we were expecting for a wormhole."
However, a close examination of the 60-day "light curve" revealed that
the caustics were subtly different from those predicted for a GNACHO. The
team concluded that what the Poles had found was not a wormhole mouth but
something almost as exotic: two black holes in orbit round each other (
Cramer says he and his colleagues have alerted the MACHO teams to look for the distinctive wormhole signature, but they are only too well aware how difficult it will be to find one. "It's a very long shot, I admit," says Morris.
With this flurry of activity, it seems that wormholes are no longer simply the province of science fiction. But unless the GNACHO searchers get very lucky, and unless the many practical and theoretical problems involved in making a wormhole are overcome, it will be some time before we can take a day trip to Alpha Centauri by travelling through one.
From issue 2022 of New Scientist magazine, 23 March 1996, page Page 29
Wormholes, time travel and quantum gravity
THE PAST few years have seen an explosion of interest in what theoretical physicists call 'wormholes'. These are tunnels in the geometry of space and time, connecting otherwise distant or completely disconnected regions of the Universe. In fact, there have been two explosions in two almost unrelated subjects. One is macroscopic wormholes, the kind that science fiction writers or sufficiently advanced civilisations might use for space travel across cosmic distances. The other is microscopic wormholes, on a scale 20 orders of magnitude smaller than an atomic nucleus. At this scale, space and time should obey the rules of quantum physics rather than classical laws.
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Neither type of wormhole is a new idea. Theorists have known about large-scale wormholes for more than 70 years - since shortly after Albert Einstein put forward the general theory of relativity, which relates gravity to the geometry of space and time. For 30 years, physicists have conjectured that microscopic wormholes might play a crucial role in understanding the structure of elementary particles or in developing a quantum theory of gravity. Recently, however, researchers have found that both types of objects may have some remarkable, previously unsuspected properties. Large-scale wormholes could provide a relatively easy means of travelling backwards in time, with all the potential complications that entails. Microscopic wormholes might, through their contribution to the quantum mechanics of the Universe, deter mine the values of all the constants in all the laws of physics. Much of this wormhole work is speculative and some very controversial, but that is why these subjects have generated such excitement lately.
Michael Morris and Ulvi Yurtsever, and their PhD thesis adviser Kip Thorne of the California Institute of Technology, began discovering new features of large-scale wormholes in 1985. They were trying to flesh out the idea of using wormholes for interstellar space travel, as described in Carl Sagan's novel Contact. What, they asked, do the known laws of physics require for such a thing to work?
A classical, large-scale wormhole is a solution of the field equations of
Einstein's general theory of relativity, a geometry of space and time, or 'space-time',
in which two regions of the Universe are connected by a short, narrow 'throat'.
The best-known such geometry is the spherically symmetrical, matter-free
solution discovered by Karl Schwarzschild in 1916. A portion of this
solution, omitting one of the exterior regions and the throat, serves to
describe the space-time around a spherical, non-rotating star, planet, or
other object. A slightly larger portion describes a non-rotating,
electrically neutral black hole. But neither of these objects connects
distant regions of the Universe. Only the full wormhole geometry does that (see
Figure 1 is deceiving, however. The wormhole shown is not a static structure; it represents the shape of space at the single instant of maximum expansion of the throat. The Schwarzschild wormhole actually expands from zero throat radius to the maximum shown in Figure 1, then shrinks back to zero. It does this so quickly that no traveller, even one moving at the speed of light, can ever pass from one mouth of the hole to the other. Such a wormhole is not 'traversable'. Any matter falling into the wormhole from the surrounding space hastens the contraction, so that the traveller cannot even come close to making a safe passage.
Other wormhole solutions to Einstein's equations - for an electrically charged or a rotating black hole, for example - while they are ostensibly traversable, suffer from the same problem. Any matter that falls in, or any radiation, is so concentrated and amplified by the gravitational fields of the hole that its own gravity alters the spacetime and closes off the hole. Moreover, all these wormholes exert tidal gravitational forces as strong as those of a black hole of the same size; a hole that is metres or kilometres in size would shred travellers of human dimensions long before they even got near it. Clearly a hole suitable for space travel requires some novel modifications.
