by Isaac Asimov
Now the quantum could be related directly to the frequency of a given radiation—that is, the number of waves emitted in 1 second. Like the quantum’s energy content, the frequency is inversely proportional to the radiation’s wavelength. The shorter the waves, the more of them can be emitted in 1 second. If both the frequency and the quantum’s energy content were inversely proportional to the wavelength, then the two were directly proportional to each other. Planck expressed this relationship by means of his now-famous equation:
e = hν
The symbol e stands for the quantum energy; ν (the Greek letter nu), for the frequency; and h for Planck’s constant; which gives the proportional relation between quantum energy and frequency.The value of h is extremely small, and so is the quantum. The units of radiation are so small, in fact, that light looks continuous to us, just as ordinary matter seems continuous. But at the beginning of the twentieth century, the same fate befell radiation as had befallen matter at the beginning of the nineteenth: both now had to be accepted as discontinuous.
Planck’s quanta cleared up the connection between temperature and the wavelengths of emitted radiation. A quantum of violet light was twice as energetic as a quantum of red light, and naturally it would take more heat energy to produce violet quanta than red quanta. Equations worked out on the basis of the quantum explained the radiation of a black body very neatly at both ends of the spectrum.
Eventually Planck’s quantum theory was to do a great deal more: it was to explain the behavior of atoms, of the electrons in atoms, and of nucleons in the atoms’ nuclei. Nowadays, physics before quantum theory is called classical physics and since quantum theory, modern physics. Planck was awarded the Nobel Prize in physics in 1918.
EINSTEIN’S PARTICLE-WAVE THEORY
Planck’s theory made little impression on physicists when it was first announced in 1900. It was too revolutionary to be accepted at once. Planck himself seemed appalled at what he had done. But five years later a young German-born Swiss physicist named Albert Einstein verified the existence of his quanta.
The German physicist Philipp Lenard had found that, when light struck certain metals, it caused the metal surface to emit electrons, as if the force of the light kicked electrons out of the atoms. The phenomenon acquired the name photoelectric effect and for its discovery Lenard received the Nobel Prize for physics in 1905. When physicists began to experiment with it, they found, to their surprise, that increasing the intensity of the light did not give the kicked-out electrons any more energy. But changing the wavelength of light did affect them: blue light, for instance, caused the electrons to fly out at greater speed than yellow light did. A very dim blue light would kick out fewer electrons than a bright yellow light would, but those few blue-light electrons would travel with greater speed than any of the yellow-light electrons. On the other hand, red light, no matter how bright, failed to knock out any electrons at all from some metals.
None of these phenomena could be explained by the old theories of light. Why should blue light do something red light cannot do?
Einstein found the answer in Planck’s quantum theory. To absorb enough energy to leave the metal surface, an electron has to be hit by a quantum of a certain minimum size. In the case of an electron held only weakly by its atom (as in cesium), even a quantum of red light will do. Where atoms hold electrons more strongly, yellow light is required, or blue light, or even ultraviolet. And in any case, the more energetic the quantum, the more speed it gives to the electron it has kicked out.
Here the quantum theory explained a physical phenomenon with perfect simplicity, whereas the prequantum view of light had remained helpless. Other applications of quantum mechanics followed thick and fast. For his explanation of the photoelectric effect (not for his theory of relativity), Einstein was awarded the Nobel Prize in physics in 1921.
In his Special Theory of Relativity, presented in 1905 and evolved in his spare time while he worked as examiner at the Swiss patent office, Einstein proposed a new fundamental view of the universe based on an extension of the quantum theory. He suggested that light travels through space in quantum form (the term photon for this unit of light was introduced by Compton in 1928), and thus resurrected the concept of light consisting of particles. But this was a new kind of particle: it has properties of a wave as well as of a particle, and sometimes it shows one set of properties and sometimes the other.
This has been made to seem a paradox, or even a kind of mysticism, as if the true nature of light passes all possible understanding. On the contrary, let me illustrate with an analogy: a man may have many aspects—husband, father, friend, businessman. Depending on circumstances and on his surroundings, he behaves like a husband, a father, a friend, or a businessman. You would not expect him to exhibit his husbandly behavior toward a customer or his businesslike beha~ior toward his wife, and yet he is neither a paradox nor more than one man.
In the same way, radiation has both corpuscular and wave properties. In some capacities, the corpuscular properties are particularly pronounced; in others, the wave properties. About 1930, Niels Bohr advanced reasons for thinking that any experiment designed to test the wave properties of radiation could not conceivably detect the particle properties, and vice versa. One could deal with one or the other, never with both at the same time. He called this the principle of complementarity. This dual set of properties gives a more satisfactory account of radiation than either set of properties alone can.
The discovery of the wave nature of light had led to all the triumphs of nineteenth-century optics, including spectroscopy. But it had also required physicists to imagine the existence of the ether. Now Einstein’s particle-wave view kept all the nineteenth-century victories (including Maxwell’s equations), but made it unnecessary to assume that the ether exists. Radiation could travel through a vacuum by virtue of its particle attributes, and the ether idea, killed by the Michelson-Morley experiment, could now be buried.
