The Theory of Everything: The Origin and Fate of the Universe - Stephen Hawking (2002)
Chapter 2. SECOND LECTURE - THE EXPANDING UNIVERSE
Our sun and the nearby stars are all part of a vast collection of stars calledthe Milky Way galaxy. For a long time it was thought that this was thewhole universe. It was only in 1924 that the American astronomer EdwinHubble demonstrated that ours was not the only galaxy. There were, in fact,many others, with vast tracks of empty space between them. In order to provethis he needed to determine the distances to these other galaxies. We candetermine the distance of nearby stars by observing how they change positionas the Earth goes around the sun. But other galaxies are so far away that, unlikenearby stars, they really do appear fixed. Hubble was forced, therefore, to useindirect methods to measure the distances.
Now the apparent brightness of a star depends on two factors-luminosity andhow far it is from us. For nearby stars we can measure both their apparentbrightness and their distance, so we can work out their luminosity. Conversely,if we knew the luminosity of stars in other galaxies, we could work out theirdistance by measuring their apparent brightness. Hubble argued that therewere certain types of stars that always had the same luminosity when they werenear enough for us to measure. If, therefore, we found such stars in anothergalaxy, we could assume that they had the same luminosity. Thus, we couldcalculate the distance to that galaxy. If we could do this for a number of starsin the same galaxy, and our calculations always gave the same distance, wecould be fairly confident of our estimate. In this way, Edwin Hubble workedout the distances to nine different galaxies.
We now know that our galaxy is only one of some hundred thousand millionthat can be seen using modern telescopes, each galaxy itself containing somehundred thousand million stars. We live in a galaxy that is about one hundredthousand light-years across and is slowly rotating; the stars in its spiral armsorbit around its center about once every hundred million years. Our sun is justan ordinary, average-sized, yellow star, near the outer edge of one of the spiralarms. We have certainly come a long way since Aristotle and Ptolemy, whenwe thought that the Earth was the center of the universe.
Stars are so far away that they appear to us to be just pinpoints of light. Wecannot determine their size or shape. So how can we tell different types of starsapart? For the vast majority of stars, there is only one correct characteristicfeature that we can observe-the color of their light. Newton discovered thatif light from the sun passes through a prism, it breaks up into its componentcolors-its spectrum-like in a rainbow. By focusing a telescope on anindividual star or galaxy, one can similarly observe the spectrum of the lightfrom that star or galaxy. Different stars have different spectra, but the relativebrightness of the different colors is always exactly what one would expect tofind in the light emitted by an object that is glowing red hot. This means thatwe can tell a star’s temperature from the spectrum of its light. Moreover, wefind that certain very specific colors are missing from stars’ spectra, and thesemissing colors may vary from star to star. We know that each chemical elementabsorbs the characteristic set of very specific colors. Thus, by matching each ofthose which are missing from a star’s spectrum, we can determine exactlywhich elements are present in the star’s atmosphere.
In the 1920s, when astronomers began to look at the spectra of stars in othergalaxies, they found something most peculiar: There were the same character-istic sets of missing colors as for stars in our own galaxy, but they were allshifted by the same relative amount toward the red end of the spectrum. Theonly reasonable explanation of this was that the galaxies were moving awayfrom us, and the frequency of the light waves from them was being reduced, orred-shifted, by the Doppler effect. Listen to a car passing on the road. As thecar is approaching, its engine sounds at a higher pitch, corresponding to ahigher frequency of sound waves; and when it passes and goes away, it soundsat a lower pitch. The behavior of light or radial waves is similar. Indeed, thepolice made use of the Doppler effect to measure the speed of cars by measur-ing the frequency of pulses of radio waves reflected off them.
In the years following his proof of the existence of other galaxies, Hubble spenthis time cataloging their distances and observing their spectra. At that timemost people expected the galaxies to be moving around quite randomly, and soexpected to find as many spectra which were blue-shifted as ones which werered-shifted. It was quite a surprise, therefore, to find that the galaxies allappeared red-shifted. Every single one was moving away from us. More surpris-ing still was the result which Hubble published in 1929: Even the size of thegalaxy’s red shift was not random, but was directly proportional to the galaxy’sdistance from us. Or, in other words, the farther a galaxy was, the faster it wasmoving away. And that meant that the universe could not be static, as every-one previously thought, but was in fact expanding. The distance between thedifferent galaxies was growing all the time.