What the team from Caltech did was to construct mathematically wormhole geometries that would allow passage, with throats that stayed open and gravitational fields such that travellers encountered only modest accelerations and tidal forces. The equations of general relativity then told them what kinds of matter and energy were needed to make up the holes. They found that the throats of their holes had to be threaded by matter or fields with enormous negative pressure, in other words, the matter would have to have a tension, rather like a stretched spring.
For a hole a kilometre or so across, the size of this tension is similar to the pressure at the centre of a neutron star (a star with about the same mass as our Sun, occupying the volume of a large mountain on Earth). For a smaller hole, the tension would be greater. Most crucially, the magnitude of the tension must be greater than the energy per unit volume (the energy density) of the matter itself.
Matter with this property is called 'exotic'. In familiar matter, tensions and pressures are always far smaller than the energy density: the breaking tension of steel, for example, is some 12 orders of magnitude (1012 times) smaller than its energy density. A tension larger than the energy density implies that some observers - moving rapidly with respect to the wormhole - will observe the matter to have negative energy density. Einstein's general theory of relativity relates the density and pressure, or tension, measured by one set of observers to those of another. The relationship guarantees a positive energy density for all observers if pressure or tension is smaller than energy density for any one observer, but not otherwise. Einstein's field equations imply that any traversable wormhole must contain some form of this exotic matter.
We do not know whether this requirement rules out the possibility of traversable wormholes or not. Physicists usually assume that matter obeys certain energy conditions, among which is the requirement that all observers measure positive energy densities. Situations exist, however, in which these conditions are known to be false.
The electromagnetic field between two metal plates can, for example, give rise to a negative energy density. Because, according to quantum mechanics, the electric and magnetic fields obey Heisenberg's uncertainty principle, they fluctuate minutely, rather than holding precise, classical values. Even the vacuum contains these field fluctuations. The energy of the fluctuations in the field between conducting plates is actually less than that in the free vacuum; that is, it is negative. This effect is named after the Dutch physicist Hendrik Casimir, who calculated it in 1948, and it has since been confirmed in the laboratory.
The evaporation of black holes, discovered by Stephen Hawking at the University of Cambridge in 1974, also involves negative energy densities.
No one knows whether exotic matter of the density and extent required to make a traversable wormhole can exist or not. If it can, and if it interacts weakly enough with other matter to avoid harming the traveller, or can be confined within the wormhole away from the traveller's path, then traversable wormholes remain a physical possibility.
The results of other theorists support this. Matt Visser of Washington University, St Louis, has found a kind of wormhole so benign that travellers can pass through it without encountering any matter, exotic or otherwise, and without feeling any forces at all. He takes two copies of what is called Minkowski space-time - this is infinite, empty space-time with no matter or gravitational fields - and excises an identical region from each, and joins them at the boundaries of the excised regions. The energy densities and pressures on the boundary surface, now the throat of the wormhole, are specified by Einstein's field equations. If, for example, the junction surface is a cube, then all the exotic matter is confined to 'struts' on the edges of the cube. A traveller can pass from one Minkowski region to the other through a face of the cube, untouched by any matter or force. Visser's work also suggests that wormholes like this could be made stable - they would neither collapse nor explode, clearly another requirement for holes usable for travel.
In a similar vein, recent work by Ian Moss, Felicity Mellor, and Paul Davies at the University of Newcastle upon Tyne indicates that in our expanding Universe, some wormholes may not be forced to collapse by the effects of infalling matter and radiation. So this may not be the problem for wormholes in our expanding Universe that it is for holes in hypothetical flat space-time.
Wormholes as time machines
The biggest surprise in all this is that if the laws of physics do permit wormholes suitable for space travel, then they provide a simple means of time travel as well. A wormhole may be turned into a time machine by keeping one mouth of the hole fixed with respect to the distant stars, while moving the other. (From outside, a wormhole mouth is an ordinary gravitating body. You could move it using the gravitational attraction of another body, or by giving it an electric charge and moving it with electric fields.) Clocks fixed to the moving mouth advance more slowly than those at the stationary mouth; they undergo 'time dilation' with respect to distant clocks, a well-known effect predicted by the special theory of relativity. However, they remain linked to clocks at the stationary mouth through the wormhole.
