There is still one problem with wormholes for any hyperspace engineers to take careful account of. The simplest calculations suggest that whatever may be going on in the universe outside, the attempted passage of a spaceship through the hole ought to make the star gate slam shut. The problem is that an accelerating object, according to the general theory of relativity, generates those ripples in the fabric of spacetime itself known as gravitational waves. Gravitational radiation itself, travelling ahead of the spaceship and into the black hole at the speed of light, could be amplified to infinite energy as it approaches the singularity inside the black hole, warping spacetime around itself and shutting the door on the advancing spaceship. Even if a natural traversable wormhole exists, it seems to be unstable to the slightest perturbation, including the disturbance caused by any attempt to pass through it.

But Thorne's team found an answer to that for Sagan. After all, the wormholes in Contact are definitely not natural, they are engineered. One of his characters explains:

There is an interior tunnel in the exact Kerr solution of the Einstein Field Equations, but it's unstable. The slightest perturbation would seal it off and convert the tunnel into a physical singularity through which nothing can pass. I have tried to imagine a superior civilization that would control the internal structure of a collapsing star to keep the interior tunnel stable. This is very difficult. The civilization would have to monitor and stabilize the tunnel forever.

But the point is that the trick, although it may be very difficult, is not impossible. It could operate by a process known as negative feedback, in which any disturbance in the spacetime structure of the wormhole creates another disturbance which cancels out the first disturbance. This is the opposite of the familiar positive feedback effect, which leads to a howl from loudspeakers if a microphone that is plugged in to those speakers through an amplifier is placed in front of them. In that case, the noise from the speakers goes into the microphone, gets amplified, comes out of the speakers louder than it was before, gets amplified . . . and so on. Imagine, instead, that the noise coming out of the speakers and into the microphone is analysed by a computer that then produces a sound wave with exactly the opposite characteristics from a second speaker. The two waves would cancel out, producing total silence.

For simple sound waves, this trick can actually be carried out, here on Earth, in the 1990s. Cancelling out more complex noise, like the roar of a football crowd, is not yet possible, but might very well be in a few years time. So it may not be completely farfetched to imagine Sagan's "superior civilization" building a gravitational wave receiver/transmitter system that sits in the throat of a wormhole and can record the disturbances caused by the passage of the spaceship through the wormhole, "playing back" a set of gravitational waves that will exactly cancel out the disturbance, before it can destroy the tunnel.

But where do the wormholes come from in the first place? The way Morris, Yurtsever and Thorne set about the problem posed by Sagan was the opposite of the way everyone before them had thought about black holes. Instead of considering some sort of known object in the Universe, like a dead massive star, or a quasar, and trying to work out what would happen to it, they started out by constructing the mathematical description of a geometry that described a traversable wormhole, and then used the equations of the general theory of relativity to work out what kinds of matter and energy would be associated with such a spacetime. What they found is almost (with hindsight) common sense. Gravity, an attractive force pulling matter together, tends to create singularities and to pinch off the throat of a wormhole. The equations said that in order for an artificial wormhole to be held open, its throat must be threaded by some form of matter, or some form of field, that exerts negative pressure, and has antigravity associated with it.

Now, you might think, remembering your school physics, that this completely rules out the possibility of constructing traversable wormholes. Negative pressure is not something we encounter in everyday life (imagine blowing negative pressure stuff in to a balloon and seeing the balloon deflate as a result). Surely exotic matter cannot exist in the real Universe? But you may be wrong.

Making antigravity

The key to antigravity was found by a Dutch physicist, Hendrik Casimir, as long ago as 1948. Casimir, who was born in The Hague in 1909, worked from 1942 onwards in the research laboratories of the electrical giant Philips, and it was while working there that he suggested what became known as the Casimir effect.

The simplest way to understand the Casimir effect is in terms of two parallel metal plates, placed very close together with nothing in between them (Figure 6). The quantum vacuum is not like the kind of "nothing" physicists imagined the vacuum to be before the quantum era. It seethes with activity, with particle-antiparticle pairs constantly being produced and annihilating one another. Among the particles popping in and out of existence in the quantum vacuum there will be many photons, the particles which carry the electromagnetic force, some of which are the particles of light. Indeed, it is particularly easy for the vacuum to produce virtual photons, partly because a photon is its own antiparticle, and partly because photons have no "rest mass" to worry about, so all the energy that has to be borrowed from quantum uncertainty is the energy of the wave associated with the particular photon. Photons with different energies are associated with electromagnetic waves of different wavelengths, with shorter wavelengths corresponding to greater energy; so another way to think of this electromagnetic aspect of the quantum vacuum is that empty space is filled with an ephemeral sea of electromagnetic waves, with all wavelengths represented.

