Time and the Universe
How to build a time machine
A beginner's guide
Ruling time travel in
Does time exist?
Another point of view
The Einstein Connection
Wormhole Engineering
The ultimate proof

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Quantum time waits for no cosmos

THE INTRIGUING notion that time might run backwards when the Universe collapses has run into difficulties. Raymond Laflamme, of the Los Alamos National Laboratory in New Mexico, has carried out a new calculation which suggests that the Universe cannot start out uniform, go through a cycle of expansion and collapse, and end up in a uniform state. It could start out disordered, expand, and then collapse back into disorder. But, since the COBE data show that our Universe was born in a smooth and uniform state, this symmetric possibility cannot be applied to the real Universe.

Physicists have long puzzled over the fact that two distinct "arrows of time" both point in the same direction. In the everyday world, things wear out -- cups fall from tables and break, but broken cups never re- assemble themselves spontaneously. In the expanding Universe at large, the future is the direction of time in which galaxies are further apart.

Many years ago, Thomas Gold suggested that these two arrows might be linked. That would mean that if and when the expansion of the Universe were to reverse, then the everyday arrow of time would also reverse, with broken cups re-assembling themselves.

More recently, these ideas have been extended into quantum physics. There, the arrow of time is linked to the so-called "collapse of the wave function", which happens, for example, when an electron wave moving through a TV tube collapses into a point particle on the screen of the TV. Some researchers have tried to make the quantum description of reality symmetric in time, by including both the original state of the system (the TV tube before the electron passes through) and the final state (the TV tube after the electron has passed through) in one mathematical description.

Murray Gell-Mann and James Hartle recently extended this idea to the whole Universe. They argued that if, as many cosmologists believe likely, the Universe was born in a Big Bang, will expand out for a finite time and then recollapse into a Big Crunch, the time-neutral quantum theory could describe time running backwards in the contracting half of its life.

Unfortunately, Laflamme has now shown that this will not work. He has proved that if there are only small inhomogeneities present in the Big Bang, then they must get larger throughout the lifetime of the Universe, in both the expanding and the contracting phases. "A low entropy Universe at the Big Bang cannot come back to low entropy at the Big Crunch" (Classical and Quantum Gravity, vol 10 p L79). He has found time-asymmetric solutions to the equations -- but only if both Big Bang and Big Crunch are highly disordered, with the Universe more ordered in the middle of its life.

Observations of the cosmic microwave background radiation show that the Universe emerged from the Big Bang in a very smooth and uniform state. This rules out the time-symmetric solutions. The implication is that even if the present expansion of the Universe does reverse, time will not run backwards and broken cups will not start re- assembling themselves.

Is time travel possible?

John and Mary Gribbin

In one of the wildest developments in serious science for decades, researchers from California to Moscow have recently been investigating the possibility of time travel. They are not, as yet, building TARDIS lookalikes in their laboratories; but they have realised that according to the equations of Albert Einstein's general theory of relativity (the best theory of time and space we have), there is nothing in the laws of physics to prevent time travel. It may be extremely difficult to put into practice; but it is not impossible.

It sounds like science fiction, but it is taken so seriously by relativists that some of them have proposed that there must be a law of nature to prevent time travel and thereby prevent paradoxes arising, even though nobody has any idea how such a law would operate. The classic paradox, of course, occurs when a person travels back in time and does something to prevent their own birth -- killing their granny as a baby, in the more gruesome example, or simply making sure their parents never get together, as in Back to the Future. It goes against commonsense, say the sceptics, so there must be a law against it. This is more or less the same argument that was used to prove that space travel is impossible.

So what do Einstein's equations tell us, if pushed to the limit? As you might expect, the possibility of time travel involves those most extreme objects, black holes. And since Einstein's theory is a theory of space and time, it should be no surprise that black holes offer, in principle, a way to travel through space, as well as through time. A simple black hole won't do, though. If such a black hole formed out of a lump of non-rotating material, it would simply sit in space, swallowing up anything that came near it. At the heart of such a black hole there is a point known as a singularity, where space and time cease to exist, and matter is crushed to infinite density. Thirty years ago, Roger Penrose (now of Oxford University) proved that anything which falls into such a black hole must be drawn into the singularity by its gravitational pull, and also crushed out of existence.