Enter the wormhole at the moving mouth when clocks there read 12:00 and you will emerge from the stationary mouth with the clocks there reading just after 12:00. The accumulated time dilation of the moving mouth, then, makes backward time travel possible. Eventually, you can travel from the stationary mouth to the moving one, through the intervening space, and reach the moving mouth when clocks there read an earlier value than those at the stationary mouth did when you left. Travel back through the hole, and you arrive at your starting point at an earlier time than you left (see Figure 2). You have made a journey, through the wormhole, back in time.
This is not the unrestricted time-travel of science fiction. There is a
surface in the wormhole space-time, defined, as shown in
Notions of causality - that causes precede effects, that the past determines the present and the future, and so on - are deeply ingrained in scientific thought. The team from Caltech, in collaboration with Igor Novikov of the Space Research Institute, Moscow, and other physicists, are examining the implications for these ideas of wormhole time machines. They supplement causality with the principle of consistency: the evolution of a physical system should be self-consistent, even when you include influences from the future acting back in time. This means that if you travel back in time and attempt to shoot your parents before your birth, your gun misfires or you miss; the sequence of events already includes the effects of your attempt.
The researchers find that a simple 'free' field in space-time with a wormhole evolves in a consistent and well-determined way from any initial conditions specified well before the wormhole's time-travel boundary. Obtaining consistent evolution from conditions specified after that boundary is more of a problem; the initial conditions may have to be restricted or specified at various times.
Interacting systems present further complications, as illustrated by the
problem of colliding 'billiard balls' in a wormhole space-time. Consistency
restricts the range of possible initial conditions
One problem that remains unsolved is that of constructing a wormhole in the first place. Theory shows, for example, that you cannot create a wormhole in a smooth, classical space-time, with well-defined time directions everywhere, unless the space-time geometry already allows travel in time. Most theorists conjecture that on very small scales the geometry of space-time fluctuates in accord with the quantum uncertainty principle, giving rise to a 'foam' of tiny wormholes. Perhaps a macroscopic wormhole could be obtained by enlarging one of these in some way. Only here does the matter of traversable wormholes and time-machines touch upon the physics of the other sort of wormhole - the microscopic wormhole.
In the late 1950s, John Wheeler, then at Princeton University, was already proposing that elementary particles might consist of microscopic wormholes threaded by electric or other field lines. This did not prove a useful description, but, since then, theorists have held that space-time should be permeated by wormholes on scales at which quantum mechanics affects gravity. This happens near the 'Planck scale', around 10-35 metres. These wormholes should play an important role in any quantum theory of gravity.
In 1987, Stephen Hawking derived some concrete consequences of this; his results indicated that such wormholes modify quantum mechanics and alter the constants of nature in unpredictable ways. In 1988, Sidney Coleman of Harvard University contested Hawking's conclusions, though not his calculations, claiming instead that quantum wormholes actually fixed the values of certain physical constants in a dramatic fashion. Other theorists quickly joined in, some supporting Coleman's conclusions, some denying them.
Hawking judged that a proper quantum-mechanical treatment of gravity (see
Box on previous page) should include the effects of microscopic wormhole
geometries such as the one shown in
Coleman's camp disputed this. They argued that Hawking's 'loss of information' would not be observable. They went on to a much more startling conclusion: that the shifting of physical constants by baby universes could solve the long-standing 'cosmological constant problem', and more.
The cosmological constant is the coefficient of a term in Einstein's gravitational field equations. It can be interpreted as an 'energy density of the vacuum', a density that remains constant as the Universe expands or contracts. (Unlike matter or radiation, the vacuum does not become more or less dense as the volume of the Universe changes; it remains the vacuum.) Einstein put the constant into his equations in order to obtain a solution describing the Universe as it was believed to be prior to the late 1920s - filled with matter, but static. Einstein's motives for using the constant have since vanished - we know the Universe is expanding - and observations show it to be very small or zero, but theorists are still having trouble getting rid of it.