Einstein introduced a second important idea in his special theory of relativity: that the speed of light in a vacuum never varies, regardless of the motion of its source. In Newton’s view of the universe, a light beam from a source moving toward an observer should seem to travel more quickly than one from a source moving in any other direction. In Einstein’s view, this would not seem to happen, and from that assumption he was able to derive the Lorentz-FitzGerald equations. He showed that the increase of mass with velocity, which Lorentz had applied only to charged particles, can be applied to all objects of any sort. Einstein reasoned further that increases in velocity would not only foreshorten length and increase mass but also slow the pace of time: in other words, clocks would slow down along with the shortening of yardsticks.
THE THEORY OF RELATIVITY
The most fundamental aspect of Einstein’s theory was its denial of the existence of absolute space and absolute time. This may sound like nonsense: How can the human mind learn anything at all about the universe if it has no point of departure? Einstein answered that all we need to do is to pick a frame of reference to which the events of the universe can be related. Any frame of reference (the earth motionless, or the sun motionless, or we ourselves motionless, for that matter) will be equally valid, and we can simply choose the frame that is most convenient. It is more convenient to calculate planetary motions in a frame of reference in which the sun is motionless than in one in which the earth is motionless—but it is no more true.
Thus measurements of space and time are “relative” to some arbitrarily chosen frame of reference—and that is the reason for naming Einstein’s idea the theory of relativity.
To illustrate. Suppose we on the earth were to observe a strange planet (Planet X), exactly like our own in size and mass, go whizzing past us at 163,000 miles per second relative to ourselves. If we could measure its dimensions as it shot past, we would find it to be foreshortened by 50 percent in the direction of its motion. It would be an ellipsoid rather than a sphere and would, on further measure
ment, seem to have twice the mass of the earth.
Yet to a man on Planet X, it would seem that he himself and his own planet were motionless. The earth would seem to be moving past him at 163,000 miles per second, and it would appear to have an ellipsoidal shape and twice the mass of his planet.
One is tempted to ask which planet would really be foreshortened and doubled in mass, but the only possible answer depends on the frame of reference. If you find that notion frustrating, consider that a man is small compared with a whale and large compared with a beetle. Is there any point in asking what a man is really—large or small?
For all its unusual consequences, relativity explains all the known phenomena of the universe at least as well as prerelativity theories do. But it goes further: it explains easily some phenomena that the Newtonian outlook explained poorly or not at all. Consequently, Einstein has been accepted over Newton, not as a replacement so much as a refinement. The Newtonian view of the universe can still be used as a simplified approximation that works well enough in ordinary life and even in ordinary astronomy, as in placing satellites in orbit. But when it comes to accelerating particles in a synchrotron, for example, we must take account of the Einsteinian increase of mass with velocity to make the machine work.
SPACE-TIME AND THE CLOCK PARADOX
Einstein’s view of the universe so mingles space and time that either concept by itself becomes meaningless. The universe is four-dimensional, with time one of the dimensions (but behaving not quite like the ordinary spatial dimensions of length, breadth, and height). The four-dimensional fusion is often referred to as space-time. This notion was first developed by one of Einstein’s teachers, the Russian-German mathematician Hermann Minkowski, in 1907.
With time as well as space up to odd tricks in relativity, one aspect of relativity that still provokes arguments among physicists is Einstein’s notion of the slowing of clocks. A clock in motion, he said, keeps time more slowly than a stationary one. In fact, all phenomena that change with time change more slowly when moving than when at rest, which is the same as saying that time itself is slowed. At ordinary speeds, the effect is negligible; but at 163,000 miles per second, a clock would seem (to an observer watching it fly past) to take two seconds to tick off one second. And at the speed of light, time would stand still.
The time-effect is more disturbing than those involving length and weight. If an object shrinks to half its length and then returns to normal, or if it doubles its weight and then returns to normal, no trace is left behind to indicate the temporary change, and opposing viewpoints need not quarrel.
Time, however, is cumulative. If a clock on Planet X seems to be running at half-time for an hour because of its great speed, and if it is then brought to rest, it will resume its ordinary time-rate, but it will bear the mark of being half an hour slow! Well then, if two ships passed each other, and each considered the other to be moving at 163,000 miles per second and to be moving at half-time, when the two ships came together again, observers on each ship would expect the clock on the other ship to be half an hour slower than their own. But it is not possible for each clock to be slower than the other. What, then, would happen? This problem is called the clock paradox.
Actually, it is not a paradox at all. If one ship just flashed by the other and both crews swore the other ship’s clock was slow, it would not matter which clock was “really” slow, because the two ships would separate forever. The two clocks would never be brought to the same place at the same time in order to be matched, and the clock paradox would never arise. Indeed, Einstein’s Special Theory of Relativity only applies to uniform motion, so it is only the steady separation we are talking about.