But, also in the 1960s, the New Zealand mathematician Roy Kerr found that things are different if the black hole is rotating. A singularity still forms, but in the form of a ring, like the mint with a hole. In principle, it would be possible to dive into such a black hole and through the ring, to emerge in another place and another time. This "Kerr solution" was the first mathematical example of a time machine, but at the time nobody took it seriously. At the time, hardly anybody took the idea of black holes seriously, and interest in the Kerr solution only really developed in the 1970s, after astronmers discovered what seem to be real black holes, both in our own Milky Way Galaxy and in the hearts of other galaxies.

This led to a rash of popular publications claiming, to the annoyance of many relativists, that time travel might be possible. In the 1980s, though, Kip Thorne, of CalTech (one of the world's leading experts in the general theory of relativity), and his colleagues set out to prove once and for all that such nonsense wasn't really allowed by Einstein's equations. They studied the situation from all sides, but were forced to the unwelcome conclusion that there really was nothing in the equations to prevent time travel, provided (and it is a big proviso) you have the technology to manipulate black holes. As well as the Kerr solution, there are other kinds of black hole time machine allowed, including setups graphically described as "wormholes", in which a black hole at one place and time is connected to a black hole in another place and time (or the same place at a different time) through a "throat". Thorne has described some of these possibilities in a recent book, Black Holes and Time Warps (Picador), which is packed with information but far from being an easy read. Now, Michio Kaku, a professor of physics in New York, has come up with a more accessible variation on the theme with his book Hyperspace (Oxford UP), which (unlike Thorne's book) at least includes some discussion of the contribution of researchers such as Robert Heinlein to the study of time travel. The Big Bang, string theory, black holes and baby universes all get a mention here; but it is the chapter on how to build a time machine that makes the most fascinating reading.

"Most scientists, who have not seriously studied Einstein's equations," says Kaku, "dismiss time travel as poppycock". And he then goes on to spell out why the few scientists who have seriously studied Einstein's equations are less dismissive. Our favourite page is the one filled by a diagram which shows the strange family tree of an individual who manages to be both his/her own father and his/her own mother, based on the Heinlein story "All you zombies --". And Kaku's description of a time machine is something fans of Dr Who and H.G. Wells would be happy with:

[It] consists of two chambers, each containing two parallel metal plates. The intense electric fields created between each pair of plates (larger than anything possible with today's technology) rips the fabric of space-time, creating a hole in space that links the two chambers.

Taking advantage of Einstein's special theory of relativity, which says that time runs slow for a moving object, one of the chambers is then taken on a long, fast journey and brought back: Time would pass at different rates at the two ends of the wormhole, [and] anyone falling into one end of the wormhole would be instantly hurled into the past or the future [as they emerge from the other end].

And all this, it is worth spelling out, has been published by serious scientists in respectable journals such as Physical Review Letters (you don't believe us? check out volume 61, page 1446). Although, as you may have noticed, the technology required is awesome, involving taking what amounts to a black hole on a trip through space at a sizeable fraction of the speed of light. We never said it was going to be easy! So how do you get around the paradoxes? The scientists have an answer to that, too. It's obvious, when you think about it; all you have to do is add in a judicious contribution from quantum theory to the time travelling allowed by relativity theory. As long as you are an expert in both theories, you can find a way to avoid the paradoxes.

It works like this. According to one interpretation of quantum physics (there are several interpretations, and nobody knows which one, if any, is "right"), every time a quantum object, such as an electron, is faced with a choice, the world divides to allow it to take every possibility on offer. In the simplest example, the electron may be faced with a wall containing two holes, so that it must go through one hole or the other. The Universe splits so that in one version of reality -- one set of relative dimensions -- it goes through the hole on the left, while in the other it goes through the hole on the right. Pushed to its limits, this interpretation says that the Universe is split into infinitely many copies of itself, variations on a basic theme, in which all possible outcomes of all possible "experiments" must happen somewhere in the "multiverse". So there is, for example, a Universe in which the Labour Party has been in power for 15 years, and is now under threat from a resurgent Tory Party led by vibrant young John Major.

How does this resolve the paradoxes? Like this. Suppose someone did go back in time to murder their granny when she was a little girl. On this multiverse picture, they have slid back to a bifurcation point in history. After killing granny, they move forward in time, but up a different branch of the multiverse. In this branch of reality, they were never born; but there is no paradox, because in he universe next door granny is alive and well, so the murderer is born, and goes back in time to commit the foul deed!