Elementary particle physics predicts vacuum energies arising from quantum fluctuations, like the Casimir energy mentioned earlier. The total energy density is typically 120 orders of magnitude larger than is consistent with the observations. To reconcile this, the theorists need to arrange for the contributions from different types of particles to cancel each other to 120 decimal places (unlikely) or to find some other way to get rid of the constant. Hawking suggested in 1984 that quantum gravity might do this; Coleman placed the idea on a firmer footing by invoking the effects of wormholes.
The wormholes contribute to what is called 'sum-over-histories' in a quantum description of gravity. This gives the probability of a physical process in terms of a sum over all possible 'paths' that the process can take. Coleman argued that if you take into account the contributions of microscopic wormholes in the 'sum-over-histories' for quantum gravity, it is completely dominated by histories in which the cosmological constant, in large regions of the Universe like our own, is zero. Any physical observation that we make to measure the constant must, therefore, give a zero result. Moreover, the completely dominant histories are also those for which Newton's gravitational constant, and other physical quantities appearing in the sum, take their minimum values. These requirements should determine all the internal states of the baby universes - all of Hawking's shifts in the physical constants - hence all the values of all those constants of nature. No wonder Coleman calls this 'the big fix'.
The possibility that quantum gravity could have such dramatic effects, and that they might be calculated, has drawn many theorists to the subject. It has become a hotbed of activity in the past two years. Many variations on the original calculations, and new calculations, have been published - to test the validity of the assumptions that were made, to examine in full detail particular models of wormholes, or to search for particular observable effects.
The arguments of Coleman and others have flaws, however, that may invalidate all their conclusions. Some careful calculations of the 'sum-over-histories' that Coleman uses indicate that the histories with zero cosmological constant do not dominate as he claims, but are actually suppressed. It is not even certain that the whole approach to quantum gravity used by Coleman, Hawking and others is well defined. William Unruh of the University of British Columbia has found particularly devastating results along these lines. He claims that Coleman and Hawking omit a whole class of histories from their sum; when these are included the sum fails to give a finite prediction for any physical process. If Unruh is correct, then microscopic wormholes become a reductio ad absurdum for this approach. (That would be nearly as significant as solving the cosmological constant problem, but it is not a result most physicists would like.)
Even if Coleman's calculations are correct, the theory could still founder when compared with observations. If the theory forces Newton's constant to a minimum value that turns out to be zero or negative, it is undone. Recent results also suggest that the theory may predict masses for elementary particles in flagrant disagreement with their measured values, or a density of wormholes in space-time large enough to conflict with well-known particle physics.
The controversy is far from over. Microscopic wormholes may provide a breakthrough in our understanding of quantum gravity, or they may completely invalidate our present models, or they may yet prove to be a dead end.
No one has ever observed a wormhole, large or small. All the ideas that I have described are extensions of theory, reasonably well-founded in the classical case, but less so in the quantum case. It is the hope of every physicist working on either subject to come up with physical effects resulting from these speculations that will bring them within reach of the experimenter or the observer.
* * *
NEW SPACE-TIME GEOMETRIES FOR QUANTUM GRAVITY
THE DESCRIPTION of quantum gravity in which wormholes appear to play so large a role is a generalisation of the formulation devised by Richard Feynman in 1948 for ordinary quantum mechanics. This is called the 'sum-over-histories' approach. For example, a classical particle travels from one position to another along a particular path, or 'history'. According to quantum mechanics, however, the particle has no definite path; the uncertainty principle forbids it. All that can be determined is the probability for the particle to go from one position to the other. This can be calculated by adding up contributions from all paths between the two positions.
For gravity, the interesting quantity, corresponding to the particle's position, is the geometry of space, three-dimensional space at some 'instant of time'. A history is the evolution of the space geometry from one such instant to another, in other words, it is the four-dimensional geometry of space-time between the instants. In quantum gravity, a sum over such four-dimensional histories should give some measure of the probability for space to evolve from one three-dimensional geometry to another. Of course this is a rough, general description. Both the precise construction of such a sum and its interpretation are problems that have occupied theorists for decades, and still are not well understood.