This irreducible vacuum activity gives the vacuum an energy, but this energy is the same everywhere, and so it cannot be detected or used. Energy can only be used to do work, and thereby make its presence known, if there is a difference in energy from one place to another.

Between two electrically conducting plates, Casimir pointed out, electromagnetic waves would only be able to form certain stable patterns. Waves bouncing around between the two plates would behave like the waves on a plucked guitar string. Such a string can only vibrate in certain ways, to make certain notes -- ones for which the vibrations of the string fit the length of the string in such a way that there are no vibrations at the fixed ends of the string. The allowed vibrations are the fundamental note for a particular length of string, and its harmonics, or overtones. In the same way, only certain wavelengths of radiation can fit into the gap between the two plates of a Casimir experiment . In particular, no photon corresponding to a wavelength greater than the separation between the plates can fit in to the gap

. This means that some of the activity of the vacuum is suppressed in the gap between the plates, while the usual activity goes on outside. The result is that in each cubic centimetre of space there are fewer virtual photons bouncing around between the plates than there are outside, and so the plates feel a force pushing them together. It may sound bizarre, but it is real. Several experiments have been carried out to measure the strength of the Casimir force between two plates, using both flat and curved plates made of various kinds of material. The force has been measured for a range of plate gaps from 1.4 nanometers to 15 nanometers (one nanometer is one billionth of a metre) and exactly matches Casimir's prediction.

In a paper they published in 1987, Morris and Thorne drew attention to such possibilities, and also pointed out that even a straightforward electric or magnetic field threading the wormhole "is right on the borderline of being exotic; if its tension were infinitesimally larger . . . it would satisfy our wormhole-building needs." In the same paper, they concluded that "one should not blithely assume the impossibility of the exotic material that is required for the throat of a traversable wormhole." The two CalTech researchers make the important point that most physicists suffer a failure of imagination when it comes to considering the equations that describe matter and energy under conditions far more extreme than those we encounter here on Earth.

hey highlight this by the example of a course for beginners in general relativity, taught at CalTech in the autumn of 1985, after the first phase of work stimulated by Sagan's enquiry, but before any of this was common knowledge, even among relativists. The students involved were not taught anything specific about wormholes, but they were taught to explore the physical meaning of spacetime metrics. In their exam, they were set a question which led them, step by step, through the mathematical description of the metric corresponding to a wormhole. "It was startling," said Morris and Thorne, "to see how hidebound were the students' imaginations. Most could decipher detailed properties of the metric, but very few actually recognised that it represents a traversable wormhole connecting two different universes."

For those with less hidebound imaginations, there are two remaining problems -- to find a way to make a wormhole large enough for people (and spaceships) to travel through, and to keep the exotic matter out of contact with any such spacefarers. Any prospect of building such a device is far beyond our present capabilities. But, as Morris and Thorne stress, it is not impossible and "we correspondingly cannot now rule out traversable wormholes." It seems to me that there's an analogy here that sets the work of such dreamers as Thorne and Visser in a context that is both helpful and intriguing. Almost exactly 500 years ago, Leonardo da Vinci speculated about the possibility of flying machines.

He designed both helicopters and aircraft with wings, and modern aeronautical engineers say that aircraft built to his designs probably could have flown if Leonardo had had modern engines with which to power them -- even though there was no way in which any engineer of his time could have constructed a powered flying machine capable of carrying a human up into the air. Leonardo could not even dream about the possibilities of jet engines and routine passenger flights at supersonic speeds. Yet Concorde and the jumbo jets operate on the same basic physical principles as the flying machines he designed. In just half a millennium, all his wildest dreams have not only come true, but been surpassed. It might take even more than half a millennium for designs for a traversable wormhole to leave the drawing board; but the laws of physics say that it is possible -- and as Sagan speculates, something like it may already have been done by a civilization more advanced than our own.