A Briefer History of Time

by Stephen Hawking

  Today we know that stars visible to the naked eye make up only a minute fraction of all the stars. We can see about five thousand stars, only about .0001 percent of all the stars in just our own galaxy, the Milky Way. The Milky Way itself is but one of more than a hundred billion galaxies that can be seen using modern telescopes—and each galaxy contains on average some one hundred billion stars. If a star were a grain of salt, you could fit all the stars visible to the naked eye on a teaspoon, but all the stars in the universe would fill a ball more than eight miles wide.

  Stars are so far away that they appear to us to be just pinpoints of light. We cannot see their size or shape. But, as Hubble noticed, there are many different types of stars, and we can tell them apart by the color of their light. Newton discovered that if light from the sun passes through a triangular piece of glass called a prism, it breaks up into its component colors as in a rainbow. The relative intensities of the various colors emitted by a given source of light are called its spectrum. By focusing a telescope on an individual star or galaxy, one can observe the spectrum of the light from that star or galaxy.

  One thing this light tells us is temperature. In 1860, the German physicist Gustav Kirchhoff realized that any material body, such as a star, will give off light or other radiation when heated, just as coals glow when they are heated. The light such glowing objects give off is due to the thermal motion of the atoms within them. It is called blackbody radiation (even though the glowing objects are not black). The spectrum of blackbody radiation is hard to mistake: it has a distinctive form that varies with the temperature of the body. The light emitted by a glowing object is therefore like a thermometer reading. The spectrum we observe from different stars is always in exactly this form: it is a postcard of the thermal state of that star.

  Stellar Spectrum

  By analyzing the component colors of starlight, one can determine both the temperature of a star and the composition of its atmosphere.

  If we look more closely, starlight tells us even more. We find that certain very specific colors are missing, and these missing colors may vary from star to star. Since we know that each chemical element absorbs a characteristic set of very specific colors, by matching these to those that are missing from a star’s spectrum we can determine exactly which elements are present in that star’s atmosphere.

  Blackbody Spectrum

  All objects-not just stars-emit radiation resulting from the thermal motion of the objects’ microscopic constituents The distribution of frequencies in this radiation is characteristic of an object’s temperature

  In the 1920s, when astronomers began to look at the spectra of stars in other galaxies, they found something most peculiar: there were the same characteristic patterns of missing colors as for stars in our own galaxy, but they were all shifted toward the red end of the spectrum by the same relative amount.

  To physicists, the shifting of color or frequency is known as the Doppler effect. We are all familiar with it in the realm of sound. Listen to a car passing on the road: as it approaches, its engine-or its horn-sounds at a higher pitch, and after it passes and is moving away, it sounds at a lower pitch. The sound of its engine or horn is a wave, a succession of crests and troughs. When a car is racing toward us, it will be progressively nearer to us as it emits each successive wave crest, so the distance between wave crests—the wavelength of the sound—will be smaller than if the car were stationary. The smaller the wavelength, the more of these fluctuations reach our ear each second, and the higher the pitch, or frequency, of the sound. Correspondingly, if the car is moving away from us, the wavelength will be greater and the waves will reach our ear with a lower frequency The faster the car is moving, the greater the effect, so we can use the Doppler effect to measure speed. The behavior of light or radio waves is similar. Indeed, the police make use of the Doppler effect to measure the speed of cars by measuring the wavelength of pulses of radio waves reflected off them.

  As we noted in Chapter 5, the wavelength of visible light is extremely small, ranging from forty-to eighty-millionths of a centimeter. The different wavelengths of light are what the human eye sees as different colors, with the longest wavelengths appearing at the red end of the spectrum and the shortest wavelengths at the blue end. Now imagine a source of light at a constant distance from us, such as a star, emitting waves of light at a constant wavelength. The wavelength of the waves we receive will be the same as the wavelength at which they are emitted. Then suppose that the source starts to move away from us. As in the case of sound, this means that the light will have its wavelength elongated, and hence its spectrum will be shifted toward the red end of the spectrum.

  In the years following his proof of the existence of other galaxies, Hubble spent his time cataloguing their distances and observing their spectra. At that time most people expected the galaxies to be moving around quite randomly, and so Hubble expected to find as many blueshifted spectra as red-shifted ones. It was quite a surprise, therefore, to find that most galaxies appeared red-shifted: nearly all were moving away from us! More surprising still was the finding that Hubble published in 1929: even the size of a galaxy’s red shift is not random but is directly proportional to the galaxy’s distance from us. In other words, the farther a galaxy is, the faster it is moving away! And that meant that the universe could not be static or unchanging in size, as everyone previously had thought. It is in fact expanding; the distance between the different galaxies is growing all the time.

