The Theory of Everything: The Origin and Fate of the Universe - Stephen Hawking (2002)
Chapter 6. SIXTH LECTURE - THE DIRECTION OF TIME
T H E D I R E C T I O N O F T I M EIn his book, The Go Between, L. P. Hartley wrote, “The past is a foreigncountry. They do things differently there-but why is the past so differentfrom the future? Why do we remember the past, but not the future?” In otherwords, why does time go forward? Is this connected with the fact that the uni-verse is expanding?
C, P, T
The laws of physics do not distinguish between the past and the future. Moreprecisely, the laws of physics are unchanged under the combination of opera-tions known as C, P, and T. (C means changing particles for antiparticles. Pmeans 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, run-ning the motion backward.) The laws of physics that govern the behavior ofmatter under all normal situations are unchanged under the operations C andP on their own. In other words, life would be just the same for the inhabitantsof another planet who were our mirror images and who were made of antimat-ter. 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 ina tremendous flash of light. If the laws of physics are unchanged by the com-bination of operations C and P, and also by the combination C, P, and T, theymust also be unchanged under the operation T alone. Yet, there is a big differ-ence 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 orbackward. If you run it backward, you will see the pieces suddenly gather them-selves 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 behavioris never observed in ordinary life. If it were, the crockery manufacturers wouldgo out of business.
THE ARROWS OF TIME
The explanation that is usually given as to why we don’t see broken cups jump-ing back onto the table is that it is forbidden by the second law of thermody-namics. This says that disorder or entropy always increases with time. In otherwords, it is Murphy’s Law-things get worse. An intact cup on the table is astate of high order, but a broken cup on the floor is a disordered state. One cantherefore go from the whole cup on the table in the past to the broken cup onthe floor in the future, but not the other way around.
The increase of disorder or entropy with time is one example of what is calledan arrow of time, something that gives a direction to time and distinguishes thepast 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 dis-order 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 inwhich we remember the past, but not the future. Third, there is the cosmolog-ical arrow of time. This is the direction of time in which the universe isexpanding rather than contracting.
I shall argue the the pyschological arrow is determined by the thermodynamicarrow and that these two arrows always point in the same direction. If one makesthe no boundary assumption for the universe, they are related to the cosmolog-ical 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 thatthere will be intelligent beings who can ask the question: Why does disorderincrease 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 ofthermodynamics is based on the fact that there are many more disorderedstates than there are ordered ones. For example, consider the pieces of a jigsawin a box. There is one, and only one, arrangement in which the pieces make acomplete picture. On the other hand, there are a very large number of arrange-ments 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. Astime goes by, the system will evolve according to the laws of physics and itsstate will change. At a later time, there is a high probability that it will be ina more disordered state, simply because there are so many more disorderedstates. Thus, disorder will tend to increase with time if the system obeys an ini-tial condition of high order.
Suppose the pieces of the jigsaw start off in the ordered arrangement in whichthey form a picture. If you shake the box, the pieces will take up anotherarrangement. This will probably be a disordered arrangement in which thepieces don’t form a proper picture, simply because there are so many moredisordered arrangements. Some groups of pieces may still form parts of thepicture, but the more you shake the box, the more likely it is that these groupswill get broken up. The pieces will take up a completely jumbled state in whichthey don’t form any sort of picture. Thus, the disorder of the pieces willprobably increase with time if they obey the initial condition that they start ina state of high order.
Suppose, however, that God decided that the universe should finish up at latetimes 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 disor-der would decrease with time. You would have broken cups gathering them-selves together and jumping back on the table. However, any human beingswho observing the cups would be living in a universe in which disorderdecreased with time. I shall argue that such beings would have a psychologicalarrow of time that was backward. That is, they would remember thence at latetimes and not remember thence at early times.
THE PSYCHOLOGICAL ARROW
It is rather difficult to talk about human memory because we don’t know howthe brain works in detail. We do, however, know all about how computermemories work. I shall therefore discuss the psychological arrow of time forcomputers. I think it is reasonable to assume that the arrow for computers isthe same as that for human. If it were not, one could make a killing on thestock 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 twostates. An example would be a superconducting loop of wire. If there is an elec-tric current flowing in the loop, it will continue to flow because there is noresistance. On the other hand, if there is no current, the loop will continuewithout 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 statewith equal probabilities for one and zero. After the memory interacts with thesystem to be remembered, it will definitely be in one state or the other, accord-ing to the state of the system. Thus, the memory passes from a disordered stateto an ordered one. However, in order to make sure that the memory is in theright state, it is necessary to use a certain amount of energy. This energy is dis-sipated as heat and increases the amount of disorder in the universe. One canshow that this increase of disorder is greater than the increase in the order ofthe memory. Thus, when a computer records an item in memory, the totalamount of disorder in the universe goes up.
The direction of time in which a computer remembers the past is the same asthat in which disorder increases. This means that our subjective sense of thedirection of time, the psychological arrow of time, is determined by the ther-modynamic arrow of time. This makes the second law of thermodynamicsalmost trivial. Disorder increases with time because we measure time in thedirection 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, theend that we call the past? Why was it not in a state of complete disorder at alltimes? After all, this might seem more probable. And why is the direction oftime in which disorder increases the same as that in which the universeexpands? One possible answer is that God simply chose that the universeshould be in a smooth and ordered state at the beginning of the expansionphase. We should not try to understand why or question His reasons becausethe beginning of the universe was the work of God. But the whole history ofthe universe can be said to be the work of God.
