by Paul Davies
How to travel faster than light
A decade after publication of van Stockum's paper, the eminent Austrian logician Kurt Gödel produced another solution of Einstein's equations of general relativity that contained time loops. Gödel was then working at Princeton's Institute for Advanced Study alongside Einstein. He discovered that if the entire universe were rotating, it would be possible to find orbits in space that spiralled back into the past. In fact, Gödel showed that in such a universe you could depart from Earth and travel to anywhere and any when you liked.
Gödel's mathematical model was intended as a curiosity only, not a serious proposal. Even in the 1940s astronomers had good reason to doubt that the universe as a whole is spinning, although individual galaxies are. Today, measurements of the heat radiation left over from the big bang can be used to determine with great accuracy any cosmic rotation, and none is observed. Despite its manifestly artificial nature, Gödel's model universe seriously disturbed Einstein, who admitted that
he'd worried about backwards time travel ever since he first formulated the general theory of relativity.
What is the secret that lets rotation open a gateway to the past? A pointer lies in the famous limerick at the start of this chapter. As I explained in chapter 1 (see p. 31), given the no-faster-than-light rule, the time order of events that can be connected by light signals is never in doubt. But if faster-than-light motion were to be allowed, causal chaos would ensue. It would be possible to reverse cause and effect or to put it another way, to move in such a way as to change ‘before’ and ‘after’ for some pairs of spatially separated events. It is but a small step from this reversal of time order actually to visit the past. In other words, ‘faster than light’ can mean ‘backwards in time’.
But there is a subtlety. Rotation does not enable an astronaut, or even a particle of matter, to break the light barrier as such, but it does affect the motion of light itself. According to the general theory of relativity, if a massive body (e.g. a cylinder, or a black hole) is spinning, it will act like a vortex in space and drag any passing light beam around with it. Normally, this dragging phenomenon is tiny, but if the body is heavy enough and spins fast enough, light can be caught up in the twisted gravitational field and pulled right round in a loop. If the intrepid astronaut ventures into this gravitational whirlpool he or she, too, will be caught up and dragged around. At all times, the astronaut travels round the spinning body slower than the light in her vicinity. But because the light itself is being whirled around, in effect the astronaut is achieving Miss Bright status relative to a distant observer. Locally, the light barrier isn't broken, but globally – considering the entire circuit – the astronaut seems to reach superluminal speed. In the 1970s, the physicist Frank Tipler showed that a superdense cylinder spinning on its axis at half the speed of light could serve in this manner as a time machine, although the scenario he outlined is not physically realistic.
Although the recent ideas for time machines don't require rotation they, too, involve a way to effectively outpace light. The most popular proposal is the ‘wormhole’. As we shall see, this is a sculpture in the structure of space that provides a short cut between two widely separated places. By travelling through the wormhole, an astronaut would be able to go from A to B before light had had a chance to get there the long way, i.e. across normal space.
So what, exactly, is a wormhole? To introduce the concept, I must first explain that better-known object, the black hole.
How to make a black hole
Black holes are certainly newsworthy, and most people are now familiar with the basic idea: dense, dark bodies in space that suck in everything around them. Small black holes a few kilometres wide form when large stars burn out, and collapse
under their own weight. Some become neutron stars, others black holes. Our sun is likely to escape either fate – being of fairly modest mass – and will probably end its days as a white dwarf. Some astronomers think the galaxy could be peppered with myriad black holes, the dead remnants of giant stars born billions of years before the solar system.
Much larger black holes lurk at the centres of galaxies. Our own Milky Way seems to harbour an object with about a million solar masses. Other galaxies are known that host central black holes 1,000 times bigger still. Sometimes material spiralling into these supermassive objects releases vast quantities of energy, creating violent disturbances. Intense radiation is given off, and jets of material spew forth at near the speed of light.
Why the term ‘black hole‘? The name was coined by the physicist John Wheeler in the late 1960s. He chose it carefully to encapsulate two defining properties: blackness and emptiness. Let me take them in reverse order. A neutron star contains what is probably the stiffest material in the universe, but even that is not completely incompressible. If it could be further squeezed, the pull of gravity would become overwhelming, and the star would collapse completely. That happens inside large stars that run out of fuel and can no longer sustain their internal pressure. The cores abruptly implode, in a fraction of a second, leaving behind a region of empty space – hence, ‘hole’. (Well, in practice the surrounding region isn't totally empty because of the ragged remains of the rest of the star. But that stuff soon either gets blown away or sucked in.)
Mystery surrounds the fate of the imploding matter. Where does it go? For now, think of the star as a precisely spherical ball. Imagine it being progressively shrunk, with all the matter retained. As I explained on p. 20, the smaller the ball gets, the higher the gravity becomes at the surface. At some point gravity would be so great that no known matter could withstand it and the ball would collapse. Because it is an exact sphere, and nothing disturbs the symmetry, the ball must remain a sphere as it implodes. In other words, all the material must move towards the precise geometrical centre. The smaller the sphere gets, the more powerful is the pull of gravity inside it and the faster it shrinks.
