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 BANGAll 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.