by Paul Davies
What can we say about the physics underlying these differently shaped paths? Since Newton, it has been known that a body will accelerate only if a force acts on it, so curved paths in a spacetime diagram demand physical forces. Path 4, for example implies a push–pull alternating force to make the body zigzag back and forth.
Gravity is one among a number of physical forces. Einstein's great insight was to spot the significance that gravity differs from other forces in a crucial respect: it affects all bodies
Caption
A spacetime diagram
equally. There is a famous story about Galileo dropping heavy and light bodies from the leaning tower of Pisa to convince sceptics that the objects will hit the ground together. Translated into a spacetime diagram, this means that when gravity is the accelerating force, all bodies (light, heavy, hot, cold, living, dead…) will follow the same path. The same would not be true of, for example, an electric field, which would accelerate charged particles, but leave uncharged particles to follow straight spacetime paths.
Einstein reasoned that if the effect of gravity on moving bodies is the same for all, it is better to represent the gravitational field not as a force, but as a geometrical property of spacetime. This mysterious-sounding idea is easily understood. In this figure I have depicted the two-dimensional sheet containing my spacetime diagram, but it is no longer flat – it is curved or warped. It should be clear that distorting the sheet in the manner shown mimics the effect of curving the paths.
(see Curved spacetime on page 59)
In other words, the wiggles in a path may be achieved either by drawing a wiggly line on a flat sheet, or a ‘straight’ line on a curvy sheet. Straight here means ‘straightest’, i.e. the shortest path between two points across a curved sheet. Einstein proposed that, in the case of gravity, it is better to think of the gravitational field geometrically, rather than as a force acting in ‘flat’ spacetime. Of course, to do this properly entails extending the notion of spacetime curvature from the two-dimensional sheetshown to four dimensions (three of space and one of time), but this is straightforward mathematically.
Wormholes: portals to another universe?
To serve as a time machine, a wormhole has to be traversable: the time traveller has to be able to pass through it and emerge intact. This is impossible with the Schwarzschild wormhole, as it pinches-off before anything can get through it. However, that entire model assumed empty space and exact spherical sym-metry. Suppose we relax these assumptions?
In the 1960s physicists and mathematicians began studying the properties of spinning black holes. These bulge around the waist in the same manner as rotating planets because of centrifugal force. Now centrifugal force acts to oppose gravity. The reason the Schwarzschild wormhole pinches-off so fast is because of the intense gravity within it. With rotation present the pinching effect is ameliorated, raising the prospect that the wormhole throat might stay open long enough for something or someone – to get through. Forty years ago it looked like spinning black holes would provide traversable wormholes, at least from the idealized mathematical models in use at the time. There was much discussion about what might await an astronaut who tumbled into a spinning black hole and emerged in another universe.
On closer inspection, several problems surfaced with this
Caption
Curved spacetime
scenario. The first is practical. Any astronaut leaping into a black hole risks being mangled by the intense gravitational forces. To see why, imagine yourself jumping out of an airplane feet first. Because the Earth's gravity diminishes with height, your feet, being closest to the ground, will be pulled down a little more strongly than your head, so your body is stretched slightly lengthwise; at the same time, your shoulders get squeezed together, because each shoulder is pulled towards the centre of the Earth, and the Earth's curvature means they try to fall on converging paths. Effectively you are stretched and crushed (slightly) at the same time.
It was just such stretch-and-squeeze gravitational forces that ripped comet Shoemaker–Levy 9 into fragments before it plunged into Jupiter in 1994. Near a solar-mass black hole the effects would be so strong they'd spaghettify an astronaut in pretty short order. Spaghettification is less likely if the hole is bigger. You could just about survive falling to the surface of a black hole with 10,000 solar masses. A supermassive black hole a billion kilometres across would be no problem, but such an object would have a mass equal to a small galaxy – not a very practical proposition for accessing another universe.
A more serious problem with traversing a spinning black hole is that the idealized model containing the wormhole ignores the effects of any matter or radiation that might be around. Not only can the astronaut fall in, but so can anything
else that comes along, such as cosmic rays and starlight. The intense gravity of the hole enormously boosts the energy of these as it sucks them in, forming an impenetrable wall across the throat of the wormhole. The gravity of the wall would probably cause the wormhole to collapse, sealing it off with a singularity.
And that's not all. The centrifugal force of a rotating black hole does combat the inward pull of gravity, but not so much that a singularity is prevented. On p. 42 I discussed how an exactly spherical ball of matter would implode to a single point of infinite density. A spinning ball wouldn't be spherical, because of the bulge around the equator. Instead, it collapses to form a ring singularity inside the hole. If we ignore the above-mentioned problems for a moment, an astronaut could fall into the hole, miss the singularity, and come out in another universe.
The idea that an astronaut might inspect a singularity and live to tell the tale strikes horror into the heart of the physicist. Taken at face value, a black hole singularity is an entity with infinite density and infinite space curvature. As such, space and time cannot be continued through it. Singularities are therefore edges or boundaries to space and/or time. There is literally nothing beyond them: they are places where physical objects and influences could leave or enter the universe. A chunk of matter hitting a singularity and ceasing to exist is bad enough, but what about a chunk of matter that spontaneously out of a singularity?
