The Theory of Everything: The Origin and Fate of the Universe - Stephen Hawking (2002)
Chapter 5. FIFTH LECTURE - THE ORIGIN AND FATE OF THE...
T H E O R I G I N A N D F A T E O F T H E U N I V E R S EThroughout the 1970s I had been working mainly on black holes. However,n 1981 my interest in questions about the origin of the universe wasreawakened when I attended a conference on cosmology in the Vatican. TheCatholic church had made a bad mistake with Galileo when it tried to laydown the law on a question of science, declaring that the sun went around theEarth. Now, centuries later, it had decided it would be better to invite a num-ber of experts to advise it on cosmology.
At the end of the conference the participants were granted an audience withthe pope. He told us that it was okay to study the evolution of the universeafter the big bang, but we should not inquire into the big bang itself becausethat was the moment of creation and therefore the work of God.I was glad then that he did not know the subject of the talk I had just given atthe conference. I had no desire to share the fate of Galileo; I have a lot of sym-pathy with Galileo, partly because I was born exactly three hundred years afterhis death.
THE HOT BIG BANG MODEL
In order to explain what my paper was about, I shall first describe the generallyaccepted history of the universe, according to what is known as the “hot bigbang model.” This assumes that the universe is described by a Friedmannmodel, right back to the big bang. In such models one finds that as the uni-verse expands, the temperature of the matter and radiation in it will go down.Since temperature is simply a measure of the average energy of the particles,this cooling of the universe will have a major effect on the matter in it. At veryhigh temperatures, particles will be moving around so fast that they can escapeany attraction toward each other caused by the nuclear or electromagneticforces. But as they cooled off, one would expect particles that attract eachother to start to clump together.
At the big bang itself, the universe had zero size and so must have been infi-nitely hot. But as the universe expanded, the temperature of the radiationwould have decreased. One second after the big bang it would have fallen toabout ten thousand million degrees. This is about a thousand times the tem-perature at the center of the sun, but temperatures as high as this are reachedin H-bomb explosions. At this time the universe would have contained mostlyphotons, electrons, and neutrinos and their antiparticles, together with someprotons and neutrons.
As the universe continued to expand and the temperature to drop, the rate atwhich electrons and the electron pairs were being produced in collisions wouldhave fallen below the rate at which they were being destroyed by annihilation.So most of the electrons and antielectrons would have annihilated each otherto produce more photons, leaving behind only a few electrons.
About one hundred seconds after the big bang, the temperature would havefallen to one thousand million degrees, the temperature inside the hotteststars. At this temperature, protons and neutrons would no longer have suffi-cient energy to escape the attraction of the strong nuclear force. They wouldstart to combine together to produce the nuclei of atoms of deuterium, orheavy hydrogen, which contain one proton and one neutron. The deuteriumnuclei would then have combined with more protons and neutrons to makehelium nuclei, which contained two protons and two neutrons. There wouldalso be small amounts of a couple of heavier elements, lithium and beryllium.One can calculate that in the hot big bang model about a quarter of the pro-tons and neutrons would have been converted into helium nuclei, along witha small amount of heavy hydrogen and other elements. The remaining neu-trons would have decayed into protons, which are the nuclei of ordinaryhydrogen atoms. These predictions agree very well with what is observed.The hot big bang model also predicts that we should be able to observe theradiation left over from the hot early stages. However, the temperature wouldhave been reduced to a few degrees above absolute zero by the expansion of theuniverse. This is the explanation of the microwave background of radiationthat was discovered by Penzias and Wilson in 1965. We are thereforethoroughly confident that we have the right picture, at least back to about onesecond after the big bang. Within only a few hours of the big bang, theproduction of helium and other elements would have stopped. And after that,for the next million years or so, the universe would have just continuedexpanding, without anything much happening. Eventually, once the tempera-ture had dropped to a few thousand degrees, the electrons and nuclei would nolonger have had enough energy to overcome the electromagnetic attractionbetween them. They would then have started combining to form atoms.