Once again, it sounds like science fiction, and once again science fiction writers have indeed been here before. But this idea of parallel universes and alternative histories as a solution to the time travel paradoxes is also now being taken seriously by some (admittedly, not many) researchers, including David Deutsch, in Oxford. Their research deals with both time, and relative dimensions in space. You could make a nice acronym for that -- TARDIS, perhaps?

Time travel for beginners

John Gribbin

Exactly one hundred years ago, in 1895, H. G. Wells classic story The Time Machine was first published in book form. As befits the subject matter, that was the minus tenth anniversary of the first publication, in 1905, of Albert Einstein's special theory of relativity. It was Einstein, as every schoolchild knows, who first described time as "the fourth dimension" -- and every schoolchild is wrong. It was actually Wells who wrote, in The Time Machine, that "there is no difference between Time and any of the three dimensions of Space, except that our consciousness moves along it".

Since the time of Wells and Einstein, there has been a continuing literary fascination with time travel, and especially with the paradoxes that seem to confront any genuine time traveller (something that Wells neglected to investigate). The classic example is the so- called "granny paradox", where a time traveller inadvertantly causes the death of his granny when she was a small girl, so that the traveller's mother, and therefore the traveller himself, were never born. In which case, he did not go back in time to kill granny . . . and so on.

A less gruesome example was entertainingly provided by the science fiction writer Robert Heinlein in his story By his bootstraps (available in several Heinlein anthologies). The protagonist in the story stumbles on a time travel device brought back to the present by a visitor from the far future. He steals it and sets up home in a deserted stretch of time, constantly worrying about being found by the old man he stole the time machine from -- until one day, many years later, he realises that he is now the old man, and carefully arranges for his younger self to "find" and "steal" the time machine. Such a narcissistic view of time travel is taken to its logical extreme in David Gerrold's The Man Who Folded Himself (Random House, 1973).

Few of the writers of Dr Who have had the imagination actually to use his time machine in this kind of way. It would, after all, make for rather dull viewing if every time the Doctor had been confronted by a disaster he popped into the TARDIS, went back in time and warned his earlier self to steer clear of the looming trouble. But the implications were thoroughly explored for a wide audience in the Back to the Future trilogy, ramming home the point that time travel runs completely counter to common sense. Obviously, time travel must be impossible. Only, common sense is about as reliable a guide to science as the well known "fact" that Einstein came up with the idea of time as the fourth dimension is to history. Sticking with Einstein's own theories, it is hardly common sense that objects get both heavier and shorter the faster they move, or that moving clocks run slow. Yet all of these predictions of relativity theory have been born out many times in experiments, to an impressive number of decimal places. And when you look closely at the general theory of relativity, the best theory of time and space we have, it turns out that there is nothing in it to forbid time travel. The theory implies that time travel may be very difficult, to be sure; but not impossible.

Perhaps inevitably, it was through science fiction that serious scientists finally convinced themselves that time travel could be made to work, by a sufficiently advanced civilization. It happened like this. Carl Sagan, a well known astronomer, had written a novel in which he used the device of travel through a black hole to allow his characters to travel from a point near the Earth to a point near the star Vega. Although he was aware that he was bending the accepted rules of physics, this was, after all, a novel. Nevertheless, as a scientist himself Sagan wanted the science in his story to be as accurate as possible, so he asked Kip Thorne, an established expert in gravitational theory, to check it out and advise on how it might be tweaked up. After looking closely at the non-commonsensical equations, Thorne realised that such a wormhole through spacetime actually could exist as a stable entity within the framework of Einstein's theory.

