SOURCE: "Einstein's Dream," in Black Holes and Baby Universes, and Other Essays, Bantam Books, 1993, pp. 69-83.

[Hawking is an English physicist, author, and educator renowned for his significant contributions to contemporary scientific theory. In the following essay, which was originally presented as a lecture at the Paradigm Session of the NTT Data Communications Systems Corporation in Tokyo in 1991, he describes relativity and quantum mechanics and explains their implications for contemporary science and culture.]

In the early years of the twentieth century, two new theories completely changed the way we think about space and time, and about reality itself. More than seventy-five years later, we are still working out their implications and trying to combine them in a unified theory that will describe everything in the universe. The two theories are the general theory of relativity and quantum mechanics. The general theory of relativity deals with space and time and how they are curved or warped on a large scale by the matter and energy in the universe. Quantum mechanics, on the other hand, deals with very small scales. Included in it is what is called the uncertainty principle, which states that one can never precisely measure the position and the velocity of a particle at the same time; the more accurately you can measure one, the less accurately you can measure the other. There is always an element of uncertainty or chance, and this affects the behavior of matter on a small scale in a fundamental way. Einstein was almost single-handedly responsible for general relativity, and he played an important part in the development of quantum mechanics. His feelings about the latter are summed up in the phrase "God does not play dice." But all the evidence indicates that God is an inveterate gambler and that He throws the dice on every possible occasion.

In this essay, I will try to convey the basic ideas behind these two theories, and why Einstein was so unhappy about quantum mechanics. I shall also describe some of the remarkable things that seem to happen when one tries to combine the two theories. These indicate that time itself had a beginning about fifteen billion years ago and that it may come to an end at some point in the future. Yet in another kind of time, the universe has no boundary. It is neither created nor destroyed. It just is.

I shall start with the theory of relativity. National laws hold only within one country, but the laws of physics are the same in Britain, the United States, and Japan. They are also the same on Mars and in the Andromeda galaxy. Not only that, the laws are the same at no matter what speed you are moving. The laws are the same on a bullet train or on a jet airplane as they are for someone standing in one place. In fact, of course, even someone who is stationary on the earth is moving at about 18.6 miles (30 kilometers) a second around the sun. The sun is also moving at several hundred kilometers a second around the galaxy, and so on. Yet all this motion makes no difference to the laws of physics; they are the same for all observers.

This independence of the speed of the system was first discovered by Galileo, who developed the laws of motion of objects like cannonballs or planets. However, a problem arose when people tried to extend this independence of the speed of the observer to the laws that govern the motion of light. It had been discovered in the eighteenth century that light does not travel instantaneously from source to observer; rather, it goes at a certain speed, about 186,000 miles (300,000 kilometers) a second. But what was this speed relative to? It seemed that there had to be some medium throughout space through which the light traveled. This medium was called the ether. The idea was that light waves traveled at a speed of 186,000 miles a second through the ether, which meant that an observer who was at rest relative to the ether would measure the speed of light to be about 186,000 miles a second, but an observer who was moving through the ether would measure a higher or lower speed. In particular, it was believed that the speed of light ought to change as the earth moves through the ether on its orbit around the sun. However, in 1887 a careful experiment carried out by Michelson and Morley showed that the speed of light was always the same. No matter what speed the observer was moving at, he would always measure the speed of light at 186,000 miles a second.

How can this be true? How can observers moving at different speeds all measure light at the same speed? The answer is they can't, not if our normal ideas of space and time hold true. However, in a famous paper written in 1905, Einstein pointed out that such observers could all measure the same speed of light if they abandoned the idea of a universal time. Instead, they would each have their own individual time, as measured by a clock each carried with him. The times measured by these different clocks would agree almost exactly if they were moving slowly with respect to each other—but the times measured by different clocks would differ significantly if the clocks were moving at high speed. This effect has actually been observed by comparing a clock on the ground with one in a commercial airliner; the clock in the airliner runs slightly slow when compared to the stationary clock. However, for normal speeds of travel, the differences between the rates of clocks are very small. You would have to fly around the world four hundred million times to add one second to your life; but your life would be reduced by more than that by all those airline meals.