The universe as a whole would have continued expanding and cooling.However, in regions that were slightly denser than average, the expansionwould have been slowed down by extra gravitational attraction. This wouldeventually stop expansion in some regions and cause them to start to recol-lapse. As they were collapsing, the gravitational pull of matter outside theseregions might start them rotating slightly. As the collapsing region gotsmaller, it would spin faster-just as skaters spinning on ice spin faster as thedraw in their arms. Eventually, when the region got small enough, it would bespinning fast enough to balance the attraction of gravity. In this way, disklikerotating galaxies were born.
As time went on, the gas in the galaxies would break up into smaller cloudsthat would collapse under their own gravity. As these contracted, the temper-ature of the gas would increase until it became hot enough to start nuclearreactions. These would convert the hydrogen into more helium, and the heatgiven off would raise the pressure, and so stop the clouds from contracting anyfurther. They would remain in this state for a long time as stars like our sun,burning hydrogen into helium and radiating the energy as heat and light.More massive stars would need to be hotter to balance their stronger gravita-tional attraction. This would make the nuclear fusion reactions proceed somuch more rapidly that they would use up their hydrogen in as little as a hun-dred million years. They would then contract slightly and, as they heated upfurther, would start to convert helium into heavier elements like carbon oroxygen. This, however, would not release much more energy, so a crisis wouldoccur, as I described in my lecture on black holes.
What happens next is not completely clear, but it seems likely that the centralregions of the star would collapse to a very dense state, such as a neutron staror black hole. The outer regions of the star may get blown off in a tremendousexplosion called a supernova, which would outshine all the other stars in thegalaxy. Some of the heavier elements produced near the end of the star’s lifewould be flung back into the gas in the galaxy. They would provide some ofthe raw material for the next generation of stars.
Our own sun contains about 2 percent of these heavier elements because it isa second- or third-generation star. It was formed some five thousand millionyears ago out of a cloud of rotating gas containing the debris of earlier super-novas. Most of the gas in that cloud went to form the sun or got blown away.However, a small amount of the heavier elements collected together to formthe bodies that now orbit the sun as planets like the Earth.
This picture of a universe that started off very hot and cooled as it expanded isin agreement with all the observational evidence that we have today.Nevertheless, it leaves a number of important questions unanswered. First, whywas the early universe so hot? Second, why is the universe so uniform on a largescale-why does it look the same at all points of space and in all directions?Third, why did the universe start out with so nearly the critical rate of expan-sion to just avoid recollapse? If the rate of expansion one second after the bigbang had been smaller by even one part in a hundred thousand millionmillion, the universe would have recollapsed before it ever reached its presentsize. On the other hand, if the expansion rate at one second had been largerby the same amount, the universe would have expanded so much that it wouldbe effectively empty now.
Fourth, despite the fact that the universe is so uniform and homogenous on alarge scale, it contains local lumps such as stars and galaxies. These are thoughtto have developed from small differences in the density of the early universefrom one region to another. What was the origin of these density fluctuations?The general theory of relativity, on its own, cannot explain these features oranswer these questions. This is because it predicts that the universe started offwith infinite density at the big bang singularity. At the singularity, general rel-ativity and all other physical laws would break down. One cannot predict whatwould come out of the singularity. As I explained before, this means that onemight as well cut any events before the big bang out of the theory, because theycan have no effect on what we observe. Space-time would have a boundary-a beginning at the big bang. Why should the universe have started off at thebig bang in just such a way as to lead to the state we observe today? Why is theuniverse so uniform, and expanding at just the critical rate to avoid recollapse?One would feel happier about this if one could show that quite a number ofdifferent initial configurations for the universe would have evolved to producea universe like the one we observe.
If this is the case, a universe that developed from some sort of random initialconditions should contain a number of regions that are like what we observe.There might also be regions that were very different. However, these regionswould probably not be suitable for the formation of galaxies and stars. Theseare essential prerequisites for the development of intelligent life, at least as weknow it. Thus, these regions would not contain any beings to observe that theywere different.
When one considers cosmology, one has to take into account the selectionprinciple that we live in a region of the universe that is suitable for intelligentlife. This fairly obvious and elementary consideration is sometimes called theanthropic principle. Suppose, on the other hand, that the initial state of theuniverse had to be chosen extremely carefully to lead to something like whatwe see around us. Then the universe would be unlikely to contain any regionin which life would appear.