Sagan gratefully accepted Thorne's modification to his fictional "star gate", and the wormhole duly featured in the novel, Contact, published in 1985. But this was still only presented as a shortcut through space. Neither Sagan nor Thorne realised at first that what they had described would also work as a shortcut through time. Thorne seems never to have given any thought to the time travel possibilities opened up by wormholes until, in December 1986, he went with his student, Mike Morris, to a symposium in Chicago, where one of the other participants casually pointed out to Morris that a wormhole could also be used to travel backwards in time. Thorne tells the story of what happened then in his own book Black Holes and Time Warps (Picador). The key point is that space and time are treated on an essentially equal footing by Einstein's equations -- just as Wells anticipated. So a wormhole that takes a shortcut through spacetime can just as well link two different times as two different places. Indeed, any naturally occurring wormhole would most probably link two different times. As word spread, other physicists who were interested in the exotic implications of pushing Einstein's equations to extremes were encouraged to go public with their own ideas once Thorne was seen to endorse the investigation of time travel, and the work led to the growth of a cottage industry of time travel investigations at the end of the 1980s and in to the 1990s. The bottom line of all this work is that while it is hard to see how any civilization could build a wormhole time machine from scratch, it is much easier to envisage that a naturally occurring wormhole might be adapted to suit the time travelling needs of a sufficiently advanced civilization. "Sufficiently advanced", that is, to be able to travel through space by conventional means, locate black holes, and manipulate them with as much ease as we manipulate the fabric of the Earth itself in projects like the Channel Tunnel.

Even then, there's one snag. It seems you can't use a time machine to go back in time to before the time machine was built. You can go anywhere in the future, and come back to where you started, but no further. Which rather neatly explains why no time travellers from our future have yet visited us -- because the time machine still hasn't been invented!

So where does that leave the paradoxes, and common sense? There is a way out of all the difficulties, but you may not like it. It involves the other great theory of physics in the twentieth century, quantum mechanics, and another favourite idea from science fiction, parallel worlds. These are the "alternative histories", in which, for example, the South won the American Civil War (as in Ward Moore's classic novel Bring the Jubilee), which are envisaged as in some sense lying "alongside" our version of reality.

According to one interpretation of quantum theory (and it has to be said that there are other interpretations), each of these parallel worlds is just as real as our own, and there is an alternative history for every possible outcome of every decision ever made. Alternative histories branch out from decision points, bifurcating endlessly like the branches and twigs of an infinite tree. Bizarre though it sounds, this idea is taken seriously by a handful of scientists (including David Deutsch, of the University of Oxford). And it certainly fixes all the time travel paradoxes.

On this picture, if you go back in time and prevent your own birth it doesn't matter, because by that decision you create a new branch of reality, in which you were never born. When you go forward in time, you move up the new branch and find that you never did exist, in that reality; but since you were still born and built your time machine in the reality next door, there is no paradox.

Hard to believe? Certainly. Counter to common sense? Of course. But the bottom line is that all of this bizarre behaviour is at the very least permitted by the laws of physics, and in some cases is required by those laws. I wonder what Wells would have made of it all.

Time travel back on the agenda

CLAIMS that time travel is impossible in principle have been shown to be in error by an Israeli researcher. Amos Ori, of the Technion-Israel Institute of Technology, in Haifa, has found a flaw in the argument put forward recently by Stephen Hawking, of Cambridge University, claiming to rule out any possibility of time travel.

This is the latest twist in a story that began in the late 1980s, when Kip Thorne and colleagues at the California Institute of Technology suggested that although there might be considerable practical difficulties in constructing a time machine, there is nothing in the laws of physics as understood at present to forbid this. Other researchers tried to find flaws in the arguments of the CalTech team, and pointed in particular to problems in satisfying a requirement known as the "weak energy condition", which says that any real observer should always measure energy distributions that are positive. This rules out some kinds of theoretical time machines, which involve travelling through black holes held open by negative energy stuff.

There are also problems with time machines that involve so-called singularities, points where space and time are crushed out of existence and the laws of physics break down. But Ori has found mathematical descriptions, within the framework of the general theory of relativity, of spacetimes which loop back upon themselves in time, but in which no singularity appears early enough to interfere with the time travel, and the weak energy condition is satisfied (Physical Review Letters, vol 71 p 2517).

"At present," he says, "one should not completely rule out the possibility of constructing a time machine from materials with positive energy densities."

Is time an illusion?

JUST because we perceive time flowing in one direction, does that mean there "really is" a difference between the past and future? The old philosophical question has been re-examined by Huw Price, of the University of Sydney, in the context of quantum mechanics. He concludes that the idea that the past is not influenced by the future is an anthropocentric illusion, a "projection of our own temporal asymmetry". By allowing signals from the future to play a part in determining the outcome of quantum experiments, he can resolve all the puzzles and paradoxes of the quantum world.

