Almost a century earlier, the Italian scientist Amedeo Avogadro (1776–1856) had developed the hypothesis—correct, as it turned out—that equal volumes of any gas, when measured at the same temperature and pressure, will have the same number of molecules. That led to a difficult quest: figuring out just how many this was.
The volume usually chosen is that occupied by a mole of the gas (its molecular weight in grams), which is 22.4 liters at standard temperature and pressure. The number of molecules under such conditions later became known as Avogadro’s number. Determining it precisely was, and still is, rather difficult. A current estimate is approximately 6.02214 x 1023. (This is a big number: that many unpopped popcorn kernels when spread across the United States would cover the country nine miles deep.)29
Most previous measurements of molecules had been done by studying gases. But as Einstein noted in the first sentence of his paper, “The physical phenomena observed in liquids have thus far not served for the determination of molecular sizes.” In this dissertation (after a few math and data corrections were later made), Einstein was the first person able to get a respectable result using liquids.
His method involved making use of data about viscosity, which is how much resistance a liquid offers to an object that tries to move through it. Tar and molasses, for example, are highly viscous. If you dissolve sugar in water, the solution’s viscosity increases as it gets more syrupy. Einstein envisioned the sugar molecules gradually diffusing their way through the smaller water molecules. He was able to come up with two equations, each containing the two unknown variables—the size of the sugar molecules and the number of them in the water—that he was trying to determine. He could then solve for these unknown variables. Doing so, he got a result for Avogadro’s number that was 2.1 x 1023.
That, unfortunately, was not very close. When he submitted his paper to the Annalen der Physik in August, right after it had been accepted by Zurich University, the editor Paul Drude (who was blissfully unaware of Einstein’s earlier desire to ridicule him) held up its publication because he knew of some better data on the properties of sugar solutions. Using this new data, Einstein came up with a result that was closer to correct: 4.15 x 1023.
A few years later, a French student tested the approach experimentally and discovered something amiss. So Einstein asked an assistant in Zurich to look at it all over again. He found a minor error, which when corrected produced a result of 6.56 x 1023, which ended up being quite respectable.30
Einstein later said, perhaps half-jokingly, that when he submitted his thesis, Professor Kleiner rejected it for being too short, so he added one more sentence and it was promptly accepted. There is no documentary evidence for this.31 Either way, his thesis actually became one of his most cited and practically useful papers, with applications in such diverse fields as cement mixing, dairy production, and aerosol products. And even though it did not help him get an academic job, it did make it possible for him to become known, finally, as Dr. Einstein.
Brownian Motion, May 1905
Eleven days after finishing his dissertation, Einstein produced another paper exploring evidence of things unseen. As he had been doing since 1901, he relied on statistical analysis of the random actions of invisible particles to show how they were reflected in the visible world.
In doing so, Einstein explained a phenomenon, known as Brownian motion, that had been puzzling scientists for almost eighty years: why small particles suspended in a liquid such as water are observed to jiggle around. And as a byproduct, he pretty much settled once and for all that atoms and molecules actually existed as physical objects.
Brownian motion was named after the Scottish botanist Robert Brown, who in 1828 had published detailed observations about how minuscule pollen particles suspended in water can be seen to wiggle and wander when examined under a strong microscope. The study was replicated with other particles, including filings from the Sphinx, and a variety of explanations was offered. Perhaps it had something to do with tiny water currents or the effect of light. But none of these theories proved plausible.
With the rise in the 1870s of the kinetic theory, which used the random motions of molecules to explain things like the behavior of gases, some tried to use it to explain Brownian motion. But because the suspended particles were 10,000 times larger than a water molecule, it seemed that a molecule would not have the power to budge the particle any more than a baseball could budge an object that was a half-mile in diameter.32
Einstein showed that even though one collision could not budge a particle, the effect of millions of random collisions per second could explain the jig observed by Brown. “In this paper,” he announced in his first sentence, “it will be shown that, according to the molecular-kinetic theory of heat, bodies of a microscopically visible size suspended in liquids must, as a result of thermal molecular motions, perform motions of such magnitudes that they can be easily observed with a microscope.”33
He went on to say something that seems, on the surface, somewhat puzzling: his paper was not an attempt to explain the observations of Brownian motion. Indeed, he acted as if he wasn’t even sure that the motions he deduced from his theory were the same as those observed by Brown: “It is possible that the motions to be discussed here are identical with so-called Brownian molecular motion; however, the data available to me on the latter are so imprecise that I could not form a judgment on the question.” Later, he distanced his work even further from intending to be an explanation of Brownian motion: “I discovered that, according to atomistic theory, there would have to be a movement of suspended microscopic particles open to observations, without knowing that observations concerning the Brownian motion were already long familiar.”34
At first glance his demurral that he was dealing with Brownian motion seems odd, even disingenuous. After all, he had written Conrad Habicht a few months earlier, “Such movement of suspended bodies has actually been observed by physiologists who call it Brownian molecular motion.” Yet Einstein’s point was both true and significant: his paper did not start with the observed facts of Brownian motion and build toward an explanation of it. Rather, it was a continuation of his earlier statistical analysis of how the actions of molecules could be manifest in the visible world.