How does having their own individual time cause people traveling at different speeds to measure the same speed of light? The speed of a pulse of light is the distance it travels between two events, divided by the time interval between the events. (An event in this sense is something that takes place at a single point in space, at a specified point in time.) People moving at different speeds will not agree on the distance between two events. For example, if I measure a car traveling down the highway, I might think it had moved only one kilometer, but to someone on the sun, it would have moved about 1,800 kilometers, because the earth would have moved while the car was going down the road. Because people moving at different speeds measure different distances between events, they must also measure different intervals of time if they are to agree on the speed of light.

Einstein's original theory of relativity, which he proposed in the paper written in 1905, is what we now call the special theory of relativity. It describes how objects move through space and time. It shows that time is not a universal quantity which exists on its own, separate from space. Rather, future and past are just directions, like up and down, left and right, forward and back, in something called space-time. You can only go in the future direction in time, but you can go at a bit of an angle to it. That is why time can pass at different rates.

The special theory of relativity combined time with space, but space and time were still a fixed background in which events happened. You could choose to move on different paths through space-time, but nothing you could do would modify the background of space and time. However, all this was changed when Einstein formulated the general theory of relativity in 1915. He had the revolutionary idea that gravity was not just a force that operated in a fixed background of space-time. Instead, gravity was a distortion of space-time, caused by the mass and energy in it. Objects like cannonballs and planets try to move on a straight line through space-time, but because space-time is curved, warped, rather than flat, their paths appear to be bent. The earth is trying to move on a straight line through space-time, but the curvature of space-time produced by the mass of the sun causes it to go in a circle around the sun. Similarly, light tries to travel in a straight line, but the curvature of space-time near the sun causes the light from distant stars to be bent if it passes near the sun. Normally, one is not able to see stars in the sky that are in almost the same direction as the sun. During an eclipse, however, when most of the sun's light is blocked off by the moon, one can observe the light from those stars. Einstein produced his general theory of relativity during the First World War, when conditions were not suitable for scientific observations, but immediately after the war a British expedition observed the eclipse of 1919 and confirmed the predictions of general relativity: Space-time is not flat, but is curved by the matter and energy in it.

This was Einstein's greatest triumph. His discovery completely transformed the way we think about space and time. They were no longer a passive background in which events took place. No longer could we think of space and time as running on forever, unaffected by what happened in the universe. Instead, they were now dynamic quantities that influenced and were influenced by events that took place in them.

An important property of mass and energy is that they are always positive. This is why gravity always attracts bodies toward each other. For example, the gravity of the earth attracts us to it even on opposite sides of the world. That is why people in Australia don't fall off the world. Similarly, the gravity of the sun keeps the planets in orbit around it and stops the earth from shooting off into the darkness of interstellar space. According to general relativity, the fact that mass is always positive means that space-time is curved back on itself, like the surface of the earth. If mass had been negative, space-time would have been curved the other way, like the surface of a saddle. This positive curvature of space-time, which reflects the fact that gravity is attractive, was seen as a great problem by Einstein. It was then widely believed that the universe was static, yet if space, and particularly time, were curved back on themselves, how could the universe continue forever in more or less the same state as it is at the present time?

Einstein's original equations of general relativity predicted that the universe was either expanding or contracting. Einstein therefore added a further term to the equations that relate the mass and energy in the universe to the curvature of space-time. This so-called cosmological term had a repulsive gravitational effect. It was thus possible to balance the attraction of the matter with the repulsion of the cosmological term. In other words, the negative curvature of space-time produced by the cosmological term could cancel the positive curvature of space-time produced by the mass and energy in the universe. In this way, one could obtain a model of the universe that continued forever in the same state. Had Einstein stuck to his original equations, without the cosmological term, he would have predicted that the universe was either expanding or contracting. As it was, no one thought the universe was changing with time until 1929, when Edwin Hubble discovered that distant galaxies are moving away from us. The universe is expanding. Einstein later called the cosmological term "the greatest mistake of my life."