Hawking and Coleman, among others, for a variety of technical reasons, do not construct their sum-over-histories from space-time geometries connecting two 'instants of time'. Instead, they use geometries with four dimensions which are all spatial, and the given three-dimensional geometries make up the boundaries of the four-dimensional one. Hawking claims this is the only way to obtain a sensible sum over-histories for quantum gravity - though not everyone agrees with him - and has used this approach to evaluate the sum for very simple models of the Universe.
The theorems that forbid the creation of a wormhole in a smooth, classical space-time, as mentioned in the main text, do not apply to these four-dimensional spatial geometries. Thus wormholes branching off and rejoining the larger space contribute to the sum-over-histories in this approach to produce the effects claimed by Hawking and Coleman. Whether the same effects are found if the sum is formulated differently - using actual space-time geometries, say - is uncertain. Some researchers contend that they are, others that they are not; the results depend sensitively on technical assumptions in the calculations.
Ian Redmount is a research associate in physics at Washington University, St Louis, Missouri. He was formerly a PhD student of Kip Thorne at Caltech.
From issue 1714 of New Scientist magazine, 28 April 1990, page
Paradox lost
21 March 1998
NewScientist.com news service
Paul Davies
NOTHING can escape from a black hole. Yet Stephen Hawking famously proved that black holes are not truly black-instead they glow with heat. But how can this heat energy possibly be flooding back out from what should be a one-way street? The answer is every bit as exotic as the phenomenon it explains: the heat radiated by a black hole is paid for, not by normal energy flowing out of the hole, but by negative energy flowing in.
A mathematical sleight-of-hand? Far from it-negative energy really does exist. The trouble is, generating and manipulating large enough quantities of negative energy could threaten some of nature's most sacrosanct laws-the second law of thermodynamics, for instance. It is easy to dream up scenarios that produce unphysical or paradoxical consequences-building perpetual motion machines, or even travelling backwards in time. To physicists, these are alarming notions. But nature seems to have a trick or two up its sleeve. Intriguingly, if you look carefully at any of these scenarios, some effect always seems to intrude in the nick of time to stymie the would-be law-breaker. It is as if there is a deeper principle of nature at work that permits negative energy, but proscribes its worst effects.
The concept of energy is familiar to all engineers and scientists. But when you are keeping track of where it goes, what normally matters is the energy differences. For example, you can calculate the energy expended in lifting a weight from A to B, without bothering about what the total energy was at A.
There is one known exception to this rule, and it concerns gravity.
Einstein's famous formula
But how is it possible to have less mass, or energy, than empty space? The secret lies with quantum theory. According to this, what appears to be empty space is in fact teeming with all manner of "virtual" particles that exist only fleetingly. The so-called quantum vacuum state cannot be stripped of these countless ghostly entities, but they reveal themselves only when something disturbs the vacuum state ("Nothing like a vacuum", 25 February 1995, p 30).
It is these ghostly particles that hold the key to creating a flux of negative energy-in effect a beam of cold and dark, rather than heat and light. The best way to picture negative energy beams is in terms of a phenomenon called interference. In a normal interference experiment, when light waves pass through two nearby slits in an opaque screen, they create a series of bright and dark stripes on the wall beyond. The bright regions occur where the waves from the two slits arrive at the wall in step and reinforce each other-so-called constructive interference. The dark regions occur where the waves are out of step and cancel each other out-destructive interference. Now think about the virtual photons in the quantum vacuum. In just the same way, you can create states in which normal "real" photons from a laser, say, interfere with virtual vacuum photons. In this case, destructive interference would form regions where the energy is less than the normal quantum vacuum-in other words, negative. If there is no wall to stop the light, you then have a beam of negative energy.
This much was known back in 1970s, and I realised along with Stephen Fulling, now at Texas A&M University, that a negative energy flux could create the power source for Hawking's black hole radiation. At a distance, the heat radiation represents positive energy streaming away from the black hole. But we knew that this energy could not be traced all the way back inside the hole, as that would violate the rule that nothing can get out. We found that negative energy continually streams into the hole from the surrounding region. Black holes create negative energy around themselves because the curvature of space-time due to their intense gravitational field disturbs the virtual particles in the quantum vacuum.