  Doppler Effect

  When a wave source moves toward an observer, its waves appear to have a shorter wavelength If the wave source moves away, its waves appear to have a longer wavelength. This is called the Doppler effect

  The discovery that the universe is expanding was one of the great intellectual revolutions of the twentieth century. With hindsight, it is easy to wonder why no one had thought of it before. Newton, and others, should have realized that a static universe would be unstable, for there is no comparable repulsive force to balance the gravitational pull that all the stars and galaxies exert upon each other. Therefore, even if at some time the universe had been static, it wouldn’t have remained static because the mutual gravitational attraction of all the stars and galaxies would soon have started it contracting. In fact, even if the universe was expanding fairly slowly, the force of gravity would cause it eventually to stop expanding, and it would start to contract. However, if the universe was expanding faster than a certain critical rate, gravity would never be strong enough to stop it, and it w ould continue to expand forever. This is a bit like what happens when you fire a rocket upward from the surface of the earth. If the rocket has a fairly low speed, gravity will eventually stop it, and it will start falling back. On the other hand, if the rocket has more than a certain critical speed (about seven miles per second), gravity will not be strong enough to pull it back, so it will keep going away from the earth forever.

  This behavior of the universe could have been predicted from Newton’s theory of gravity at any time in the nineteenth, the eighteenth, or even the late seventeenth century. Yet so strong was the belief in a static universe that it persisted into the early twentieth century. Even Einstein, when he formulated the general theory of relativity in 1915, was so sure that the universe had to be static that he modified his theory to make this possible by introducing a fudge factor, called the cosmological constant, into his equations. The cosmological constant had the effect of a new “antigravity” force, which, unlike other forces, did not come from any particular source but was built into the very fabric of space-time. As a result of this new force, space-time had an inbuilt tendency to expand. By adjusting the cosmological constant, Einstein could adjust the strength of this tendency. He found he could adjust it to exactly balance the mutual attraction of all the matter in the universe, so a static universe would result. He later disavowed the cosmological constant, calling this fudge factor his “greatest mistake.” As we’ll soon see, today we have reason to believe that he mi
ght have been right to introduce it after all. But what must have disappointed Einstein was that he had allowed his belief in a static universe to override what his theory seemed to be predicting: that the universe is expanding. Only one man, it seems, was willing to take this prediction of general relativity at face value. While Einstein and other physicists were looking for ways of avoiding general relativity’s nonstatic universe, the Russian physicist and mathematician Alexander Friedmann instead set about explaining it.

  Friedmann made two very simple assumptions about the universe: that the universe looks identical in whichever direction we look, and that this would also be true if we were observing the universe from anywhere else. From these two ideas alone, Friedmann showed, by solving the equations of general relativity, that we should not expect the universe to be static. In fact, in 1922, several years before Edwin Hubble’s discovery, Friedmann predicted exactly what Hubble later found!

  The assumption that the universe looks the same in every direction is clearly not exactly true in reality. For example, as we have noted, the other stars in our galaxy form a distinct band of light across the night sky, called the Milky Way. But if we look at distant galaxies, there seems to be more or less the same number of them in every direction. So the universe does appear to be roughly the same in every direction, provided we view it on a large scale compared to the distance between galaxies, and ignore the differences on small scales. Imagine standing in a forest in which the trees are growing in random locations. If you look in one direction, you may see the nearest tree at a distance of one meter. In another direction, the nearest tree might be three meters away. In a third direction, you might see a clump of trees at two meters. It doesn’t seem as if the forest looks the same in every direction, but if you were to take into account all the trees within a one-mile radius, these kinds of differences would average out and you would find that the forest is the same in whichever direction you look.

  Isotropic Forest

  Even if the trees in a forest are uniformly distributed, nearby trees may appear bunched. Similarly, the universe does not look uniform in our local neighborhood, yet on large scales our view appears identical in whichever direction we look.

  For a long time, the uniform distribution of stars was sufficient justification for Friedmann’s assumption—as a rough approximation to the real universe. But more recently a lucky accident uncovered another respect in which Friedmann’s assumption is in fact a remarkably accurate description of our universe. In 1965, two American physicists at the Bell Telephone Laboratories in New Jersey, Arno Penzias and Robert Wilson, were testing a very sensitive microwave detector. (Recall that microwaves are just like light waves, but with a wavelength of around a centimeter.) Penzias and Wilson were worried when they found that their detector was picking up more noise than it ought to. They discovered bird droppings in their detector and checked for other possible malfunctions, but they soon ruled these out. The noise was peculiar in that it remained the same day and night and throughout the year, even though the earth was rotating on its axis and orbiting around the sun. Since the earth’s rotation and orbit pointed the detector in different directions in space, Penzias and Wilson concluded that the noise was coming from beyond the solar system and even from beyond the galaxy. It seemed to be coming equally from every direction in space. We now know that in whichever direction we look, this noise never varies by more than a tiny fraction, so Penzias and Wilson had unwittingly stumbled across a striking example of Friedmann’s first assumption that the universe is the same in every direction.