But with or without the cosmological term, the fact that matter caused space-time to curve in on itself remained a problem, though it was not generally recognized as such. What it meant was that matter could curve a region in on itself so much that it would effectively cut itself off from the rest of the universe. The region would become what is called a black hole. Objects could fall into the black hole, but nothing could escape. To get out, they would need to travel faster than the speed of light, which is not allowed by the theory of relativity. Thus the matter inside the black hole would be trapped and would collapse to some unknown state of very high density.

Einstein was deeply disturbed by the implications of this collapse, and he refused to believe that it happened. But Robert Oppenheimer showed in 1939 that an old star of more than twice the mass of the sun would inevitably collapse when it had exhausted all its nuclear fuel. Then war intervened, Oppenheimer became involved in the atom bomb project, and he lost interest in gravitational collapse. Other scientists were more concerned with physics that could be studied on earth. They distrusted predictions about the far reaches of the universe because it did not seem they could be tested by observation. In the 1960s, however, the great improvement in the range and quality of astronomical observations led to new interest in gravitational collapse and in the early universe. Exactly what Einstein's general theory of relativity predicted in these situations remained unclear until Roger Penrose and I proved a number of theorems. These showed that the fact that space-time was curved in on itself implied that there would be singularities, places where space-time had a beginning or an end. It would have had a beginning in the big bang, about fifteen billion years ago, and it would come to an end for a star that collapsed and for anything that fell into the black hole the collapsing star left behind.

The fact that Einstein's general theory of relativity turned out to predict singularities led to a crisis in physics. The equations of general relativity, which relate the curvature of space-time with the distribution of mass and energy, cannot be defined as a singularity. This means that general relativity cannot predict what comes out of a singularity. In particular, general relativity cannot predict how the universe should begin at the big bang. Thus, general relativity is not a complete theory. It needs an added ingredient in order to determine how the universe should begin and what should happen when matter collapses under its own gravity.

The necessary extra ingredient seems to be quantum mechanics. In 1905, the same year he wrote his paper on the special theory of relativity, Einstein also wrote about a phenomenon called the photoelectric effect. It had been observed that when light fell on certain metals, charged particles were given off. The puzzling thing was that if the intensity of the light was reduced, the number of particles emitted diminished, but the speed with which each particle was emitted remained the same. Einstein showed this could be explained if light came not in continuously variable amounts, as everyone had assumed, but rather in packets of a certain size. The idea of light coming only in packets, called quanta, had been introduced a few years earlier by the German physicist Max Planck. It is a bit like saying one can't buy sugar loose in a supermarket but only in kilogram bags. Planck used the idea of quanta to explain why a red-hot piece of metal doesn't give off an infinite amount of heat; but he regarded quanta simply as a theoretical trick, one that didn't correspond to anything in physical reality. Einstein's paper showed that you could directly observe individual quanta. Each particle emitted corresponded to one quantum of light hitting the metal. It was widely recognized to be a very important contribution to quantum theory, and it won him the Nobel Prize in 1922. (He should have won a Nobel Prize for general relativity, but the idea that space and time were curved was still regarded as too speculative and controversial, so they gave him a prize for the photoelectric effect instead—not that it was not worth the prize on its own account.)

The full implications of the photoelectric effect were not realized until 1925, when Werner Heisenberg pointed out that it made it impossible to measure the position of a particle exactly. To see where a particle is, you have to shine light on it. But Einstein had shown that you couldn't use a very small amount of light; you had to use at least one packet, or quantum. This packet of light would disturb the particle and cause it to move at a speed in some direction. The more accurately you wanted to measure the position of the particle, the greater the energy of the packet you would have to use and thus the more it would disturb the particle. However you tried to measure the particle, the uncertainty in its position, times the uncertainty in its speed, would always be greater than a certain minimum amount.

This uncertainty principle of Heisenberg showed that one could not measure the state of a system exactly, so one could not predict exactly what it would do in the future. All one could do is predict the probabilities of different outcomes. It was this element of chance, or randomness, that so disturbed Einstein. He refused to believe that physical laws should not make a definite, unambiguous prediction for what would happen. But however one expresses it, all the evidence is that the quantum phenomenon and the uncertainty principle are unavoidable and that they occur in every branch of physics.