In other words, Einstein wanted to assert that he had produced a theory that was deduced from grand principles and postulates, not a theory that was constructed by examining physical data (just as he had made plain that his light quanta paper had not started with the photo-electric effect data gathered by Philipp Lenard). It was a distinction he would also make, as we shall soon see, when insisting that his theory of relativity did not derive merely from trying to explain experimental results about the speed of light and the ether.
Einstein realized that a bump from a single water molecule would not cause a suspended pollen particle to move enough to be visible. However, at any given moment, the particle was being hit from all sides by thousands of molecules. There would be some moments when a lot more bumps happened to hit one particular side of the particle. Then, in another moment, a different side might get the heaviest barrage.
The result would be random little lurches that would result in what is known as a random walk. The best way for us to envision this is to imagine a drunk who starts at a lamppost and lurches one step in a random direction every second. After two such lurches he may have gone back and forth to return to the lamp. Or he may be two steps away in the same direction. Or he may be one step west and one step northeast. A little mathematical plotting and charting reveals an interesting thing about such a random walk: statistically, the drunk’s distance from the lamp will be proportional to the square root of the number of seconds that have elapsed.35
Einstein realized that it was neither possible nor necessary to measure each zig and zag of Brownian motion, nor to measure the particle’s velocity at any moment. But it was rather easy to measure the total distances of randomly lurching particles as these distances grew over time.
Einstein wanted concrete predictions that could be tested, so he used both his theoretical knowledge and experimental data about viscosity and diffusion rates to come up with precise predictions showing the distance a particle should move depending on its size and the temperature of the liquid. For example, he predicted, in the case of a particle with a diameter of one thousandth of a millimeter in water at 17 degrees centigrade, “the mean displacement in one minute would be about 6 microns.”
Here was something that could actually be tested, and with great consequence. “If the motion discussed here can be observed,” he wrote, “then classical thermodynamics can no longer be viewed as strictly valid.” Better at theorizing than at conducting experiments, Einstein ended his paper with a charming exhortation: “Let us hope that a researcher will soon succeed in solving the problem presented here, which is so important for the theory of heat.”
Within months, a German experimenter named Henry Seidentopf, using a powerful microscope, confirmed Einstein’s predictions. For all practical purposes, the physical reality of atoms and molecules was now conclusively proven. “At the time atoms and molecules were still far from being regarded as real,” the theoretical physicist Max Born later recalled. “I think that these investigations of Einstein have done more than any other work to convince physicists of the reality of atoms and molecules.”36
As lagniappe, Einstein’s paper also provided yet another way to determine Avogadro’s number. “It bristles with new ideas,” Abraham Pais said of the paper. “The final conclusion, that Avogadro’s number can essentially be determined from observations with an ordinary microscope, never fails to cause a moment of astonishment even if one has read the paper before and therefore knows the punch line.”
A strength of Einstein’s mind was that it could juggle a variety of ideas simultaneously. Even as he was pondering dancing particles in a liquid, he had been wrestling with a different theory that involved moving bodies and the speed of light. A day or so after sending in his Brownian motion paper, he was talking to his friend Michele Besso when a new brainstorm struck. It would produce, as he wrote Habicht in his famous letter of that month, “a modification of the theory of space and time.”
CHAPTER SIX
SPECIAL RELATIVITY 1905
The Bern Clock Tower
The Background
Relativity is a simple concept. It asserts that the fundamental laws of physics are the same whatever your state of motion.
For the special case of observers moving at a constant velocity, this concept is pretty easy to accept. Imagine a man in an armchair at home and a woman in an airplane gliding very smoothly above. Each can pour a cup of coffee, bounce a ball, shine a flashlight, or heat a muffin in a microwave and have the same laws of physics apply.
In fact, there is no way to determine which of them is “in motion” and which is “at rest.” The man in the armchair could consider himself at rest and the plane in motion. And the woman in the plane could consider herself at rest and the earth as gliding past. There is no experiment that can prove who is right.