The idea of having a region of space from which anything at all might emerge without cause and without warning is pretty startling. It would represent nothing less than a breakdown of the rational order of the cosmos. For this reason, Sir Roger Penrose proposed a law of nature to ban such unwelcome invasions. He conjectured that singularities are so obscene they will always be decently clothed by black holes. In that way, nobody in the outside universe would be able to see a singularity. No uncaused physical influences could emanate into the wider cosmos to wreak havoc. Never would an edge to spacetime be exposed to public gaze.
(see Sir Roger Penrose on page 64)
Penrose called this ban his cosmic censorship hypothesis:
Let there be no naked singularities!
And this is where the trouble lies concerning spinning black holes. If you could fall into one, zoom past the ring singularity, and come out again in another universe, then all the gremlins from the singularity could come out with you. The singularity would be naked to the other universe, in defiance of cosmic censorship.
Now it has to be said that this isn't a watertight no-go the-orem for traversing spinning black holes. Nobody has proved
Caption
Sir Roger Penrose
the cosmic censorship hypothesis; it might be wrong. Also, a singularity could be a mathematical fiction: perhaps the gene-al theory of relativity, or even the concepts of space and time, break down before a singularity forms. Still, for all the above reasons, using a spinning black hole as a gateway to another universe looks decidedly suspect If the aim is to find a safely traversable wormhole, something else is called for – something to combat gravity with more oomph.
How to make a traversable wormhole
The concept of time travel began with a work of science fiction, and it has remained firmly i
n the realm of fiction until recently. Curiously, the trigger that transformed time travel into the business of serious science was another work of fiction. In the mid 1980s the astrophysicist Carl Sagan wrote his novel Contact, which later became a Hollywood movie starring Jodie Foster. The story is not actually about time travel, but concerns a radio message received from an advanced alien civilization. The message contains the design of a machine to create a wormhole in space between Earth and the star Vega, 26 light years away. The wormhole is then used by a team of scientists to travel and meet the aliens. Sagan employed the wormhole idea as a fictional device to get around that old bugbear of sci-fi – the finite speed of light. In Contact, the scientists reach Vega in only a few minutes.
(see Carl Sagan on page 66)
Caption
Carl Sagan
Sagan's wormhole differs in one important detail from the ones I discussed above. The black hole wormholes were conjectured to be a gateway to another universe. Sagan's wormhole is a tunnel linking two points in the same universe. Sagan gave scant details about how such a wormhole was to be constructed. In the movie version Jodie Foster climbs aboard a capsule and gets dropped into what looks like a gigantic kitchen mixer, whereupon she zooms through a narrow tunnel and emerges part way across the galaxy. It looks great, but is it feasible? Sagan was intrigued to know whether using a wormhole as a short cut through interstellar space had any scientific credibility, so he approached his friend the theoretical physicist Kip Thorne at the California Institute of Technology.
Thorne and his colleagues agreed to check out what would be needed to make Sagan's vision a reality. They did this by adopting a sort of reverse engineering approach to gravitational theory. Normally a physicist considers a gravitating object – a star, say – and uses the general theory of relativity to work out the gravitational field it generates, which in turn predicts how the space near the object will curve.
For this project, Thorne started by writing down the answer first. He knew the sort of geometry of space needed – something shaped like a wormhole with two spherical mouths. But it had to be a benign wormhole – one that stayed open long enough to allow Jodie Foster to get through, and not to rip her apart with gravitational forces or incinerate her with surfaces of infinite energy. Obviously, the sort of wormholes I discussed above wouldn't do. Then Thorne asked what type of matter would be needed to generate this benign wormhole.
It soon became clear that any familiar form of matter (water, diamond, hydrogen, light, neutrinos…) was out of the question. In all cases it would make the throat of the wormhole collapse before anything could traverse it. Clearly, some exotic form of matter would be needed.
It's not hard to figure out what. If a wormhole is traversable, it must have an exit as well as an entrance. In that case, it should be possible to shine light through it. The reason a black hole has no way out is because its gravity bends light inwards, trapping it and focusing it down on to a singularity. As the wormhole allows light to come out the other end, somewhere inside it light would have to be defocused, i.e. bent outwards.
Thorne realized the way to do this was to use some sort of antigravity. This is no surprise. Something powerful is needed to shore up the wormhole, to combat gravity's inexorable tendency to crush the wormhole and pinch it off in a singularity. Antigravitating matter is the answer. But does it exist?
Well, it has long existed in folklore. Levitation is an ancient myth, and features in many world religions and mystical beliefs. Antigravity remains a favourite idea among UFO buffs for alien spacecraft propulsion. It also attracts a variety of
independent thinkers, wacky inventors and visionary venture capitalists, fixated by the dream of nullifying the Earth's gravity and floating to the stars without the need for rockets. Antigravity also crops up in science fiction: H. G. Wells envisaged a sort of gravity shield (called cavorite) in The First Men in the Moon.