This approach has a long (if not entirely respectable) history, but the implications have never been spelled out as clearly as Price does in an article to be published in the journal Mind. It is one of the curiosities of Maxwell's equations, for example, that they allow two sets of solutions for the effect of a moving electric charge, one describing an electromagnetic wave moving out from the particle into the future at the speed of light (a retarded wave) and the other describing waves from the future converging on the particle at the speed of light (advanced waves). The advanced wave solutions have been largely ignored since Maxwell developed his equations in the 19th century, but a few researchers, including Richard Feynman and Fred Hoyle, have considered the implications of taking such waves to be physically real.

More recently, the idea has been investigated in a quantum context by the American researcher John Cramer. He envisages a quantum entity such as an electron that is about to be involved in an interaction (from the everyday point of view) sending out an "offer" wave into the future. The particle that the electron is about to interact with picks up the offer wave, and sends a response echoing backwards in time to the electron. The advanced and retarded waves combine to create a "handshake" between the two particles which, in a sense atemporally, determines the outcome of the interaction at the instant the electron starts to make the offer .

As Price discusses, this kind of approach solves the classic quantum puzzles, such as the electron faced with two holes in a screen, "deciding" which hole to go through. Experiments show that, even though an individual electron can only go through one hole, its behaviour is affected by whether or not the second hole is open or closed. The offer wave goes out through both holes, but the echo comes back only through one hole, the one the electron then goes through. So the handshake process does take account of the presence of both holes, even though the electron only goes through one of them.

Many physicists find such ideas abhorrent, because they run counter to "common sense". They would, for example, encourage speculations like those of Henry Stapp (see Science, XX August), that our own minds can influence things that have already happened. The power of Price's approach, though, is that it offers a framework for understanding how the world can include both forward and backward causation at a fundamental level, but appear to have a unique direction of time from a human perspective.

His argument is complex, but in words it boils down to an argument that the reason why the things we do in the present do not seem to have altered the past is that the past has already taken account of what we are doing! If we decide to do something different, the past already knows -- so "to say that if we suppose the present to be different, while the past remains the same, it will follow that the past is different . . . is untrue, of course, but simply on logical grounds. No physical asymmetry is required to explain it".

For the more mathematically inclined, Price offers a discussion of John Bell's famous inequality, in which two widely separated quantum systems seem to be connected by what Albert Einstein called a "spooky action at a distance". The action at a distance is real, on this picture, and is essentially Cramer's handshaking process. But there is no limitation on free will, according to Price. We are free to make any decisions we please, and to take any actions we choose. The past already knows what those decisions will be, but that does not affect our freedom in making them, and "we shouldn't expect to 'see' backward influence in action," which may be bad news for Stapp, after all. "It is time," says Price, "that this neglected approach [to quantum mechanics] received the attention it so richly deserves."

Time Machines

Paul J. Nahin American Institute of Physics p408 ?? Distributed in UK by OUP; ISBN 0883189356

John Gribbin

TIME TRAVEL has become, if not respectable, then certainly fashionable in some quarters of the physics world over the past decade or so. Much of the blame can be laid at the door of the astronomer Carl Sagan, who was writing a science fiction novel in the summer of 1985, and asked the relativist Kip Thorne, of CalTech, to come up with some plausible sounding scientific mumbo-jumbo to "explain" the literary device of a wormhole through space which could enable his characters to travel between the stars. Encouraged to look at the equations of the general theory of relativity in a new light, Thorne and his colleagues first found that there is nothing in those equations to prevent the existence of such wormholes, and then realised that any tunnel through space is also, potentially, a tunnel through time. The laws of physics do not forbid time travel.

This realisation had two consequences. When Sagan's novel, Contact, appeared in 1986 it contained a passage that read like pure Sf hokum, but which was (although few readers realised it at the time) a serious science factual description of a spacetime wormhole. And as Thorne and his colleagues began to publish scientific papers about time machines and time travel, the spreading ripples have stimulated a cottage industry of similar studies.

Curiously, this anecdote does not feature in Paul Nahin's otherwise remarkably comprehensive account of the fact and fiction of time travel. Nahin is a professor of electrical engineering at the University of New Hampshire, and the author of several published science fiction stories, some dealing with the puzzles and paradoxes of time travel. He tells us how he discovered, and "devoured" science fiction stories at the age of ten, and this book is clearly a labour of love. The approach is scholarly, with 36 pages of footnotes, nine technical (but not overly mathematical) appendices, and a no-holds-barred bibliography. Nahin's style is distinctly more sober than the material he deals with, but what he lacks in sparkle he certainly makes up for in comprehensiveness.