Indeed, there is no absolute right. All that can be said is that each is moving relative to the other. And of course, both are moving very rapidly relative to other planets, stars, and galaxies.*
The special theory of relativity that Einstein developed in 1905 applies only to this special case (hence the name): a situation in which the observers are moving at a constant velocity relative to one another—uniformly in a straight line at a steady speed—referred to as an “inertial reference system.”1
It’s harder to make the more general case that a person who is accelerating or turning or rotating or slamming on the brakes or moving in an arbitrary manner is not in some form of absolute motion, because coffee sloshes and balls roll away in a different manner than for people on a smoothly gliding train, plane, or planet. It would take Einstein a decade more, as we shall see, to come up with what he called a general theory of relativity, which incorporated accelerated motion into a theory of gravity and attempted to apply the concept of relativity to it.2
The story of relativity best begins in 1632, when Galileo articulated the principle that the laws of motion and mechanics (the laws of electromagnetism had not yet been discovered) were the same in all constant-velocity reference frames. In his Dialogue Concerning the Two Chief World Systems, Galileo wanted to defend Copernicus’s idea that the earth does not rest motionless at the center of the universe with everything else revolving around it. Skeptics contended that if the earth was moving, as Copernicus said, we’d feel it. Galileo refuted this with a brilliantly clear thought experiment about being inside the cabin of a smoothly sailing ship:
Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully, have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.3
There is no better description of relativity, or at least of how that principle applies to systems that are moving at a constant velocity relative to each other.
Inside Galileo’s ship, it is easy to have a conversation, because the air that carries the sound waves is moving smoothly along with the people in the chamber. Likewise, if one of Galileo’s passengers dropped a pebble into a bowl of water, the ripples would emanate the same way they would if the bowl were resting on shore; that’s because the water propagating the ripples is moving smoothly along with the bowl and everything else in the chamber.
Sound waves and water waves are easily explained by classical mechanics. They are simply a traveling disturbance in some medium. That is why sound cannot travel through a vacuum. But it can travel through such things as air or water or metal. For example, sound waves move through room temperature air, as a vibrating disturbance that compresses and rarefies the air, at about 770 miles per hour.
Deep inside Galileo’s ship, sound and water waves behave as they do on land, because the air in the chamber and the water in the bowls are moving at the same velocity as the passengers. But now imagine that you go up on deck and look at the waves out in the ocean, or that you measure the speed of the sound waves from the horn of another boat. The speed at which these waves come toward you depends on your motion relative to the medium (the water or air) propagating them.
In other words, the speed at which an ocean wave reaches you will depend on how fast you are moving through the water toward or away from the source of the wave. The speed of a sound wave relative to you will likewise depend on your motion relative to the air that’s propagating the sound wave.
Those relative speeds add up. Imagine that you are standing in the ocean as the waves come toward you at 10 miles per hour. If you jump on a Jet Ski and head directly into the waves at 40 miles per hour, you will see them moving toward you and zipping past you at a speed (relative to you) of 50 miles per hour. Likewise, imagine that sound waves are coming at you from a distant boat horn, rippling through still air at 770 miles per hour toward the shore. If you jump on your Jet Ski and head toward the horn at 40 miles per hour, the sound waves will be moving toward you and zipping past you at a speed (relative to you) of 810 miles per hour.
All of this led to a question that Einstein had been pondering since age 16, when he imagined riding alongside a light beam:
Does light behave the same way?
Newton had conceived of light as primarily a stream of emitted particles. But by Einstein’s day, most scientists accepted the rival theory, propounded by Newton’s contemporary Christiaan Huygens, that light should be considered a wave.
A wide variety of experiments had confirmed the wave theory by the late nineteenth century. For example, Thomas Young did a famous experiment, now replicated by high school students, showing how light passing through two slits produces an interference pattern that resembles that of water waves going through two slits. In each case, the crests and troughs of the waves emanating from each slit reinforce each other in some places and cancel each other out in some places.
James Clerk Maxwell helped to enshrine this wave theory when he successfully conjectured a connection between light, electricity, and magnetism. He came up with equations that described the behavior of electric and magnetic fields, and when they were combined they predicted electromagnetic waves. Maxwell found that these electromagnetic waves had to travel at a certain speed: approximately 186,000 miles per second.* That was the speed that scientists had already measured for light, and it was obviously not a mere coincidence.4
It became clear that light was the visible manifestation of a whole spectrum of electromagnetic waves. This includes what we now call AM radio signals (with a wavelength of 300 yards), FM radio signals (3 yards), and microwaves (3 inches). As the wavelengths get shorter (and the frequency of the wave cycles thus increases), they produce the spectrum of visible light, ranging from red (25 millionths of an inch) to violet (14 millionths of an inch). Even shorter wavelengths produce ultraviolet rays, X-rays, and gamma rays. When we speak of “light” and the “speed of light,” we mean all electromagnetic waves, not just the ones that are visible to our eyes.