The discovery that the universe was expanding was one of the great intellec-tual revolutions of the twentieth century. With hindsight, it is easy to wonderwhy no one had thought of it before. Newton and others should have realizedthat a static universe would soon start to contract under the influence ofgravity. But suppose that, instead of being static, the universe was expanding.If it was expanding fairly slowly, the force of gravity would cause it eventuallyto stop expanding and then to start contracting. However, if it was expandingat more than a certain critical rate, gravity would never be strong enough tostop it, and the universe would continue to expand forever. This is a bit likewhat happens when one fires a rocket upward from the surface of the Earth. Ifit has a fairly low speed, gravity will eventually stop the rocket and it will startfalling back. On the other hand, if the rocket has more than a certain criticalspeed-about seven miles a second-gravity will not be strong enough to pull itback, so it will keep going away from the Earth forever.
This behavior of the universe could have been predicted from Newton’s theoryof gravity at any time in the nineteenth, the eighteenth, or even the late sev-enteenth centuries. Yet so strong was the belief in a static universe that it per-sisted into the early twentieth century. Even when Einstein formulated thegeneral theory of relativity in 1915, he was sure that the universe had to bestatic. He therefore modified his theory to make this possible, introducing a so-called cosmological constant into his equations. This was a new “antigravity”force, which, unlike other forces, did not come from any particular source, butwas built into the very fabric of space-time. His cosmological constant gavespace-time an inbuilt tendency to expand, and this could be made to exactlybalance the attraction of all the matter in the universe so that a static universewould result.
Only one man, it seems, was willing to take general relativity at face value.While Einstein and other physicists were looking for ways of avoiding generalrelativity’s prediction of a nonstatic universe, the Russian physicist AlexanderFriedmann instead set about explaining it.
THE FRIEDMANN MODELSThe equations of general relativity, which determined how the universeevolves in time, are too complicated to solve in detail. So what Friedmanndid, instead, was to make two very simple assumptions about the universe:that the universe looks identical in whichever direction we look, and thatthis would also be true if we were observing the universe from anywhere else.On the basis of general relativity and these two assumptions, Friedmannshowed that we should not expect the universe to be static. In fact, in 1922,several years before Edwin Hubble’s discovery, Friedmann predicted exactlywhat Hubble found.
The assumption that the universe looks the same in every direction is clearly nottrue in reality. For example, the other stars in our galaxy form a distinct band oflight across the night sky called the Milky Way. But if we look at distant galax-ies, there seems to be more or less the same number of them in each direction.So the universe does seem to be roughly the same in every direction, providedone views it on a large scale compared to the distance between galaxies.For a long time this was sufficient justification for Friedmann’s assumption-as a rough approximation to the real universe. But more recently a lucky acci-dent uncovered the fact that Friedmann’s assumption is in fact a remarkablyaccurate description of our universe. In 1965, two American physicists, ArnoPenzias and Robert Wilson, were working at the Bell Labs in New Jersey onthe design of a very sensitive microwave detector for communicating withorbiting satellites. They were worried when they found that their detector waspicking up more noise than it ought to, and that the noise did not appear tobe coming from any particular direction. First they looked for bird droppingson their detector and checked for other possible malfunctions, but soon ruledthese out. They knew that any noise from within the atmosphere would bestronger when the detector is not pointing straight up than when it is, becausethe atmosphere appears thicker when looking at an angle to the vertical.The extra noise was the same whichever direction the detector pointed, so itmust have come from outside the atmosphere. It was also the same day andnight throughout the year, even though the Earth was rotating on its axis andorbiting around the sun. This showed that the radiation must come frombeyond the solar system, and even from beyond the galaxy, as otherwise itwould vary as the Earth pointed the detector in different directions.In fact, we know that the radiation must have traveled to us across most ofthe observable universe. Since it appears to be the same in different direc-tions, the universe must also be the same in every direction, at least on a largescale. We now know that whichever direction we look in, this noise nevervaries by more than one part in ten thousand. So Penzias and Wilson hadunwittingly stumbled across a remarkably accurate confirmation ofFriedmann’s first assumption.