Firing negative energy into a black hole
Unfortunately, beams of negative energy don't just solve problems. They create them too. Suppose you directed such a beam at a hot object-an oven, say, with an opening protected by a shutter. The contents of the oven would lose energy and cool down. But this would be a clear breach of the celebrated second law of thermodynamics, which concerns a property called entropy. Roughly speaking, entropy is a measure of how much disorder there is in a given system, and the rule is that, taken as a whole, the total entropy can never fall. If the oven cooled down, its entropy would drop. Usually, this process would be accompanied by a rise in entropy elsewhere, but the negative energy beam has no associated entropy and nothing changes in the world outside the oven. This would mean that the total entropy of the Universe would fall, violating the second law. But since this law is the linchpin of thermodynamics, any violation would open the way to serious problems-such as the ability to create a perpetual motion machine, which could conjure up useful work from a system without any cost.
Negative energy also threatens to create difficulties in a quite different situation. If you drop an object into a black hole, you generally lose all sight of its physical properties. One exception is electric charge; any charge carried by a sacrificed object is retained by the hole, and sets up an electric field around it. This electric field has energy, which modifies the gravitational field of the hole. If a black hole has an electric charge so great that its electric force rivals its gravitational force, then you have a problem. According to the general theory of relativity, if the charge-to-mass ratio exceeds a critical value, the black hole will abruptly disappear.
This rings alarm bells, because lurking at the heart of a black hole is a so-called singularity, a point with infinite density where space-time is infinitely curved. A singularity is effectively a boundary or edge to space-time. If this edge were exposed- a "naked"singularity-unknown and unpredictable physical influences could emerge and invade the Universe. Fortunately, when the singularity is safely trapped inside a black hole, the wider Universe is safe. But if the black hole were removed, and the singularity let loose, then all bets would be off. A negative energy beam could do just that by lowering the mass of a black hole without affecting its electric charge until the critical limit is reached.
Using negative energy to cool down a cooker
Cosmic flashing
Larry Ford of Tufts University in Massachusetts has studied these disturbing scenarios, and found something curious. Although you can imagine ways of sending negative energy into ovens and black holes, in practice you can't keep it up for long enough to cause real trouble. Ford used a simple example of a negative energy flux formed by interference between real photons and virtual photons in the vacuum. Just as every dark stripe in a conventional interference experiment has a neighbouring bright stripe, so a region of negative energy created by destructive interference has a nearby region of positive energy created by constructive interference. So in a beam like this, every pulse of negative energy will be followed soon after by a pulse of positive energy.
It's true that a pulse of negative energy could temporarily lower the entropy of an oven, but the next positive pulse would raise it again. Although the beam would create a fluctuation in the total entropy, the second law of thermodynamics is a statistical law that permits small fluctuations. Ford managed to prove that, for the states he investigated, the strength and duration of the entropy fluctuations stayed safely within permitted statistical boundaries. In the black hole scenario the situation is rather less clear cut. It looks as if a space-time singularity might be momentarily exposed, a situation that Ford describes as "cosmic flashing". Whether such flashes would compromise the rationality of the cosmos is still uncertain.
You might wonder whether it is possible to improve on Ford's scenario, for example by opening the oven's shutter only when a pulse of negative energy approaches, and shutting out the positive energy pulses. That way the negative energy would gradually accumulate inside the oven and the positive energy would be kept out. There is, however, yet another snag. Just operating the shutter will itself create a disturbance in the quantum vacuum, and a short calculation shows that the disturbance serves to create a burst of photons from the vacuum. When the sums are done, you find that the entropy of the newly made photons more than outweighs the reduction in entropy due to the negative energy. More elaborate strategies designed to chop out the negative energy parts of a beam and stockpile them all run into similar obstacles. Once again, nature seems to find ways of confounding experimenters' attempts to use negative energy to achieve more than a token reduction in entropy.