In the hot big bang model that I described earlier, there was not enough timein the early universe for heat to have flowed from one region to another. Thismeans that different regions of the universe would have had to have startedout with exactly the same temperature in order to account for the fact that themicrowave background has the same temperature in every direction we look.Also, the initial rate of expansion would have had to be chosen very preciselyfor the universe not to have recollapsed before now. This means that the ini-tial state of the universe must have been very carefully chosen indeed if thehot big bang model was correct right back to the beginning of time. It wouldbe very difficult to explain why the universe should have begun in just thisway, except as the act of a God who intended to create beings like us.
THE INFLATIONARY MODEL
In order to avoid this difficulty with the very early stages of the hot big bangmodel, Alan Guth at the Massachusetts Institute of Technology put forward anew model. In this, many different initial configurations could have evolved tosomething like the present universe. He suggested that the early universe mighthave had a period of very rapid, or exponential, expansion. This expansion issaid to be inflationary-an analogy with the inflation in prices that occurs to agreater or lesser degree in every country. The world record for price inflationwas probably in Germany after the first war, when the price of a loaf of breadwent from under a mark to millions of marks in a few months. But the inflationwe think may have occurred in the size of the universe was much greater eventhan that-a million million million million million times in only a tiny frac-tion of a second. Of course, that was before the present government.
Guth suggested that the universe started out from the big bang very hot. Onewould expect that at such high temperatures, the strong and weak nuclearforces and the electromagnetic force would all be unified into a single force.As the universe expanded, it would cool, and particle energies would go down.Eventually there would be what is called a phase transition, and the symmetrybetween the forces would be broken. The strong force would become differentfrom the weak and electromagnetic forces. One common example of a phasetransition is the freezing of water when you cool it down. Liquid water is sym-metrical, the same at every point and in every direction. However, when icecrystals form, they will have definite positions and will be lined up in somedirection. This breaks the symmetry of the water.
In the case of water, if one is careful, one can “supercool” it. That is, one canreduce the temperature below the freezing point-0 degrees centigrade-with-out ice forming. Guth suggested that the universe might behave in a similarway: The temperature might drop below the critical value without the symme-try between the forces being broken. If this happened, the universe would bein an unstable state, with more energy than if the symmetry had been broken.This special extra energy can be shown to have an antigravitational effect. Itwould act just like a cosmological constant.
Einstein introduced the cosmological constant into general relativity when hewas trying to construct a static model of the universe. However,in this case,the universe would already be expanding. The repulsive effect of this cosmo-logical constant would therefore have made the universe expand at an ever-increasing rate. Even in regions where there were more matter particles thanaverage, the gravitational attraction of the matter would have been out-weighed by the repulsion of the effective cosmological constant. Thus, theseregions would also expand in an accelerating inflationary manner.
As the universe expanded, the matter particles got farther apart. One would beleft with an expanding universe that contained hardly any particles. It wouldstill be in the supercooled state, in which the symmetry between the forces isnot broken. Any irregularities in the universe would simply have beensmoothed out by the expansion, as the wrinkles in a balloon are smoothedaway when you blow it up. Thus, the present smooth and uniform state of theuniverse could have evolved from many different nonuniform initial states.The rate of expansion would also tend toward just the critical rate needed toavoid recollapse.
Moreover, the idea of inflation could also explain why there is so much matterin the universe. There are something like 1,080 particles in the region of theuniverse that we can observe. Where did they all come from? The answer isthat, in quantum theory, particles can be created out of energy in the form ofparticle/antiparticle pairs. But that just raises the question of where the energycame from. The answer is that the total energy of the universe is exactly zero.The matter in the universe is made out of positive energy. However, the mat-ter is all attracting itself by gravity. Two pieces of matter that are close to eachother have less energy than the same two pieces a long way apart. This isbecause you have to expend energy to separate them. You have to pull againstthe gravitational force attracting them together. Thus, in a sense, the gravita-tional field has negative energy. In the case of the whole universe, one canshow that this negative gravitational energy exactly cancels the positive ener-gy of the matter. So the total energy of the universe is zero.
