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SIXTH LECTURE - THE DIRECTION OF TIME

I n his book, The Go Between, L. P. Hartley wrote, “The past is a foreign country. They do things differently there–but why is the past so different from the future? Why do we remember the past, but not the future?” In other words, why does time go forward? Is this connected with the fact that the universe is expanding?

C, P, T

The laws of physics do not distinguish between the past and the future. More precisely, the laws of physics are unchanged under the combination of operations known as C, P, and T. (C means changing particles for antiparticles. P means taking the mirror image so left and right are swapped for each other. And T means reversing the direction of motion of all particles—in effect, running the motion backward.) The laws of physics that govern the behavior of matter under all normal situations are unchanged under the operations C and P on their own. In other words, life would be just the same for the inhabitants of another planet who were our mirror images and who were made of antimatter. If you meet someone from another planet and he holds out his left hand, don’t shake it. He might be made of antimatter. You would both disappear in a tremendous flash of light. If the laws of physics are unchanged by the combination of operations C and P, and also by the combination C, P, and T, they must also be unchanged under the operation T alone. Yet, there is a big difference between the forward and backward directions of time in ordinary life. Imagine a cup of water falling off a table and breaking in pieces on the floor. If you take a film of this, you can easily tell whether it is being run forward or backward. If you run it backward, you will see the pieces suddenly gather themselves together off the floor and jump back to form a whole cup on the table. You can tell that the film is being run backward because this kind of behavior is never observed in ordinary life. If it were, the crockery manufacturers would go out of business.

THE ARROWS OF TIME

The explanation that is usually given as to why we don’t see broken cups jumping back onto the table is that it is forbidden by the second law of thermodynamics. This says that disorder or entropy always increases with time. In other words, it is Murphy’s Law—things get worse. An intact cup on the table is a state of high order, but a broken cup on the floor is a disordered state. One can therefore go from the whole cup on the table in the past to the broken cup on the floor in the future, but not the other way around.

The increase of disorder or entropy with time is one example of what is called an arrow of time, something that gives a direction to time and distinguishes the past from the future. There are at least three different arrows of time. First, there is the thermodynamic arrow of time—the direction of time in which disorder or entropy increases. Second, there is the psychological arrow of time. This is the direction in which we feel time passes—the direction of time in which we remember the past, but not the future. Third, there is the cosmological arrow of time. This is the direction of time in which the universe is expanding rather than contracting.

I shall argue the the pyschological arrow is determined by the thermodynamic arrow and that these two arrows always point in the same direction. If one makes the no boundary assumption for the universe, they are related to the cosmological arrow of time, though they may not point in the same direction. However, I shall argue that it is only when they agree with the cosmological arrow that there will be intelligent beings who can ask the question: Why does disorder increase in the same direction of time as that in which the universe expands?

THE THERMODYNAMIC ARROW

I shall talk first about the thermodynamic arrow of time. The second law of thermodynamics is based on the fact that there are many more disordered states than there are ordered ones. For example, consider the pieces of a jigsaw in a box. There is one, and only one, arrangement in which the pieces make a complete picture. On the other hand, there are a very large number of arrangements in which the pieces are disordered and don’t make a picture.

Suppose a systems starts out in one of the small number of ordered states. As time goes by, the system will evolve according to the laws of physics and its state will change. At a later time, there is a high probability that it will be in a more disordered state, simply because there are so many more disordered states. Thus, disorder will tend to increase with time if the system obeys an initial condition of high order.

Suppose the pieces of the jigsaw start off in the ordered arrangement in which they form a picture. If you shake the box, the pieces will take up another arrangement. This will probably be a disordered arrangement in which the pieces don’t form a proper picture, simply because there are so many more disordered arrangements. Some groups of pieces may still form parts of the picture, but the more you shake the box, the more likely it is that these groups will get broken up. The pieces will take up a completely jumbled state in which they don’t form any sort of picture. Thus, the disorder of the pieces will probably increase with time if they obey the initial condition that they start in a state of high order.