Suppose, though, the two ships did come together after the flash-past, so that the clocks could be compared. In order for that to happen, there must be some new factor. At least one ship must accelerate. Suppose ship B did so—slowing down, traveling in a huge curve to point itself in the direction of A, then speeding up until it catches up with A. Of course, B might choose to consider itself at rest; by its chosen frame of reference, it is A that does all the changing, speeding up backward to come to B. If the two ships were all there were to the universe, then indeed the symmetry would keep the clock paradox in being.
However, A and B are not all there is to the universe—and that upsets the symmetry. When B accelerates, it is doing so with reference not only to A but to all the rest of the universe besides. If B chooses to consider itself at rest, it must consider not only A, but all the galaxies without exception, to be accelerating with respect to itself. It is B against the universe, in short. Under these circumstances, it is B’s clock that ends up half an hour slow; not A’s.
This phenomenon affects notions of space travel. If astronauts leaving Earth speed up to near the speed of light, their rate of time passage would be much slower than ours. They might reach a distant destination and return in what seemed to them weeks, though on the earth many centuries would have passed. If time really slows in motion, one might journey even to a distant star in one’s own lifetime. But of course one would have to say good-bye to one’s own generation and the world one knew, and return to a world of the future.
GRAVITY AND EINSTEIN’S GENERAL THEORY
In the Special Theory of Relativity, Einstein did not deal with accelerated motion or gravitation: These were treated in his General Theory of Relativity, published in 1915. The General Theory presented a completely altered view of gravitation. It was viewed as a property of space rather than as a force between bodies. As the result of the presence of matter, space becomes curved, and bodies follow the line of least resistance among the curves, so to speak. Strange as Einstein’s idea seemed, it was able to explain something that the Newtonian law of gravity had not been able to explain.
The greatest triumph of Newton’s law of gravity had come in 1846 with the discovery of Neptune (see chapter 3). After that, nothing seemed capable of shaking Newton’s law of gravity. And yet one planetary motion remained unexplained. The planet Mercury’s point of nearest approach to the sun, its perihelion, changes from one trip to the next; it advances steadily in the course of the planet’s revolutions around the sun. Astronomers were able to account for most of this irregularity as due to perturbations of its orbit by the pull of the neighboring planets.
Indeed, there had been some feeling in the early days of work with the theory of gravitation that perturbations arising from the shifting pull of one planet on another might eventually act to break up the delicate mechanism of the solar system. In the earliest decades of the nineteenth century, however, Laplace showed that the solar system was not so delicate. The perturbations are all cyclic, and orbital irregularities never increase to more than a certain amount in any direction. In the long run, the solar system is stable, and astronomers were more certain than ever that all particular irregularities could be worked out by taking perturbations into account.
This assumption, however, did not work for Mercury. After all the perturbations had been allowed for, there was still an unexplained advance of Mercury’s perihelion by an amount equal to 43 seconds of arc per century. This motion, discovered by Leverrier in 1845, is not much: in 4,000 years it adds up only to the width of the moon. It was enough, however, to upset astronomers.
Leverrier suggested that this deviation might be caused by a small, undiscovered planet closer to the sun than Mercury. For decades astronomers searched for the supposed planet (called Vulcan), and many were the reports of its discovery. All the reports turned out to be mistaken. Finally it was agreed that Vulcan did not exist.
Then Einstein’s General Theory of Relativity supplied the answer. It showed that the perihelion of any revolving body should have a motion beyond that predicted by Newton’s law. When this new calculation was applied to Mercury, the planet’s shift of perihelion fit it exactly. Planets farther from the sun than Mercury should show a progressively smaller shift of perihelion. In 1960, the perihelion of Venus’s orbit had been fou
nd to be advancing about 8 seconds of arc per century; this shift fits Einstein’s theory almost exactly.
More impressive were two unexpected new phenomena that only Einstein’s theory predicted. First, Einstein maintained that an intense gravitational field should slow down the vibrations of atoms. The slowdown would be evidenced by a shift of spectral lines toward the red (the Einstein shift). Casting about for a gravitational field strong enough to produce this effect, Eddington suggested the white dwarfs: light leaving such a condensed star against its powerful surface gravity might lose a detectable amount of energy. In 1925, W. S. Adams, who had been the first to demonstrate the enormous density of such stars, studied the spectral lines in the light of white dwarfs and found the necessary red shift.
The verification of Einstein’s second prediction was even more dramatic. His theory said a gravitational field would bend light-rays. Einstein calculated that a ray of light just skimming the sun’s surface would be bent out of a straight line by 1.75 seconds of arc (figure 8.4). How could that be checked? Well, if stars beyond the sun and just off its edge could be observed during an eclipse of the sun and their positions compared with what they were against the background when the sun did not interfere, any shift resulting from bending of their light should show up. Since Einstein had published his paper on general relativity in 1915, the test had to wait until after the end of the First World War. In 1919, the British Royal Astronomical Society organized an expedition to make the test by witnessing a total eclipse visible from the island of Principe, a small Portuguese-owned island off West Africa. The stars did shift position. Einstein had been verified again.
Figure 8.4. The gravitational bending of light waves, postulated by Einstein in the General Theory of Relativity.