The approach, in line with the author's background, is from the fiction and towards the fact. Old favourites, such as H. G. Wells and Frank Tipler, make their expected appearances, as do less familiar time travel fictions from the nineteenth century (comfortably predating Albert Einstein's theories) and more obscure scientists and philosophers. And, of course, the familiar time travel paradoxes get a thorough airing.

There are, though, two major weaknesses in Nahin's treatment of the science. The lesser is his discussion of black holes, which is weak and sometimes a little confused. Much more importantly, though, he fails to appreciate how the "many worlds" interpretation of quantum mechanics allows a time traveller to go back in time and alter the past without producing problems such as the notorious grandfather paradox. In the conventional version of the paradox, a traveller goes back and murders his grandfather as a young boy, so the traveller could never have been born, so grandfather never died -- and so on. But in the many worlds version (championed today by David Deutsch, of the University of Oxford), the act of killing grandad creates a new reality, so that when the traveller then goes forward in time he is no longer in his own world, but in the universe "next door". This explains, for example, some of the more subtle touches in the "Back to the Future" trilogy of movies, which Nahin comments on while missing their point entirely. But although the book is flawed, it is still welcome. It does not lend itself to being read from front to back like a novel, but is ideal to dip in to and hop around in, like a time traveller dipping in to history. It is also a first class reference book for anyone interested in the Sf side of time travel, and one that will be welcomed by the fans -- at least, they will welcome it when and if it becomes available in paperback at a sensible price.

Hyperspace connections: Black holes, white holes, wormholes

John Gribbin

When astronomer Carl Sagan decided to write a science fiction novel, he needed a fictional device that would allow his characters to travel great distances across the Universe. He knew, of course, that it is impossible to travel faster than light; and he also knew that there was a common convention in science fiction that allowed writers to use the gimmick of a shortcut through "hyperspace" as a means around this problem. But, being a scientist, Sagan wanted something that would seem to be more substantial than a conventional gimmick for his story. Was there any way to dress up the mumbo-jumbo of Sf hyperspace in a cloak of respectable sounding science? Sagan didn't know. He isn't an expert on black holes and general relativity -- his background specialty is planetary studies. But he knew just the person to turn to for some advice on how to make the obviously impossible idea of hyperspace connections through spacetime sound a bit more scientifically plausible in his book Contact.

The man Sagan turned to for advice, in the summer of 1985, was Kip Thorne, at CalTech. Thorne was sufficiently intrigued to set two of his PhD students, Michael Morris and Ulvi Yurtsever, the task of working out some details of the physical behaviour of what the relativists know as "wormholes". At that time, in the mid-1980s, relativists had long been aware that the equations of the general theory provided for the possibility of such hyperspace connections. Indeed, Einstein himself, working at Princeton with Nathan Rosen in the 1930s, had discovered that the equations of relativity -- Karl Schwarzschild's solution to Einstein's equations -- actually represent a black hole as a bridge between two regions of flat spacetime -- an "Einstein-Rosen bridge". A black hole always has two "ends", a property ignored by everyone except a few mathematicians until the mid-1980s. Before Sagan set the ball rolling again, it had seemed that such hyperspace connections had no physical significance and could never, even in principle, be used as shortcuts to travel from one part of the Universe to another. Morris and Yurtsever found that this widely held belief was wrong. By starting out from the mathematical end of the problem, they constructed a spacetime geometry that matched Sagan's requirement of a wormhole that could be physically traversed by human beings. Then they investigated the physics, to see if there was any way in which the known laws of physics could conspire to produce the required geometry. To their own surprise, and the delight of Sagan, they found that there is.

To be sure, the physical requirements seem rather contrived and implausible. But that isn't the point. What matters is that it seems that there is nothing in the laws of physics that forbids travel through wormholes. The science fiction writers were right -- hyperspace connections do, at least in theory, provide a means to travel to far distant regions of the Universe without spending thousands of years pottering along through ordinary space at less than the speed of light. The conclusions reached by the CalTech team duly appeared as the scientifically accurate window dressing in Sagan's novel when it was published in 1986, although few readers can have appreciated that most of the "mumbo-jumbo" was soundly based on the latest discoveries made by mathematical relativists. Since then, the discovery of equations that describe physically permissible, traversable wormholes has led to a booming cottage industry of mathematicians investigating these strange phenomena. It all starts with the Einstein-Rosen bridge.