Suppose, however, that God decided that the universe should finish up at late times in a state of high order but it didn’t matter what state it started in. Then, at early times the universe would probably be in a disordered state, and disorder would decrease with time. You would have broken cups gathering themselves together and jumping back on the table. However, any human beings who observing the cups would be living in a universe in which disorder decreased with time. I shall argue that such beings would have a psychological arrow of time that was backward. That is, they would remember thence at late times and not remember thence at early times.

THE PSYCHOLOGICAL ARROW

It is rather difficult to talk about human memory because we don’t know how the brain works in detail. We do, however, know all about how computer memories work. I shall therefore discuss the psychological arrow of time for computers. I think it is reasonable to assume that the arrow for computers is the same as that for human. If it were not, one could make a killing on the stock exchange by having a computer that would remember tomorrow’s prices.

A computer memory is basically some device that can be in either one of two states. An example would be a superconducting loop of wire. If there is an electric current flowing in the loop, it will continue to flow because there is no resistance. On the other hand, if there is no current, the loop will continue without a current. One can label the two states of the memory “one” and “zero.”

Before an item is recorded in the memory, the memory is in a disordered state with equal probabilities for one and zero. After the memory interacts with the system to be remembered, it will definitely be in one state or the other, according to the state of the system. Thus, the memory passes from a disordered state to an ordered one. However, in order to make sure that the memory is in the right state, it is necessary to use a certain amount of energy. This energy is dissipated as heat and increases the amount of disorder in the universe. One can show that this increase of disorder is greater than the increase in the order of the memory. Thus, when a computer records an item in memory, the total amount of disorder in the universe goes up.

The direction of time in which a computer remembers the past is the same as that in which disorder increases. This means that our subjective sense of the direction of time, the psychological arrow of time, is determined by the thermodynamic arrow of time. This makes the second law of thermodynamics almost trivial. Disorder increases with time because we measure time in the direction in which disorder increases. You can’t have a safer bet than that.

THE BOUNDARY CONDITIONS

OF THE UNIVERSE

But why should the universe be in a state of high order at one end of time, the end that we call the past? Why was it not in a state of complete disorder at all times? After all, this might seem more probable. And why is the direction of time in which disorder increases the same as that in which the universe expands? One possible answer is that God simply chose that the universe should be in a smooth and ordered state at the beginning of the expansion phase. We should not try to understand why or question His reasons because the beginning of the universe was the work of God. But the whole history of the universe can be said to be the work of God.

It appears that the universe evolves according to well-defined laws. These laws may or may not be ordained by God, but it seems that we can discover and understand them. Is it, therefore, unreasonable to hope that the same or similar laws may also hold at the beginning of the universe? In the classical theory of general relativity, the beginning of the universe has to be a singularity of infinite density in space-time curvature. Under such conditions, all the known laws of physics would break down. Thus, one could not use them to predict how the universe would begin.

The universe could have started out in a very smooth and ordered state. This would have led to well-defined thermodynamic and cosmological arrows of time, like we observe. But it could equally well have started out in a very lumpy and disordered state. In this case, the universe would already be in a state of complete disorder, so disorder could not increase with time. It would either stay constant, in which case there would be no well-defined thermodynamic arrow of time, or it would decrease, in which case the thermodynamic arrow of time would point in the opposite direction to the cosmological arrow.

Neither of these possibilities would agree with what we observe.

As I mentioned, the classical theory of general relativity predicts that the universe should begin with a singularity where the curvature of space-time is infinite. In fact, this means that classical general relativity predicts its own downfall. When the curvature of space-time becomes large, quantum gravitational effects will become important and the classical theory will cease to be a good description of the universe. One has to use the quantum theory of gravity to understand how the universe began.

In a quantum theory of gravity, one considers all possible histories of the universe. Associated with each history, there are a couple of numbers. One represents the size of a wave and the other the face of the wave, that is, whether the wave is at a crest or a trough. The probability of the universe having a particular property is given by adding up the waves for all the histories with that property. The histories would be curved spaces that would represent the evolution of the universe in time. One would still have to say how the possible histories of the universe would behave at the boundary of space–time in the past. We do not and cannot know the boundary conditions of the universe in the past. However, one could avoid this difficulty if the boundary condition of the universe is that it has no boundary. In other words, all the possible histories are finite in extent but have no boundaries, edges, or singularities. They are like the surface of the Earth, but with two more dimensions. In that case, the beginning of time would be a regular smooth point of space–time. This means that the universe would have begun its expansion in a very smooth and ordered state. It could not have been completely uniform because that would violate the uncertainty principle of quantum theory. There had to be small fluctuations in the density and velocities of particles. The no boundary condition, however, would imply that these fluctuations were as small as they could be, consistent with the uncertainty principle.