Now, twice zero is also zero. Thus, the universe can double the amount of pos-itive matter energy and also double the negative gravitational energy withoutviolation of the conservation of energy. This does not happen in the normalexpansion of the universe in which the matter energy density goes down as theuniverse gets bigger. It does happen, however, in the inflationary expansion,because the energy density of the supercooled state remains constant while theuniverse expands. When the universe doubles in size, the positive matter ener-gy and the negative gravitational energy both double, so the total energyvery large amount. Thus, the total amount of energy available to make parti-cles becomes very large. As Guth has remarked, “It is said that there is no suchthing as a free lunch. But the universe is the ultimate free lunch.”
THE END OF INFLATION
The universe is not expanding in an inflationary way today. Thus, there hadto be some mechanism that would eliminate the very large effective cosmolog-ical constant. This would change the rate of expansion from an acceleratedone to one that is slowed down by gravity, as we have today. As the universeexpanded and cooled, one might expect that eventually the symmetry betweenthe forces would be broken, just as supercooled water always freezes in the end.The extra energy of the unbroken symmetry state would then be released andwould reheat the universe. The universe would then go on to expand and cool,just like the hot big bang model. However, there would now be an explanationof why the universe was expanding at exactly the critical rate and why differ-ent regions had the same temperature.
In Guth’s original proposal, the transition to broken symmetry was supposed tooccur suddenly, rather like the appearance of ice crystals in very cold water.The idea was that “bubbles” of the new phase of broken symmetry would haveformed in the old phase, like bubbles of steam surrounded by boiling water.The bubbles were supposed to expand and meet up with each other until thewhole universe was in the new phase. The trouble was, as I and several otherpeople pointed out, the universe was expanding so fast that the bubbles wouldbe moving away from each other too rapidly to join up. The universe would beleft in a very nonuniform state, with some regions having symmetry betweenthe different forces. Such a model of the universe would not correspond towhat we see.
In October 1981 I went to Moscow for a conference on quantum gravity. Afterthe conference, I gave a seminar on the inflationary model and its problems atthe Sternberg Astronomical Institute. In the audience was a young Russian,Andrei Linde. He said that the difficulty with the bubbles not joining up couldbe avoided if the bubbles were very big. In this case, our region of the universecould be contained inside a single bubble. In order for this to work, the changefrom symmetry to broken symmetry must have taken place very slowly insidethe bubble, but this is quite possible according to grand unified theories.Linde’s idea of a slow breaking of symmetry was very good, but I pointed outthat his bubbles would have been bigger than the size of the universe at thetime. I showed that instead the symmetry would have broken everywhere atthe same time, rather than just inside bubbles. This would lead to a uniformuniverse, like we observe. The slow symmetry breaking model was a goodattempt to explain why the universe is the way it is. However, I and severalother people showed that it predicted much greater variations in themicrowave background radiation than are observed. Also, later work castdoubt on whether there would have been the right kind of phase transition inthe early universe. A better model, called the chaotic inflationary model, wasintroduced by Linde in 1983. This doesn’t depend on phase transitions, and itcan give us the right size of variations of the microwave background. The infla-tionary model showed that the present state of the universe could have arisenfrom quite a large number of different initial configurations. It cannot be thecase, however, that every initial configuration would have led to a universelike the one we observe. So even the inflationary model does not tell us whythe initial configuration was such as to produce what we observe. Must we turnto the anthropic principle for an explanation? Was it all just a lucky chance?That would seem a counsel of despair, a negation of all our hopes of under-standing the underlying order of the universe.
In order to predict how the universe should have started off, one needs laws thathold at the beginning of time. If the classical theory of general relativity wascorrect, the singularity theorem showed that the beginning of time would havebeen a point of infinite density and curvature. All the known laws of sciencewould break down at such a point. One might suppose that there were new lawsthat held at singularities, but it would be very difficult even to formulate lawsat such badly behaved points and we would have no guide from observations asto what those laws might be. However, what the singularity theorems reallyindicate is that the gravitational field becomes so strong that quantum gravita-tional effects become important: Classical theory is no longer a good descrip-tion of the universe. So one has to use a quantum theory of gravity to discussthe very early stages of the universe. As we shall see, it is possible in the quan-tum theory for the ordinary laws of science to hold everywhere, including at thebeginning of time. It is not necessary to postulate new laws for singularities,because there need not be any singularities in the quantum theory.