The Einstein connection

It's one of the intriguing curiosities of the history of science that spacetime wormholes were actually investigated by mathematical relativists in great detail long before anybody took the notion of black holes seriously. As early as 1916, less than a year after Einstein had formulated his equations of the general theory, the Austrian Ludwig Flamm had realised that Schwarzschild's solution to Einstein's equations actually describes a wormhole connecting two regions of flat spacetime -- two universes, or two parts of the same universe. Speculation about the nature of wormholes continued intermittently for decades. What the pioneering relativists did establish, very early on, was that Schwarzschild wormholes provide no means of communicating from one universe to the other.

The problem is that in order to traverse an Einstein-Rosen bridge from one universe to the other, a traveller would have to move faster than light at some stage of the journey. And there is another problem with this kind of wormhole -- it is unstable. If you imagine the "dent" in spacetime made by a large mass such as the Sun, squeezed into a volume only slightly bigger than its corresponding Schwarzschild sphere, you would get an "embedding diagram", like Figure 1. The surprise about the Schwarzschild geometry is that when you shrink the mass down to within its Schwarzschild radius, you don't just get a bottomless pit, as in Figure 2; instead, the bottom of the embedding diagram opens out to make the connection with another region of flat spacetime (Figure 3). But this beautiful, open throat, offering the tantalising prospect of travel between universes, exists for only a tiny fraction of a second before it snaps shut. The wormhole itself does not even exist for long enough for light to cross from one universe to the other. In effect, gravity slams shut the door between universes. This is especially disappointing, because if you ignore the rapid evolution of the wormhole and only look at the geometry corresponding to the instant when the throat is wide open, it seems as if such wormholes might even connect, not separate universes but separate regions of our own Universe. Space may be flat near each mouth of the wormhole, but bent around in a gentle curve, far away from the wormhole, so that the connection really is a shortcut from one part of the Universe to another (Figure 4). If you imagine unfolding this geometry to make the entire Universe flat except in the vicinity of the wormhole mouths, you get something like Figure 5, in which a curved wormhole connects two separate regions of a completely flat Universe -- and don't be fooled by the fact that in this drawing the distance from one mouth to the other through the wormhole itself seems to be longer than the distance from one mouth to the other through ordinary space; in the proper four-dimensional treatment, even such a curved wormhole can still provide a shortcut from A to B.

Or at least, it could if the wormhole stayed open for long enough, and if passage through the wormhole didn't involve travelling at speeds faster than that of light. But this is not the end of the story of hyperspace connections. A simple Schwarzschild black hole has no overall electric charge, and it does not rotate. Intriguingly, adding either electric charge or rotation to a black hole transforms the nature of the singularity, thereby opening the gateway to other universes, and makes the journey possible while travelling at speeds less than that of light.

Adding electric charge to a black hole provides it with a second field of force, in addition to gravity. Because charges with the same sign repel one another, this electric field acts in the opposite sense to gravity, trying to blow the black hole apart, not pulling it more tightly together. Rotation does much the same. There is a force, in either case, that opposes the inward tug of gravity.

Although gravity still tries to slam shut the door opening to other universes, the electric field, or rotation, holds the door open for travellers to get through. But there is still a sense in which this is a one way door; you could not get back to the universe you started from - - you would inevitably emerge into another region of spacetime, usually interpreted as another universe. What goes in one end (the black hole) comes out of the other end (sometimes dubbed a white hole). Turning around to go back the way you came would require travelling faster than light.

Until Sagan made his innocent enquiry about wormholes to Thorne, this was the nearest the mathematicians had come to describing a plausible traversable, macroscopic wormhole.

New speculations, encouraged by Sagan's wishful thinking and developed by the CalTech researchers and others, suggest that it might indeed be possible to construct traversable wormholes artificially, just as Sf writers have been telling us for decades, given a suitably advanced technological civilization.

Wormhole engineering

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 (Figure 7). 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. They 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.

Why time travel is possible

John Gribbin

Physicists have found the law of nature which prevents time travel paradoxes, and thereby permits time travel. It turns out to be the same law that makes sure light travels in straight lines, and which underpins the most straightforward version of quantum theory, developed half a century ago by Richard Feynman.