Where does it all end? Under these conditions it can end only with the entire contents of the ball concentrated at a single point at the centre. Obviously, this dot of matter would have infinite density and its gravity would also be infinite there. Mathematicians refer to such an entity as a singularity. When infinity looms in a physical theory it is an alarm signal, suggesting something drastic happens, but in this case nobody is quite sure what. Shortly I shall have more to say about singularities, but for now it is enough to note that whatever is the ultimate fate of the ball, it does not affect the gravitational field outside it. The ball's gravity doesn't go away just because it has imploded: the collapsed object still retains its mass. Like the fading grin of the Cheshire cat, the ball's erstwhile existence leaves an imprint in the surrounding universe in the form of its ferocious gravitational field.
Let me now turn to the other property that characterizes black holes – their blackness.
In the last chapter 1 explained how gravity slows time; the stronger the gravity, the bigger the timewarp. Think what happens to time at the surface of the ball as it contracts. The slowdown factor rises as the radius gets less. When the ball approaches a certain critical radius – about 3 kilometres for an object containing one solar mass – the timewarp becomes infinite, which is to say that the march of time at the surface of the ball grinds to a halt, relative to, say, Earth time. A clock on the surface of the ball would appear from afar to be frozen into total immobility.
Of course, no man-made clock could withstand the huge forces involved here, but light waves can be regarded as a type of clock; their undulations mimic the swing of a pendulum. So the light from a shrinking star gets lower and lower in frequency as the escalating timewarp retards its oscillations. Translated into colour, the light from the contracting ball gets redder and redder, until it fades away completely, like the dying embers of a cooling fire. Eventually, the last of the light from the star gets out; after that, all is black. Thus, the region of space around the collapsed object is both black and empty – hence, black hole.
The way I've described it makes the ball's disappearance as viewed from Earth – by our twin Sally, say – seem rather slow-going. In fact, for a solar mass star the fade-out time is as short as a few hundredths of a millisecond. Sally would see the core of the star vanish in an instant (assuming she could see the core anyway), and the region of space where the ball of matter used to be would be occupied by a featureless black sphere – a black hole.
An observer – Sam, let's say – standing on the surface of the contracting star, and unlucky enough to accompany it into the black hole, would experience events very differently. No temporal slowdown for him. (Remember, time is relative.) In fact, in this case the two accounts given by Sally and Sam are infinitely different, because of the infinite timewarp. Sally sees the star collapse to a 3-kilometre black ball and freeze – permanently – while Sam sees the entire star shrink to nothing in the twinkling of an eye. As far as Sam is concerned, in the fraction of a second it takes the star to implode across the critical radius, all of eternity will have passed by in the outside universe.
The formation of an infinite timewarp around an imploding ball of matter leads to an arresting conclusion:
A black hole is a one-way journey to nowhere.
You can't fall into one and come out again, because the region inside the black hole is beyond the end of time as far as the outside universe is concerned. If you did somehow manage to emerge from a black hole, you would have to come out before you fell in. Which is another way of saying that you would be projected back in time.
So there's a clue here. A black hole has an entrance but no exit; it is a one-way fast track to the end of time.
What if there existed something like a black hole, but with an exit as well as an entrance – a wormhole? Maybe it could be used to reach the past.
Wormholes and curved space
In order to explain what a wormhole is I need to describe how gravity affects space as well as time. The theory of relativity demands that both space and time are elastic. That means space can stretch, too. In fact, the expansion of the universe is more or less just that: the galaxies move apart because space between them stretches. Because space has three dimensions, however, its elasticity can produce a broader range of distortions than simple stretching and shrinking; space can also be curved.
So what does curved space mean? At school we learn the rules of geometry compiled by Euclid. To give a simple example, the three angles of a triangle add up to two right angles (180 degrees). Euclid's rules apply to geometrical shapes drawn on blackboards and exercise books, which are flat. But on a curved surface, the rules of geometry are different. For example, on a spherical surface such as the Earth it is possible to draw a triangle with three right angles (270 degrees). The apex of the triangle is at the North Pole and the opposite side lies along the Equator. Navigators are familiar with the fact that on the Earth's surface different rules of geometry are needed.
Similar rules can apply in three-dimensional space if it is appropriately warped. To give an example, imagine drawing a flat triangle around the sun. What would the angles add up to? Most people would guess 180 degrees. The theory of relativity predicts the answer should be a little bit more than 180 degrees, because the sun's gravity curves the space around it. The effect is very small – a few seconds of arc for a triangle that just encloses the sun, still less for a bigger one. Nevertheless, the distortion can be measured, not by literally drawing a triangle, of course, but by observing light rays or radar signals passing close to the sun. Sometimes the effect is described by saying that the sun's gravity bends light rays, but it is more accurate to think of space itself as bent, with light following the shortest path through the curved geometry.