At roughly the same time, two American physicists at nearby PrincetonUniversity, Bob Dicke and Jim Peebles, were also taking an interest inmicrowaves. They were working on a suggestion made by George Gamow,once a student of Alexander Friedmann, that the early universe should havebeen very hot and dense, glowing white hot. Dicke and Peebles argued that weshould still be able to see this glowing, because light from very distant partsof the early universe would only just be reaching us now. However, theexpansion of the universe meant that this light should be so greatly red-shift-ed that it would appear to us now as microwave radiation. Dicke and Peebleswere looking for this radiation when Penzias and Wilson heard about theirwork and realized that they had already found it. For this, Penzias andWilson were awarded the Nobel Prize in 1978, which seems a bit hard onDicke and Peebles.
Now at first sight, all this evidence that the universe looks the same whichev-er direction we look in might seem to suggest there is something special aboutour place in the universe. In particular, it might seem that if we observe allother galaxies to be moving away from us, then we must be at the center of theuniverse. There is, however, an alternative explanation: The universe mightalso look the same in every direction as seen from any other galaxy. This, as wehave seen, was Friedmann’s second assumption.
We have no scientific evidence for or against this assumption. We believe itonly on grounds of modesty. It would be most remarkable if the universelooked the same in every direction around us, but not around other points inthe universe. In Friedmann’s model, all the galaxies are moving directly awayfrom each other. The situation is rather like steadily blowing up a balloonwhich has a number of spots painted on it. As the balloon expands, the dis-tance between any two spots increases, but there is no spot that can be said tobe the center of the expansion. Moreover, the farther apart the spots are, thefaster they will be moving apart. Similarly, in Friedmann’s model the speed atwhich any two galaxies are moving apart is proportional to the distancebetween them. So it predicted that the red shift of a galaxy should be directlyproportional to its distance from us, exactly as Hubble found.
Despite the success of his model and his prediction of Hubble’s observations,Friedmann’s work remained largely unknown in the West. It became knownonly after similar models were discovered in 1935 by the American physicistHoward Robertson and the British mathematician Arthur Walker, in responseto Hubble’s discovery of the uniform expansion of the universe.
Although Friedmann found only one, there are in fact three different kinds ofmodels that obey Friedmann’s two fundamental assumptions. In the firstkind-which Friedmann found-the universe is expanding so sufficientlyslowly that the gravitational attraction between the different galaxies causesthe expansion to slow down and eventually to stop. The galaxies then start tomove toward each other and the universe contracts. The distance between twoneighboring galaxies starts at zero, increases to a maximum, and then decreasesback down to zero again.
In the second kind of solution, the universe is expanding so rapidly that thegravitational attraction can never stop it, though it does slow it down a bit.The separation between neighboring galaxies in this model starts at zero, andeventually the galaxies are moving apart at a steady speed.
Finally, there is a third kind of solution, in which the universe is expandingonly just fast enough to avoid recollapse. In this case the separation also startsat zero, and increases forever. However, the speed at which the galaxies aremoving apart gets smaller and smaller, although it never quite reaches zero.
A remarkable feature of the first kind of Friedmann model is that the universeis not infinite in space, but neither does space have any boundary. Gravity isso strong that space is bent round onto itself, making it rather like the surfaceof the Earth. If one keeps traveling in a certain direction on the surface of theEarth, one never comes up against an impassable barrier or falls over the edge,but eventually comes back to where one started. Space, in the first Friedmannmodel, is just like this, but with three dimensions instead of two for the Earth’ssurface. The fourth dimension-time-is also finite in extent, but it is like aline with two ends or boundaries, a beginning and an end. We shall see laterthat when one combines general relativity with the uncertainty principle ofquantum mechanics, it is possible for both space and time to be finite withoutany edges or boundaries. The idea that one could go right around the universeand end up where one started makes good science fiction, but it doesn’t havemuch practical significance because it can be shown that the universe wouldrecollapse to zero size before one could get round. You would need to travelfaster than light in order to end up where you started before the universe cameto an end-and that is not allowed.
But which Friedmann model describes our universe? Will the universe eventu-ally stop expanding and start contracting, or will it expand forever? To answerthis question we need to know the present rate of expansion of the universeand its present average density. If the density is less than a certain criticalvalue, determined by the rate of expansion, the gravitational attraction will betoo weak to halt the expansion. If the density is greater than the critical value,gravity will stop the expansion at some time in the future and cause theuniverse to recollapse.