The universe would have started off with a period of exponential or “inflationary” expansion. In this, it would have increased its size by a very large factor. During this expansion, the density fluctuations would have remained small at first, but later would have started to grow. Regions in which the density was slightly higher than average would have had their expansion slowed down by the gravitational attraction of the extra mass. Eventually, such regions would stop expanding, and would collapse to form galaxies, stars, and beings like us.

The universe would have started in a smooth and ordered state and would become lumpy and disordered as time went on. This would explain the existence of the thermodynamic arrow of time. The universe would start in a state of high order and would become more disordered with time. As I showed earlier, the psychological arrow of time points in the same direction as the thermodynamic arrow. Our subjective sense of time would therefore be that in which the universe is expanding, rather than the opposite direction, in which it would be contracting.

DOES THE ARROW OF TIME REVERSE?

But what would happen if and when the universe stopped expanding and began to contract again? Would the thermodynamic arrow reverse and disorder begin to decrease with time? This would lead to all sorts of science–fiction–like possibilities for people who survived from the expanding to the contracting phase. Would they see broken cups gathering themselves together off the floor and jumping back on the table? Would they be able to remember tomorrow’s prices and make a fortune on the stock market?

It might seem a bit academic to worry about what would happen when the universe collapses again, as it will not start to contract for at least another ten thousand million years. But there is a quicker way to find out what will happen: Jump into a black hole. The collapse of a star to form a black hole is rather like the later stages of the collapse of the whole universe. Thus, if disorder were to decrease in the contracting phase of the universe, one might also expect it to decrease inside a black hole. So perhaps an astronaut who fell into a black hole would be able to make money at roulette by remembering where the ball went before he placed his bet. Unfortunately, however, he would not have long to play before he was turned to spaghetti by the very strong gravitational fields. Nor would he be able to let us know about the reversal of the thermodynamic arrow, or even bank his winnings, because he would be trapped behind the event horizon of the black hole.

At first, I believed that disorder would decrease when the universe recollapsed. This was because I thought that the universe had to return to a smooth and ordered state when it became small again. This would have meant that the contracting phase was like the time reverse of the expanding phase. People in the contracting phase would live their lives backward. They would die before they were born and would get younger as the universe contracted. This idea is attractive because it would mean a nice symmetry between the expanding and contracting phases. However, one cannot adopt it on its own, independent of other ideas about the universe. The question is: Is it implied by the no boundary condition or is it inconsistent with that condition?

As I mentioned, I thought at first that the no boundary condition did indeed imply that disorder would decrease in the contracting phase. This was based on work on a simple model of the universe in which the collapsing phase looked like the time reverse of the expanding phase. However, a colleague of mine, Don Page, pointed out that the no boundary condition did not require the contracting phase necessarily to be the time reverse of the expanding phase. Further, one of my students, Raymond Laflamme, found that in a slightly more complicated model, the collapse of the universe was very different from the expansion. I realized that I had made a mistake. In fact, the no boundary condition implied that disorder would continue to increase during the contraction. The thermodynamic and psychological arrows of time would not reverse when the universe begins to recontract or inside black holes.

What should you do when you find you have made a mistake like that? Some people, like Eddington, never admit that they are wrong. They continue to find new, and often mutually inconsistent, arguments to support their case. Others claim to have never really supported the incorrect view in the first place or, if they did, it was only to show that it was inconsistent. I could give a large number of examples of this, but I won’t because it would make me too unpopular. It seems to me much better and less confusing if you admit in print that you were wrong. A good example of this was Einstein, who said that the cosmological constant, which he introduced when he was trying to make a static model of the universe, was the biggest mistake of his life.