(see A spacewarp round the sun on page 48)
The curvature of space around the sun is barely discernible. A really big spacewarp demands a much stronger gravitational
Caption
A spacewarp round the sun
field, such as that of an entire galaxy containing hundreds of billions of stars. Sometimes, by chance, one galaxy will line up exactly in front of another as seen from Earth. In this case the intervening galaxy's gravitational field serves as a type of lens, bending the light from the more distant galaxy and focusing it, producing a halo effect known as an Einstein ring. Another place where very large spacewarps occur is near a black hole. At the surface of a solar-mass black hole gravity is about 100 billion times stronger than at the surface of the sun, and space is spectacularly warped.
(see Einstein ring on page 50)
One way to picture elastic space near a massive object is by analogy with a rubber sheet. The sheet is laid horizontally with a pit in the middle made by placing a heavy ball there. This might represent the sun's spacewarp, for example. A smaller ball rolled across the sheet will, in negotiating its way across the warped surface, travel in a curved path around the pit, just as the Earth travels in a curved orbit around the sun. Of course, the sheet here represents only two space dimensions; in reality the sun's gravity curves space in all three dimensions, but that is hard to show in a picture.
Imagine that instead of the sun, there is a black hole. The rubber now curves away dramatically, into an apparently bottomless pit. A lot of thought has been given to what lies at the bottom of the pit, or whether there is any bottom at all It may end in a singularity – an edge of spacetime (see p. 62). As
(see Black hole spacewarp on page 52)
Caption
Einstein ring
early as 1916 the Austrian physicist Ludwig Flamm studied the geometry of space in and around what we would now call a black hole, although that term wasn't coined until 1968, by John Wheeler.
Later, in 1935, Einstein and his collaborator Nathan Rosen revisited this topic, and the shape shown on page 53 became known as the Einstein-Rosen bridge. Today, a structure of this general form is called a wormhole, and the narrow bit in the middle is termed the ‘throat’. Far from the black hole the sheet is nearly flat, because gravity is weak there. As the hole is approached, the curvature rises, and the sheet falls away into a pit. But instead of plunging down for ever, it opens out again to form a second surface underneath.
(see Einstein–Rosen bridge on page 53)
This is unexpected. What are we to make of the lower part? What is the significance of the region of space down there? The best way of describing the lower surface is as ‘another universe’, although a proper understanding of these features didn't come until 1960, with the work of George Szekeres in Australia and Martin Kruskal in the United States.
Intriguing though it is, the ‘other universe’ that links to ours via the wormhole shouldn't be taken too seriously, being the product of an idealized mathematical model. It traces back to 1916, and a solution of Einstein's equations discovered by Karl Schwarzschild to represent the gravitational field in the empty space outside a star. It doesn't apply to the interior of the star.
Caption
Black hole spacewarp
Caption
Einstein–Rosen bridge
If you try patching together Schwarzschild's solution with another describing the star's innards, the whole bottom half of the wormhole is eliminated. Even when you allow the star to collapse to a singularity, you don't create the bottom sheet.
The only way the entire wormhole could make physical sense is if somehow the universe was made like this, with the worm-hole embedded in it, care of Mother Nature. Even then, there is a complication, because the wormhole doesn't just sit there: it changes with time. Initially, the two universes are detached. Then they connect together at a single point corresponding to a singularity, where the curvature of space is infinite. This is just like the singularity that results when a ball collapses to a point of infinite density, only in this case there is no collapsing ball, only empty space.
From this singular beginning, the wormhole throat then opens out, but only for a limited duration, after which it closes up again, and the two universes disconnect. Crucially, th
is sequence happens so fast that nothing can get through the wormhole before the throat pinches off. Not even light can pass from one universe to another, so an observer in our universe wouldn't even be able to see the ‘other universe’, let alone visit it. This makes the existence of the ‘other universe’ rather hypothetical, since the two universes – top and bottom sheets of the wormhole – couldn't affect each other in any way. Any astronaut stupid enough to jump into the black hole would end up hitting the singularity at the centre and being totally obliterated.
Curved spacetime
I've described what it means for space and time individually to be warped by gravity, but it's more accurate to consider them together in a unified description.
The concept of ‘spacetime’ is not especially hard to envisage, and in this diagram time is drawn vertically and space horizontally. For ease of representation, I have shown only one space dimension. The white lines on the diagram are the paths in spacetime of physical objects. Path 1 is a body that remains at rest as time goes along. Path 2 is a body that moves at constant speed to the right. Path 3 shows a body accelerating to the right. The wiggly line labelled 4 depicts a body that moves back and forth.
(see A spacetime diagram on page 56)