Einstein's general relativity is what is called a classical theory; that is, it does not incorporate the uncertainty principle. One therefore has to find a new theory that combines general relativity with the uncertainty principle. In most situations, the difference between this new theory and classical general relativity will be very small. This is because, as noted earlier, the uncertainty predicted by quantum effects is only on very small scales, while general relativity deals with the structure of space-time on very large scales. However, the singularity theorems that Roger Penrose and I proved show that space-time will become highly curved on very small scales. The effects of the uncertainty principle will then become very important and seem to point to some remarkable results.

Part of Einstein's problems with quantum mechanics and the uncertainty principle arose from the fact that he used the ordinary, commonsense notion that a system has a definite history. A particle is either in one place or in another. It can't be half in one and half in another. Similarly, an event like the landing of astronauts on the moon either has taken place or it hasn't. It cannot have half-taken place. It's like the fact that you can't be slightly dead or slightly pregnant. You either are or you aren't. But if a system has a single definite history, the uncertainty principle leads to all sorts of paradoxes, like the particles being in two places at once or astronauts being only half on the moon.

An elegant way to avoid these paradoxes that had so troubled Einstein was put forward by the American physicist Richard Feynman. Feynman became well known in 1948 for work on the quantum theory of light. He was awarded the Nobel Prize in 1965 with another American, Julian Schwinger, and the Japanese physicist Shinichiro Tomonaga. But he was a physicist's physicist, in the same tradition as Einstein. He hated pomp and humbug, and he resigned from the National Academy of Sciences because he found that they spent most of their time deciding which other scientists should be admitted to the Academy. Feynman, who died in 1988, is remembered for his many contributions to theoretical physics. One of these was the diagrams that bear his name, which are the basis of almost every calculation in particle physics. But an even more important contribution was his concept of a sum over histories. The idea was that a system didn't have just a single history in space-time, as one would normally assume it did in a classical nonquantum theory. Rather, it had every possible history. Consider, for example, a particle that is at a point A at a certain time. Normally, one would assume that the particle will move on a straight line away from A. However, according to the sum over histories, it can move on any path that starts at A. It is like what happens when you place a drop of ink on a piece of blotting paper. The particles of ink will spread through the blotting paper along every possible path. Even if you block the straight line between two points by putting a cut in the paper, the ink will get around the corner.

Associated with each path or history of the particle will be a number that depends on the shape of the path. The probability of the particle traveling from A to B is given by adding up the numbers associated with all the paths that take the particle from A to B. For most paths, the number associated with the path will nearly cancel out the numbers from paths that are close by. Thus, they will make little contribution to the probability of the particle's going from A to B. But the numbers from the straight paths will add up with the numbers from paths that are almost straight. Thus the main contribution to the probability will come from paths that are straight or almost straight. That is why the track a particle makes when going through a bubble chamber looks almost straight. But if you put something like a wall with a slit in it in the way of the particle, the particle paths can spread out beyond the slit. There can be a high probability of finding the particle away from the direct line through the slit.

In 1973 I began investigating what effect the uncertainty principle would have on a particle in the curved spacetime near a black hole. Remarkably enough, I found that the black hole would not be completely black. The uncertainty principle would allow particles and radiation to leak out of the black hole at a steady rate. This result came as a complete surprise to me and everyone else, and it was greeted with general disbelief. But with hindsight, it ought to have been obvious. A black hole is a region of space from which it is impossible to escape if one is traveling at less than the speed of light. But the Feynman sum over histories says that particles can take any path through space-time. Thus it is possible for a particle to travel faster than light. The probability is low for it to move a long distance at more than the speed of light, but it can go faster than light for just far enough to get out of the black hole, and then go slower than light. In this way, the uncertainty principle allows particles to escape from what was thought to be the ultimate prison, a black hole. The probability of a particle getting out of a black hole of the mass of the sun would be very low because the particle would have to travel faster than light for several kilometers. But there might be very much smaller black holes, which were formed in the early universe. These primordial black holes could be less than the size of the nucleus of an atom, yet their mass could be a billion tons, the mass of Mount Fuji. They could be emitting as much energy as a large power station. If only we could find one of these little black holes and harness its energy! Unfortunately, there don't seem to be many around in the universe.