We can determine the present rate of expansion by measuring the velocities atwhich other galaxies are moving away from us, using the Doppler effect. Thiscan be done very accurately. However, the distances to the galaxies are notvery well known because we can only measure them indirectly. So all we knowis that the universe is expanding by between 5 percent and 10 percent everythousand million years. However, our uncertainty about the present averagedensity of the universe is even greater.
If we add up the masses of all the stars that we can see in our galaxy and othergalaxies, the total is less than one-hundredth of the amount required to haltthe expansion of the universe, even in the lowest estimate of the rate of expan-sion. But we know that our galaxy and other galaxies must contain a largeamount of dark matter which we cannot see directly, but which we know mustbe there because of the influence of its gravitational attraction on the orbits ofstars and gas in the galaxies. Moreover, most galaxies are found in clusters, andwe can similarly infer the presence of yet more dark matter in between thegalaxies in these clusters by its effect on the motion of the galaxies. When weadd up all this dark matter, we still get only about one-tenth of the amountrequired to halt the expansion. However, there might be some other form ofmatter which we have not yet detected and which might still raise the averagedensity of the universe up to the critical value needed to halt the expansion.The present evidence, therefore, suggests that the universe will probablyexpand forever. But don’t bank on it. All we can really be sure of is that evenif the universe is going to recollapse, it won’t do so for at least another tenthousand million years, since it has already been expanding for at least thatlong. This should not unduly worry us since by that time, unless we havecolonies beyond the solar system, mankind will long since have died out,extinguished along with the death of our sun.
THE BIG BANG
All of the Friedmann solutions have the feature that at some time in thepast, between ten and twenty thousand million years ago, the distancebetween neighboring galaxies must have been zero. At that time, which wecall the big bang, the density of the universe and the curvature of space-timewould have been infinite. This means that the general theory of relativity-on which Friedmann’s solutions are based-predicts that there is a singularpoint in the universe.
All our theories of science are formulated on the assumption that space-timeis smooth and nearly flat, so they would all break down at the big bang singu-larity, where the curvature of space-time is infinite. This means that even ifthere were events before the big bang, one could not use them to determinewhat would happen afterward, because predictability would break down at thebig bang. Correspondingly, if we know only what has happened since the bigbang, we could not determine what happened beforehand. As far as we areconcerned, events before the big bang can have no consequences, so theyshould not form part of a scientific model of the universe. We should thereforecut them out of the model and say that time had a beginning at the big bang.Many people do not like the idea that time has a beginning, probably becauseit smacks of divine intervention. (The Catholic church, on the other hand, hadseized on the big bang model and in 1951 officially pronounced it to be inaccordance with the Bible.) There were a number of attempts to avoid the con-clusion that there had been a big bang. The proposal that gained widest supportwas called the steady state theory. It was suggested in 1948 by two refugees fromNazi-occupied Austria, Hermann Bondi and Thomas Gold, together with theBriton Fred Hoyle, who had worked with them on the development of radarduring the war. The idea was that as the galaxies moved away from each other,new galaxies were continually forming in the gaps in between, from newmatter that was being continually created. The universe would therefore lookroughly the same at all times as well as at all points of space.
The steady state theory required a modification of general relativity to allowfor the continual creation of matter, but the rate that was involved was solow-about one particle per cubic kilometer per year-that it was not in con-flict with experiment. The theory was a good scientific theory, in the sensethat it was simple and it made definite predictions that could be tested byobservation. One of these predictions was that the number of galaxies or sim-ilar objects in any given volume of space should be the same wherever andwhenever we look in the universe.
In the late 1950s and early 1960s, a survey of sources of radio waves from outerspace was carried out at Cambridge by a group of astronomers led by MartinRyle. The Cambridge group showed that most of these radio sources must lieoutside our galaxy, and also that there were many more weak sources thanstrong ones. They interpreted the weak sources as being the more distant ones,and the stronger ones as being near. Then there appeared to be fewer sourcesper unit volume of space for the nearby sources than for the distant ones.