We don’t yet have a complete and consistent theory that combines quantummechanics and gravity. However, we are thoroughly certain of some featuresthat such a unified theory should have. One is that it should incorporateFeynman’s proposal to formulate quantum theory in terms of a sum over histo-ries. In this approach, a particle going from A to B does not have just a singlehistory as it would in a classical theory. Instead, it is supposed to follow everypossible path in space-time. With each of these histories, there are associateda couple of numbers, one representing the size of a wave and the other repre-senting its position in the cycle-its phase.The probability that the particle, say, passes through some particular point isfound by adding up the waves associated with every possible history thatpasses through that point. When one actually tries to perform these sums,however, one runs into severe technical problems. The only way around theseis the following peculiar prescription: One must add up the waves for particlehistories that are not in the real time that you and I experience but take placein imaginary time.
Imaginary time may sound like science fiction, but it is in fact a well-definedmathematical concept. To avoid the technical difficulties with Feynman’s sumover histories, one must use imaginary time. This has an interesting effect onspace-time: The distinction between time and space disappears completely. Aspace-time in which events have imaginary values of the time coordinate issaid to be Euclidean because the metric is positive definite.In Euclidean space-time there is no difference between the time direction anddirections in space. On the other hand, in real space-time, in which events arelabeled by real values of the time coordinate, it is easy to tell the difference. Thetime direction lies within the light cone, and space directions lie outside. Onecan regard the use of imaginary time as merely a mathematical device-ortrick-to calculate answers about real space-time. However, there may be moreto it than that. It may be that Euclidean space-time is the fundamental conceptand what we think of as real space-time is just a figment of our imagination.When we apply Feynman’s sum over histories to the universe, the analogue ofthe history of a particle is now a complete curved space-time which representsthe history of the whole universe. For the technical reasons mentioned above,these curved space-times must be taken to be Euclidean. That is, time isimaginary and is indistinguishable from directions in space. To calculate theprobability of finding a real space-time with some certain property, one addsup the waves associated with all the histories in imaginary time that have thatproperty. One can then work out what the probable history of the universewould be in real time.
THE NO BOUNDARY CONDITION
In the classical theory of gravity, which is based on real space-time, there areonly two possible ways the universe can behave. Either it has existed for an infi-nite time, or else it had a beginning at a singularity at some finite time in thepast. In fact, the singularity theorems show it must be the second possibility. Inthe quantum theory of gravity, on the other hand, a third possibility arises.Because one is using Euclidean space-times, in which the time direction is onthe same footing as directions in space, it is possible for space-time to be finitein extent and yet to have no singularities that formed a boundary or edge.Space-time would be like the surface of the Earth, only with two more dimen-sions. The surface of the Earth is finite in extent but it doesn’t have a boundaryor edge. If you sail off into the sunset, you don’t fall off the edge or run into asingularity. I know, because I have been around the world.
If Euclidean space-times direct back to infinite imaginary time or else startedat a singularity, we would have the same problem as in the classical theory ofspecifying the initial state of the universe. God may know how the universebegan, but we cannot give any particular reason for thinking it began one wayrather than another. On the other hand, the quantum theory of gravity hasopened up a new possibility. In this, there would be no boundary tospace-time. Thus, there would be no need to specify the behavior at theboundary. There would be no singularities at which the laws of science brokedown and no edge of space-time at which one would have to appeal to God orsome new law to set the boundary conditions for space-time. One could say:”The boundary condition of the universe is that it has no boundary.” The uni-verse would be completely self-contained and not affected by anything outsideitself. It would be neither created nor destroyed. It would just be.
It was at the conference in the Vatican that I first put forward the suggestionthat maybe time and space together formed a surface that was finite in size butdid not have any boundary or edge. My paper was rather mathematical, how-ever, so its implications for the role of God in the creation of the universe werenot noticed at the time-just as well for me. At the time of the Vatican confer-ence, I did not know how to use a no boundary idea to make predictions aboutthe universe. However, I spent the following summer at the University ofCalifornia, Santa Barbara. There, a friend and colleague of mine, Jim Hartle,worked out with me what conditions the universe must satisfy if space-timehad no boundary.