  What is the origin of this cosmic background noise? At roughly the same time as Penzias and Wilson were investigating noise in their detector, two American physicists at nearby Princeton University, Bob Dicke and Jim Peebles, were also taking an interest in microwaves. They were working on a suggestion, made by George Gamow (once a student of Alexander Friedmann), that the early universe should have been very hot and dense, glowing white hot. Dicke and Peebles argued that we should still be able to see the glow of the early universe, because light from very distant parts of it would only just be reaching us now. However, the expansion of the universe meant that this light should be so greatly red-shifted that it would appear to us now as microwave radiation, rather than visible light. Dicke and Peebles were preparing to look for this radiation when Penzias and Wilson heard about their work and realized that they had already found it. For this, Penzias and Wilson were awarded the Nobel Prize in 1978 (which seems a bit hard on Dicke and Peebles, not to mention Gamow).

  At first sight, all this evidence that the universe appears the same whichever direction we look in might seem to suggest there is something distinctive about our place in the universe. In particular, it might seem that if we observe all other galaxies to be moving away from us, then we must be at the center of the universe. There is, however, an alternative explanation: the universe might look the same in every direction as seen from any other galaxy too. This, as we have seen, was Friedmann’s second assumption.

  We have no scientific evidence for or against that second assumption. Centuries ago, the church would have considered the assumption heresy, since church doctrine stated that we do occupy a special place at the center of the universe. But today we believe Friedmann’s assumption for almost the opposite reason, a kind of modesty: we feel it would be most remarkable if the universe looked the same in every direction around us but not around other points in the universe!

  In Friedmann’s model of the universe, all the galaxies are moving directly away from each other. The situation is rather like a balloon with a number of spots painted on it being steadily blown up. As the balloon expands, the distance between any two spots increases, but there is no spot that can be said to be the center of the expansion. Moreover, as the radius of the balloon steadily increases, the farther apart the spots on the balloon, the faster they will be moving apart. For example, suppose that the radius of the balloon doubles in one second. Two spots that were previously one centimeter apart will now be two centimeters apart (as measured along the surface of the balloon), so their relative speed is one centimeter per second. On the other hand, a pair of spots that were separated by ten centimeters will now be separated by twenty, so their relative speed will be ten centimeters per second. Similarly, in Friedmann’s model the speed at which any two galaxies are moving apart is proportional to the distance between them, so he predicted that the red shift of a galaxy should be directly proportional 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 until similar models were discovered in 1935 by the American physicist Howard Robertson and the British mathematician Arthur Walker, in response to Hubble’s discovery of the uniform expansion of the universe.

  The Expanding Balloon Universe

  As a result of the expansion of the universe, all galaxies are moving directly away from each other Over time, like spots on an inflating balloon, galaxies that are farther apart increase their separation more than nearer galaxies Hence, to an observer in any given galaxy, the more distant a galaxy is, the faster it appears to be moving

  Friedmann derived only one model of the universe. But if his assumptions are correct, there are actually three possible types of solutions to Einstein’s equations, that is, three different kinds of Friedmann models—and three different ways the universe can behave.

  In the first kind of solution (which Friedmann found), the universe is expanding sufficiently slowly that the gravitational attraction between the different galaxies causes the expansion to slow down and eventually to stop. The galaxies then start to move toward each other, and the universe contracts. In the second kind of solution, the universe is expanding so rapidly that the gravitational attraction can never stop it, though it does slow it down a bit. Finally, there is a third kind of solution, in which the universe is expanding only just fast enough to avoid collapse. The speed at which th
e galaxies are moving apart gets smaller and smaller, but it never quite reaches zero.

  A remarkable feature of the first kind of Friedmann model is that in it the universe is not infinite in space, but neither does space have any boundary. Gravity is so strong that space is bent round onto itself. This is rather like the surface of the earth, which is finite but has no boundary. If you keep traveling in a certain direction on the surface of the earth, you never come up against an impassable barrier or fall over the edge, and you eventually come back to where you started. In this model, space is just like this, but with three dimensions instead of two for the earth’s surface. The idea that you could go right round the universe and end up where you started makes good science fiction, but it doesn’t have much practical significance, because it can be shown that the universe would collapse to zero size before you could get around. It is so large, you would need to travel faster than light in order to end up where you started before the universe came to an end—and that is not allowed! Space is also curved in the second Friedmann model, though in a different way. Only the third Friedmann model corresponds to a universe whose large-scale geometry is flat (though space is still curved, or warped, in the vicinity of massive objects).

  Which Friedmann model describes our universe? Will the universe eventually stop expanding and start contracting, or will it expand forever?

  It turns out the answer to this question is more complicated than scientists first thought. The most basic analysis depends on two things: the present rate of expansion of the universe, and its present average density (the amount of matter in a given volume of space). The faster the current rate of expansion, the greater the gravitational force required to stop it, and thus the greater the density of matter needed. If the average density is greater than a certain critical value (determined by the rate of expansion), the gravitational attraction of the matter in the universe will succeed in halting its expansion and cause it to collapse—corresponding to the first Friedmann model. If the average density is less than the critical value, there is not enough gravitational pull to stop the expansion, and the universe will expand forever— corresponding to Friedmann’s second model. And if the average density of the universe is exactly the critical number, then the universe will forever slow its expansion, ever more gradually approaching, but not ever reaching, a static size. This corresponds to the third Friedmann model.