Relativists have been trying to come to terms with time travel for the past seven years, since Kip Thorne and his colleagues at Caltech discovered -- much to their surprise -- that there is nothing in the laws of physics (specifically, the general theory of relativity) to forbid it. Among several different ways in which the laws allow a time machine to exist, the one that has been most intensively studied mathematically is the "wormhole". This is like a tunnel through space and time, connecting different regions of the Universe -- different spaces and different times. The two "mouths" of the wormhole could be next to each other in space, but separated in time, so that it could literally be used as a time tunnel.

Building such a device would be very difficult -- it would involve manipulating black holes, each with many times the mass of our Sun. But they could conceivably occur naturally, either on this scale or on a microscopic scale.

The worry for physicists is that this raises the possibility of paradoxes, familiar to science fiction fans. For example, a time traveller could go back in time and accidentally (or even deliberately) cause the death of her granny, so that neither the time traveller's mother nor herself was ever born. People are hard to describe mathematically, but the equivalent paradox in the relativists' calculations involves a billiard ball that goes in to one mouth of a wormhole, emerges in the past from the other mouth, and collides with its other self on the way in to the first mouth, so that it is knocked out of the way and never enters the time tunnel at all. But, of course, there are many possible "self consistent" journeys through the tunnel, in which the two versions of the billiard ball never disturb one another.

If time travel really is possible -- and after seven years' intensive study all the evidence says that it is -- there must, it seems, be a law of nature to prevent such paradoxes arising, while permitting the self- consistent journeys through time. Igor Novikov, who holds joint posts at the P. N. Lebedev Institute, in Moscow, and at NORDITA (the Nordic Institute for Theoretical Physics), in Copenhagen, first pointed out the need for a "Principle of Self-consistency" of this kind in 1989 (Soviet Physics JETP, vol 68 p 439). Now, working with a large group of colleagues in Denmark, Canada, Russia and Switzerland, he has found the physical basis for this principle.

It involves something known as the Principle of least action (or Principle of minimal action), and has been known, in one form or another, since the early seventeenth century. It describes the trajectories of things, such as the path of a light ray from A to B, or the flight of a ball tossed through an upper story window. And, it now seems, the trajectory of a billiard ball through a time tunnel. Action, in this sense, is a measure both of the energy involved in traversing the path and the time taken. For light (which is always a special case), this boils down to time alone, so that the principle of least action becomes the principle of least time, which is why light travels in straight lines.

You can see how the principle works when light from a source in air enters a block of glass, where it travels at a slower speed than in air. In order to get from the source A outside the glass to a point B inside the glass in the shortest possible time, the light has to travel in one straight line up to the edge of the glass, then turn through a certain angle and travel in another straight line (at the slower speed) on to point B. Travelling by any other route would take longer.

The action is a property of the whole path, and somehow the light (or "nature") always knows how to choose the cheapest or simplest path to its goal. In a similar fashion, the principle of least action can be used to describe the entire curved path of the ball thrown through a window, once the time taken for the journey is specified. Although the ball can be thrown at different speeds on different trajectories (higher and slower, or flatter and faster) and still go through the window, only trajectories which satisfy the Principle of least action are possible. Novikov and his colleagues have applied the same principle to the "trajectories" of billiard balls around time loops, both with and without the kind of "self collision" that leads to paradoxes. In a mathematical tour de force, they have shown that in both cases only self-consistent solutions to the equations satisfy the principle of least action -- or in their own words, "the whole set of classical trajectories which are globally self-consistent can be directly and simply recovered by imposing the principle of minimal action" (NORDITA Preprint, number 95/49A).

The word "classical" in this connection means that they have not yet tried to include the rules of quantum theory in their calculations. But there is no reason to think that this would alter their conclusions. Feynman, who was entranced by the principle of least action, formulated quantum physics entirely on the basis of it, using what is known as the "sum over histories" or "path integral" formulation, because, like a light ray seemingly sniffing out the best path from A to B, it takes account of all possible trajectories in selecting the most efficient.

So self-consistency is a consequence of the Principle of least action, and nature can be seen to abhor a time travel paradox. Which removes the last objection of physicists to time travel in principle -- and leaves it up to the engineers to get on with the job of building a time machine.