I should emphasize that this idea that time and space should be finite withoutboundary is just a proposal. It cannot be deduced from some other principle.Like any other scientific theory, it may initially be put forward for aesthetic ormetaphysical reasons, but the real test is whether it makes predictions thatagree with observation. This, however, is difficult to determine in the case ofquantum gravity, for two reasons. First, we are not yet sure exactly which the-ory successfully combines general relativity and quantum mechanics, thoughwe know quite a lot about the form such a theory must have. Second, anymodel that described the whole universe in detail would be much too compli-cated mathematically for us to be able to calculate exact predictions. Onetherefore has to make approximations-and even then, the problem ofextracting predictions remains a difficult one.
One finds, under the no boundary proposal, that the chance of the universebeing found to be following most of the possible histories is negligible. Butthere is a particular family of histories that are much more probable than theothers. These histories may be pictured as being like the surface of the Earth,with a distance from the North Pole representing imaginary time; the size of acircle of latitude would represent the spatial size of the universe. The universestarts at the North Pole as a single point. As one moves south, the circle of lat-itude get bigger, corresponding to the universe expanding with imaginary time.The universe would reach a maximum size at the equator and would contractagain to a single point at the South Pole. Even though the universe wouldhave zero size at the North and South poles, these points would not be singu-larities any more than the North and South poles on the Earth are singular.The laws of science will hold at the beginning of the universe, just as they doat the North and South poles on the Earth.
The history of the universe in real time, however, would look very different. Itwould appear to start at some minimum size, equal to the maximum size of thehistory in imaginary time. The universe would then expand in real time likethe inflationary model. However, one would not now have to assume that theuniverse was created somehow in the right sort of state. The universe wouldexpand to a very large size, but eventually it would collapse again into whatlooks like a singularity in real time. Thus, in a sense, we are still all doomed,even if we keep away from black holes. Only if we could picture the universein terms of imaginary time would there be no singularities.
The singularity theorems of classical general relativity showed that the uni-verse must have a beginning, and that this beginning must be described interms of quantum theory. This in turn led to the idea that the universe couldbe finite in imaginary time, but without boundaries or singularities. When onegoes back to the real time in which we live, however, there will still appear tobe singularities. The poor astronaut who falls into a black hole will still cometo a sticky end. It is only if he could live in imaginary time that he wouldencounter no singularities.
This might suggest that the so-called imaginary time is really the fundamen-tal time, and that what we call real time is something we create just in ourminds. In real time, the universe has a beginning and an end at singularitiesthat form a boundary to space-time and at which the laws of science breakdown. But in imaginary time, there are no singularities or boundaries. Somaybe what we call imaginary time is really more basic, and what we call realtime is just an idea that we invent to help us describe what we think the uni-verse is like. But according to the approach I described in the first lecture, ascientific theory is just a mathematical model we make to describe our obser-vations. It exists only in our minds. So it does not have any meaning to ask:Which is real, “real” or “imaginary” time? It is simply a matter of which is amore useful description.
The no boundary proposal seems to predict that, in real time, the universeshould behave like the inflationary models. A particularly interesting problemis the size of the small departures from uniform density in the early universe.These are thought to have led to the formation first of the galaxies, then ofstars, and finally of beings like us. The uncertainty principle implies that theearly universe cannot have been completely uniform. Instead, there must havebeen some uncertainties or fluctuations in the positions and velocities of theparticles. Using the no boundary condition, one finds that the universe musthave started off with just the minimum possible nonuniformity allowed by theuncertainty principle.
The universe would have then undergone a period of rapid expansion, like inthe inflationary models. During this period, the initial nonuniformities wouldhave been amplified until they could have been big enough to explain the ori-gin of galaxies. Thus, all the complicated structures that we see in the universemight be explained by the no boundary condition for the universe and theuncertainty principle of quantum mechanics.
The idea that space and time may form a closed surface without boundary alsohas profound implications for the role of God in the affairs of the universe.With the success of scientific theories in describing events, most people havecome to believe that God allows the universe to evolve according to a set oflaws. He does not seem to intervene in the universe to break these laws.However, the laws do not tell us what the universe should have looked likewhen it started. It would still be up to God to wind up the clockwork andchoose how to start it off. So long as the universe had a beginning that was asingularity, one could suppose that it was created by an outside agency. But ifthe universe is really completely self-contained, having no boundary or edge,it would be neither created nor destroyed. It would simply be. What place,then, for a creator?