It appears that the universe evolves according to well-defined laws. These lawsmay or may not be ordained by God, but it seems that we can discover andunderstand them. Is it, therefore, unreasonable to hope that the same or simi-lar laws may also hold at the beginning of the universe? In the classicaltheory of general relativity, the beginning of the universe has to be a singular-ity of infinite density in space-time curvature. Under such conditions, all theknown laws of physics would break down. Thus, one could not use them topredict how the universe would begin.
The universe could have started out in a very smooth and ordered state. Thiswould have led to well-defined thermodynamic and cosmological arrows oftime, like we observe. But it could equally well have started out in a verylumpy and disordered state. In this case, the universe would already be in astate of complete disorder, so disorder could not increase with time. It wouldeither stay constant, in which case there would be no well-defined thermody-namic arrow of time, or it would decrease, in which case the thermodynamicarrow 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 theuniverse should begin with a singularity where the curvature of space-time isinfinite. In fact, this means that classical general relativity predicts its owndownfall. When the curvature of space-time becomes large, quantum gravita-tional effects will become important and the classical theory will cease to be agood description of the universe. One has to use the quantum theory ofgravity to understand how the universe began.
In a quantum theory of gravity, one considers all possible histories of theuniverse. Associated with each history, there are a couple of numbers. Onerepresents 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 universehaving a particular property is given by adding up the waves for all the histo-ries with that property. The histories would be curved spaces that wouldrepresent the evolution of the universe in time. One would still have to sayhow the possible histories of the universe would behave at the boundary ofspace-time in the past. We do not and cannot know the boundary conditionsof the universe in the past. However, one could avoid this difficulty if theboundary 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, orsingularities. They are like the surface of the Earth, but with two more dimen-sions. In that case, the beginning of time would be a regular smooth point ofspace-time. This means that the universe would have begun its expansion ina very smooth and ordered state. It could not have been completely uniformbecause that would violate the uncertainty principle of quantum theory. Therehad to be small fluctuations in the density and velocities of particles. The noboundary condition, however, would imply that these fluctuations were assmall as they could be, consistent with the uncertainty principle.
The universe would have started off with a period of exponential or “inflation-ary” expansion. In this, it would have increased its size by a very large factor.During this expansion, the density fluctuations would have remained small atfirst, but later would have started to grow. Regions in which the density wasslightly higher than average would have had their expansion slowed down bythe gravitational attraction of the extra mass. Eventually, such regions wouldstop expanding, and would collapse to form galaxies, stars, and beings like us.The universe would have started in a smooth and ordered state and wouldbecome lumpy and disordered as time went on. This would explain the exis-tence of the thermodynamic arrow of time. The universe would start in a stateof high order and would become more disordered with time. As I showed ear-lier, the psychological arrow of time points in the same direction as the ther-modynamic arrow. Our subjective sense of time would therefore be that inwhich the universe is expanding, rather than the opposite direction, in whichit would be contracting.
DOES THE ARROW OF TIME REVERSE?
But what would happen if and when the universe stopped expanding andbegan to contract again? Would the thermodynamic arrow reverse anddisorder begin to decrease with time? This would lead to all sorts ofscience-fiction-like possibilities for people who survived from the expandingto the contracting phase. Would they see broken cups gathering themselvestogether off the floor and jumping back on the table? Would they be able toremember 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 uni-verse collapses again, as it will not start to contract for at least another tenthousand million years. But there is a quicker way to find out what will hap-pen: Jump into a black hole. The collapse of a star to form a black hole is ratherlike the later stages of the collapse of the whole universe. Thus, if disorder wereto decrease in the contracting phase of the universe, one might also expect itto decrease inside a black hole. So perhaps an astronaut who fell into a blackhole would be able to make money at roulette by remembering where the ballwent before he placed his bet. Unfortunately, however, he would not have longto 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 thermodynamicarrow, or even bank his winnings, because he would be trapped behind theevent 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 andordered state when it became small again. This would have meant that thecontracting phase was like the time reverse of the expanding phase. People inthe contracting phase would live their lives backward. They would die beforethey were born and would get younger as the universe contracted. This idea isattractive because it would mean a nice symmetry between the expanding andcontracting phases. However, one cannot adopt it on its own, independent ofother ideas about the universe. The question is: Is it implied by the no bound-ary condition or is it inconsistent with that condition?
As I mentioned, I thought at first that the no boundary condition did indeedimply that disorder would decrease in the contracting phase. This was basedon work on a simple model of the universe in which the collapsing phaselooked like the time reverse of the expanding phase. However, a colleague ofmine, Don Page, pointed out that the no boundary condition did not requirethe contracting phase necessarily to be the time reverse of the expandingphase. Further, one of my students, Raymond Laflamme, found that in a slightlymore complicated model, the collapse of the universe was very different fromthe expansion. I realized that I had made a mistake. In fact, the no boundarycondition implied that disorder would continue to increase during the con-traction. The thermodynamic and psychological arrows of time would notreverse when the universe begins to recontract or inside black holes.What should you do when you find you have made a mistake like that? Somepeople, like Eddington, never admit that they are wrong. They continue tofind new, and often mutually inconsistent, arguments to support their case.Others claim to have never really supported the incorrect view in the firstplace or, if they did, it was only to show that it was inconsistent. I could givea large number of examples of this, but I won’t because it would make me toounpopular. It seems to me much better and less confusing if you admit in printthat you were wrong. A good example of this was Einstein, who said that thecosmological constant, which he introduced when he was trying to make astatic model of the universe, was the biggest mistake of his life.