However, there are other ways to create sustained negative fluxes. One of
these was discovered by Fulling and me, and uses a single moving mirror
Such a beam would be continuous rather than pulsed, so does that mean you could you use it to cool down the oven and violate the second law? Curiously, you would still run into problems sustaining the negative flux. Picture the experiment: the accelerating mirror produces a beam of negative energy as it heads towards the oven, but eventually it collides with the oven, halting the experiment. The greater the acceleration the bigger the flux, but the shorter the duration before collision. Again, the sums show that the total accumulated negative energy at the oven is not enough to exceed the allowed entropy fluctuations.
A different approach is to station an oven close to the surface of a
black hole, in the hope of scooping up some of the negative energy before
it's sucked in
It's not just fluxes of negative energy that threaten paradoxes-static negative energy could cause trouble too. One such scenario is a wormhole in space. Wormholes are hypothetical tubes of space that create a shortcut between distant points ("Planes, trains and wormholes", 23 March 1996, p 28). If they exist they can be used as time machines-astronauts who pass through the wormhole and return home via normal space, could get back before they left.
Opening up
Kip Thorne and his colleagues at Caltech have shown that wormholes are theoretically possible, but they need something to keep their throats open. A wormhole is basically an adaptation of a black hole, and has an intense gravitational field. Under normal circumstances, with only ordinary positive energy around, any wormhole that formed would collapse under gravity before anything could pass through it. But because negative energy has negative mass it exerts a negative gravitational force, so it could oppose the pinching effect of positive energy and keep the throat open. If unrestricted negative energy states were possible, they could be used to make a time machine that would enable us to change the past-an absurd prospect in many people's eyes.
But here, the problem is how to make the static negative energy. One way could be to exploit the Casimir effect, discovered by Dutch physicist Hendrick Casimir back in 1948. Two mirrors placed face-to-face trap a slab of quantum vacuum between them. While mirrors reflect real photons of light, they also reflect ghostly virtual photons too. According to quantum theory, every photon is associated with an electromagnetic wave whose wavelength corresponds to the photon's energy. Electromagnetic waves sandwiched between Casimir mirrors form patterns of standing waves, which are restricted to certain values-in the same way that plucked guitar strings play only certain notes. Because of this, many virtual photons that would exist in unbounded empty space cannot be trapped between the mirrors because their wavelengths don't fit. The energy associated with all these "missing" photons is absent from the region between the plates, and the total energy of the quantum vacuum is lower there than in unbounded empty space. In other words, a static negative energy state exists between the plates.
But this would probably not be good enough to keep a wormhole open. You might think that the Casimir effect would offer unrestricted negative energies, if the mirrors have a large enough area and a small enough separation. However, the mirrors themselves are made of normal, positive-energy matter. Moreover, no mirror is perfectly flat, infinitely smooth and perfectly reflecting. It is unlikely that the total energy of the entire Casimir system could ever be negative. Whether or not there are any configurations of matter that can form large, stable, regions of negative energy is an open question still being investigated by theoretical physicists interested in time travel. My hunch is that the same deep principle that protects the second law of thermodynamics from negative energy abuse will intervene here too.
So what is this deep principle and where does it come from? I think that it all stems from the concept of information. All the paradoxical scenarios we have explored relate to the subject of information. Reducing entropy is like creating order, which is equivalent to creating information, and naked singularities are a source of gratuitous information. Wormhole time travel is paradoxical because it enables information to be generated out of thin air.
Consider the professor who travels ahead to the year 2000 and jots down a new theorem from a journal. The professor then returns to 1998 and promptly writes up the theorem in a paper. When published, this work constitutes the very paper that the professor inspected in the year 2000. The question is, where did the information in the new theorem come from? Not from the professor, who read it in the journal. But not from anyone else. It came into existence spontaneously and paradoxically. So if there is a hidden law at work in the Universe banning excessive negative energy, then it would seem to ban information from appearing out of nothing. Since the spontaneous appearance of information is tantamount to a miracle, and deeply irrational, such a principle goes to the very heart of the scientific description of nature.
If this principle does exist, it will continue to protect us from the vagaries of negative energy. But nobody has yet proved for sure that it's impossible to create sustained negative energy fluxes that you could manipulate to make mischief. Until a no-go theorem is proved, the subject will continue to be a fertile ground for the imaginative inventor.
From issue 2126 of New Scientist magazine, 21 March 1998, page 26
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