The first appearance of antigravity in science was provided by Einstein. In 1917, he doctored his own general theory of relativity to incorporate a repulsive form of gravitation. He did this to produce a model of the universe. At that time, nobody knew the universe was expanding. Einstein was puzzled (as was Newton) about how the universe could be static when the only truly cosmic force is gravitation, which is universally attractive. So he added an extra term to his gravitational field equations to describe a type of antigravity. By balancing the attractive force of normal gravity with the repulsive force of antigravity, a static universe might result.
Once Einstein discovered the universe was not static but expanding, he abandoned the repulsive force, calling it the biggest blunder of his life. Ironically, he could have been right after all. Although antigravity may not be needed for a static universe any more, the force may yet exist, and recent astronomical evidence suggests that, in fact, it does. However, in its pervasive cosmic form, Einstein's antigravity is far too feeble to help make a traversable wormhole.
Antigravity crops up in other branches of physics too, though only under unusual circumstances. The basic idea is easy to grasp. In normal matter, mass is the source of its gravity. Because of the link between mass and energy (E = mc2) all forms of energy gravitate. If what you want is antigravity, this can be produced by negative energy:
Positive energy gravitates, negative energy antigravitates.
At first sight, negative energy sounds as mysterious as a negative lunch. Surely, either you have some lunch or you don't? How can you have less than no lunch?
The answer lies with the definition of zero energy. Because energy gravitates, zero energy must correspond to a state with no gravitational field whatsoever. In the general theory of relativity that condition implies no spacewarps and no timewarps – spacetime is precisely flat. So if we can engineer a physical situation in which the energy is less than such a zero energy state, the energy will be negative, and the state will antigravi-tate.
Imagine a box made of ordinary matter filled with enough negative energy to make the total mass-energy negative. Would it then fly upwards instead of falling downwards? Unfortunately not. True, the box would feel an upward gravitational force, but because its mass is negative, it would actually move in the opposite direction, i.e. downward! So negative energy falls just like positive energy. It can't be used to soar to the stars.
However, the gravity field that the negative energy itself creates is certainly repulsive. A ball of normal matter placed near the box would be accelerated away from it. If the Earth were made of negative energy, we should all be shot into space.
In chapter 3 I shall explain how negative energy states can be created, but for now let's assume that some suitable exotic matter is available, and is stuffed into the throat of the wormhole. If it antigravitates powerfully enough it will stop the throat collapsing and allow light and perhaps even astronauts to pass through. The final form of the wormhole can then be represented as a flexible two-dimensional sheet, but this time the sheet is bent right round until the two ends come close together and then get connected through the wormhole. In this manner, points A and B that lie far apart in space - perhaps many light years – can be joined by a short wormhole, exactly as in Contact.
Bending the sheet around in the manner shown seems like drastically curving a large portion of the universe until it is almost folded back on itself, a task that would tax even a supercivilization. In fact, the representation is misleading in this respect. It is true that gravity curves space, but the
folding-over curvature here is not a gravitational spacewarp. The act of folding the sheet, or even rolling it into a cylinder, does not affect the geometrical properties within the sheet itself.
To see this, imagine drawing geometrical figures on the sheet – triangles and circles, say. When the sheet is simply folded over, there is no stretching or shrinking. Nothing changes within the surface – all the angles remain the same, squares stay squares etc. Contrast this with trying to paste the sheet on to the surface of a sphere. In that case you
would have to stretch or crease the sheet, thereby altering angles, deforming squares, etc. (The reverse is also true; just think of the distortions involved with Mercator's projection of a map of the Earth.) Cylindrical surfaces have no intrinsic curvature, but spherical surfaces do. Similar statements can be made about their three-dimensional equivalents.
When it comes to the wormhole depicted here, there is no intrinsic curvature in the region of ordinary or ‘outer’ space between A and B. In spite of the bending back, the geometry there remains as it was – more or less flat – with points in space staying the same distance apart, angles unchanged, and so on. You would not know by inspecting the geometry of this ordinary region that there is a wormhole linking two widely separated places.
The research by Thorne and his colleagues didn't uncover
anything fundamentally wrong with the idea of a traversable wormhole, so long as some form of exotic matter could be deployed. And the matter needed was not too exotic – some known physical systems are believed actually to possess it, albeit in tiny amounts. This was a very significant discovery. It did not prove that traversable wormholes definitely could exist, but it didn't rule them out either.
Already that was exciting enough. More was to follow, though. Once the researchers digested the possibility of a wormhole in space, it dawned on them that if one were somehow made, it could also serve as a time machine As with a black hole, the gravitational field of a wormhole can act as a means to reach the future. However the wormhole can do more: it can also be used to go to the past. By passing through the wormhole from A to B, it is possible to go backwards in time. And by returning rapidly across ordinary space, you could get back to A before you left. At last physicists had found a plausible way to travel both back and forth in time. But how might a wormhole time machine be made?
(see Using a wormhole as a time machine on page 75)