The prediction of radiation from black holes was the first nontrivial result of combining Einstein's general relativity with the quantum principle. It showed that gravitational collapse was not as much of a dead end as it had appeared to be. The particles in a black hole need not have an end of their histories at a singularity. Instead, they could escape from the black hole and continue their histories outside. Maybe the quantum principle would mean that one could also avoid the histories having a beginning in time, a point of creation, at the big bang.

This is a much more difficult question to answer, because it involves applying the quantum principle to the structure of time and space themselves and not just to particle paths in a given space-time background. What one needs is a way of doing the sum over histories not just for particles but for the whole fabric of space and time as well. We don't know yet how to do this summation properly, but we do know certain features it should have. One of these is that it is easier to do the sum if one deals with histories in what is called imaginary time rather than in ordinary, real time. Imaginary time is a difficult concept to grasp, and it is probably the one that has caused the greatest problems for readers of my book. I have also been criticized fiercely by philosophers for using imaginary time. How can imaginary time have anything to do with the real universe? I think these philosophers have not learned the lessons of history. It was once considered obvious that the earth was flat and that the sun went around the earth, yet since the time of Copernicus and Galileo, we have had to adjust to the idea that the earth is round and that it goes around the sun. Similarly, it was long obvious that time went at the same rate for every observer, but since Einstein, we have had to accept that time goes at different rates for different observers. It also seemed obvious that the universe had a unique history, yet since the discovery of quantum mechanics, we have had to consider the universe as having every possible history. I want to suggest that the idea of imaginary time is something that we will also have to come to accept. It is an intellectual leap of the same order as believing that the world is round. I think that imaginary time will come to seem as natural as a round earth does now. There are not many Flat Earthers left in the educated world.

You can think of ordinary, real time as a horizontal line, going from left to right. Early times are on the left, and late times are on the right. But you can also consider another direction of time, up and down the page. This is the so-called imaginary direction of time, at right angles to real time.

What is the point of introducing the concept of imaginary time? Why doesn't one just stick to the ordinary, real time that we understand? The reason is that, as noted earlier, matter and energy tend to make space-time curve in on itself. In the real time direction, this inevitably leads to singularities, places where space-time comes to an end. At the singularities, the equations of physics cannot be defined; thus one cannot predict what will happen. But the imaginary time direction is at right angles to real time. This means that it behaves in a similar way to the three directions that correspond to moving in space. The curvature of space-time caused by the matter in the universe can then lead to the three space directions and the imaginary time direction meeting up around the back. They would form a closed surface, like the surface of the earth. The three space directions and imaginary time would form a space-time that was closed in on itself, without boundaries or edges. It wouldn't have any point that could be called a beginning or end, any more than the surface of the earth has a beginning or end.

In 1983, Jim Hartle and I proposed that the sum over histories for the universe should not be taken over histories in real time. Rather, it should be taken over histories in imaginary time that were closed in on themselves, like the surface of the earth. Because these histories didn't have any singularities or any beginning or end, what happened in them would be determined entirely by the laws of physics. This means that what happened in imaginary time could be calculated. And if you know the history of the universe in imaginary time, you can calculate how it behaves in real time. In this way, you could hope to get a complete unified theory, one that would predict everything in the universe. Einstein spent the later years of his life looking for such a theory. He did not find one because he distrusted quantum mechanics. He was not prepared to admit that the universe could have many alternative histories, as in the sum over histories. We still do not know how to do the sum over histories properly for the universe, but we can be fairly sure that it will involve imaginary time and the idea of space-time closing up on itself. I think these concepts will come to seem as natural to the next generation as the idea that the world is round. Imaginary time is already a commonplace of science fiction. But it is more than science fiction or a mathematical trick. It is something that shapes the universe we live in.