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.
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.
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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).FIG-mg17144501.GIF
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 Figure 2, by the first cyclic journey when no time has elapsed, before which no backward time-travel can take place. But in the space-time beyond this boundary cyclic trips through time are possible. Hence we must face all the paradoxes associated with time-travel, or re-evaluate our familiar ideas of causality and time evolution.FIG-mg17144502.GIF
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 (see Figure 3 a and b). With the right initial velocity, a ball can be knocked into the wormhole and travel back in time to knock itself into the hole, in a consistent way. With the wrong velocity, the ball emerging from the time machine fails to knock itself into the hole; this is excluded by the principle of consistency. But more is needed. Some initial conditions imply more than one distinct, consistent solution, as in Figure 3 c and d. A ball can pass undisturbed between the wormhole mouths, or it can be deflected into the hole, travel back in time, and emerge to cause the deflection. Both are consistent outcomes of the same initial conditions. The researchers from Caltech and Moscow resolve this paradox by treating the balls according to the laws of quantum mechanics. In this light, both outcomes occur with some probability, but the probability distribution for the balls evolves in a unique and consistent way. While much more remains to be understood, it appears that time travel may be more physically reasonable than everyone had thought.FIG-mg17144503.GIF
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 Figure 4. Such a geometry represents a ‘baby universe’ (the interior of the wormhole) branching off and rejoining the larger Universe. He calculated that the contribution of these to physical processes was equivalent to a simple interaction between elementary particles and baby universes. He concluded that such an interaction would cause a loss of information (down the wormholes) from quantum systems in the larger Universe. The interaction would also shift the values of the constants in the original theory describing the elementary particles, by amounts that depended on the internal states of all possible baby universes. This means that even if we had a comprehensive ‘theory of everything’, it would be useless. Nothing could be calculated from it without first making an infinite number of measurements to determine all the shifts caused by the baby universes.FIG-mg17144504.GIF
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.
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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.