This could have meant that we were at the center of a great region in the uni-verse in which the sources were fewer than elsewhere. Alternatively, it couldhave meant that the sources were more numerous in the past, at the time thatthe radio waves left on their journey to us, than they are now. Either explana-tion contradicted the predictions of the steady state theory. Moreover, thediscovery of the microwave radiation by Penzias and Wilson in 1965 also indi-cated that the universe must have been much denser in the past. The steadystate theory therefore had regretfully to be abandoned.
Another attempt to avoid the conclusion that there must have been a big bangand, therefore, a beginning of time, was made by two Russian scientists,Evgenii Lifshitz and Isaac Khalatnikov, in 1963. They suggested that the bigbang might be a peculiarity of Friedmann’s models alone, which after all wereonly approximations to the real universe. Perhaps, of all the models that wereroughly like the real universe, only Friedmann’s would contain a big bang sin-gularity. In Friedmann’s models, the galaxies are all moving directly away fromeach other. So it is not surprising that at some time in the past they were all atthe same place. In the real universe, however, the galaxies are not just movingdirectly away from each other-they also have small sideways velocities. So inreality they need never have been all at exactly the same place, only very closetogether. Perhaps, then, the current expanding universe resulted not from a bigbang singularity, but from an earlier contracting phase; as the universe had col-lapsed, the particles in it might not have all collided, but they might haveflown past and then away from each other, producing the present expansion ofthe universe. How then could we tell whether the real universe should havestarted out with a big bang?
What Lifshitz and Khalatnikov did was to study models of the universe whichwere roughly like Friedmann’s models but which took account of the irregular-ities and random velocities of galaxies in the real universe. They showed thatsuch models could start with a big bang, even though the galaxies were nolonger always moving directly away from each other. But they claimed thatthis was still only possible in certain exceptional models in which the galaxieswere all moving in just the right way. They argued that since there seemed tobe infinitely more Friedmann-like models without a big bang singularity thanthere were with one, we should conclude that it was very unlikely that therehad been a big bang. They later realized, however, that there was a much moregeneral class of Friedmann-like models which did have singularities, and inwhich the galaxies did not have to be moving in any special way. They there-fore withdrew their claim in 1970.
The work of Lifshitz and Khalatnikov was valuable because it showed that theuniverse could have had a singularity-a big bang-if the general theory of rel-ativity was correct. However, it did not resolve the crucial question: Does gen-eral relativity predict that our universe should have the big bang, a beginningof time? The answer to this came out of a completely different approach start-ed by a British physicist, Roger Penrose, in 1965. He used the way light conesbehave in general relativity, and the fact that gravity is always attractive, toshow that a star that collapses under its own gravity is trapped in a region whoseboundary eventually shrinks to zero size. This means that all the matter in thestar will be compressed into a region of zero volume, so the density of matterand the curvature of space-time become infinite. In other words, one has a sin-gularity contained within a region of space-time known as a black hole.
At first sight, Penrose’s result didn’t have anything to say about the questionof whether there was a big bang singularity in the past. However, at the timethat Penrose produced his theorem, I was a research student desperately look-ing for a problem with which to complete my Ph.D. thesis. I realized that if onereversed the direction of time in Penrose’s theorem so that the collapse becamean expansion, the conditions of his theorem would still hold, provided theuniverse were roughly like a Friedmann model on large scales at the presenttime. Penrose’s theorem had shown that any collapsing star must end in asingularity; the time-reversed argument showed that any Friedmann-likeexpanding universe must have begun with a singularity. For technical reasons,Penrose’s theorem required that the universe be infinite in space. So I coulduse it to prove that there should be a singularity only if the universe wasexpanding fast enough to avoid collapsing again, because only that Friedmannmodel was infinite in space.
During the next few years I developed new mathematical techniques toremove this and other technical conditions from the theorems that provedthat singularities must occur. The final result was a joint paper by Penroseand myself in 1970, which proved that there must have been a big bang singu-larity provided only that general relativity is correct and that the universecontains as much matter as we observe.
There was a lot of opposition to our work, partly from the Russians, whofollowed the party line laid down by Lifshitz and Khalatnikov, and partly frompeople who felt that the whole idea of singularities was repugnant and spoiledthe beauty of Einstein’s theory. However, one cannot really argue with themathematical theorem. So it is now generally accepted that the universe musthave a beginning.