by Michio Kaku
Then it suddenly hit him, the key to the entire problem. Einstein recalled, “A storm broke loose in my mind.” The answer was simple and elegant: time can beat at different rates throughout the universe, depending on how fast you moved. Imagine clocks scattered at different points in space, each one announcing a different time, each one ticking at a different rate. One second on Earth was not the same length as one second on the moon or one second on Jupiter. In fact, the faster you moved, the more time slowed down. (Einstein once joked that in relativity theory, he placed a clock at every point in the universe, each one running at a different rate, but in real life he didn’t have enough money to buy even one.) This meant that events that were simultaneous in one frame were not necessarily simultaneous in another frame, as Newton thought. He had finally tapped into “God’s thoughts.” He would recall excitedly, “The solution came to me suddenly with the thought that our concepts and laws of space and time can only claim validity insofar as they stand in a clear relation to our experiences…. By a revision of the concept of simultaneity into a more malleable form, I thus arrived at the theory of relativity.”
For example, remember that in the paradox of the speeding motorist, the police officer was traveling neck and neck with the speeding light beam, while the officer himself claimed that the light beam was speeding away from him at precisely the speed of light, no matter how much he gunned his engines. The only way to reconcile these two pictures is to have the brain of the officer slow down. Time slows down for the policeman. If we could have seen the officer’s wristwatch from the roadside, we would have seen that it nearly stopped and that his facial expressions were frozen in time. Thus, from our point of view, we saw him speeding neck and neck with the light beam, but his clocks (and his brain) were nearly stopped. When we interviewed the officer later, we found that he perceived the light beam to be speeding away, only because his brain and clocks were running much slower.
To complete his theory, Einstein also incorporated the Lorentz-FitzGerald contraction, except that it was space itself that was contracted, not the atoms, as Lorentz and FitzGerald thought. (The combined effect of space contraction and time dilation is today called the “Lorentz transformation.”) Thus, he could dispense entirely with the aether theory. Summarizing the path that he took to relativity, he would write, “I owe more to Maxwell than to anyone.” Apparently, although he was dimly aware of the Michelson-Morley experiment, the inspiration for relativity did not come from the aether wind, but directly from Maxwell’s equations.
The day after this revelation, Einstein went back to Besso’s home and, without even saying hello, he blurted out, “Thank you, I’ve completely solved the problem.” He would proudly recall, “An analysis of the concept of time was my solution. Time cannot be absolutely defined, and there is an inseparable relation between time and signal velocity.” For the next six weeks, he furiously worked out every mathematical detail of his brilliant insight, leading to a paper that is arguably one of the most important scientific papers of all time. According to his son, he then went straight to bed for two weeks after giving the paper to Mileva to check for any mathematical errors. The final paper, “On the Electrodynamics of Moving Bodies,” was scribbled on thirty-one handwritten pages, but it changed world history.
In the paper, he does not acknowledge any other physicist; he only gives thanks to Michele Besso. (Einstein was aware of Lorentz’s early work on the subject, but not the Lorentz contraction itself, which Einstein found independently.) It was finally published in Annalen der Physik in September 1905, in volume 17. In fact, Einstein would publish three of his pathbreaking papers in that famous volume 17. His colleague Max Born has written, volume 17 is “one of the most remarkable volumes in the whole scientific literature. It contains three papers by Einstein, each dealing with a different subject and each today acknowledged to be a masterpiece.” (Copies of that famous volume recently sold for $15,000 at an auction in 1994.)
With almost breathtaking sweep, Einstein began his paper by proclaiming that his theories worked not just for light, but were truths about the universe itself. Remarkably, he derived all his work from two simple postulates applying to inertial frames (i.e., objects that move with constant velocity with respect to each other):
The laws of physics are the same in all inertial frames.
The speed of light is a constant in all inertial frames.
These two deceptively simple principles mark the most profound insights into the nature of the universe since Newton’s work. From them, one can derive an entirely new picture of space and time.
First, in one masterful stroke, Einstein elegantly proved that if the speed of light was indeed a constant of nature, then the most general solution was the Lorentz transformation. He then showed that Maxwell’s equations did indeed respect that principle. Last, he showed that velocities add in a peculiar way. Although Newton, observing the motion of sailing ships, concluded that velocities could add without limit, Einstein concluded that the speed of light was the ultimate velocity in the universe. Imagine, for the moment, that you are in a rocket speeding at 90% the speed of light away from Earth. Now fire a bullet inside the rocket that is also going at 90% the speed of light. According to Newtonian physics, the bullet should be going at 180% the speed of light, thus exceeding light velocity. But Einstein showed that because meter sticks are shortening and time is slowing down, the sum of these velocities is actually close to 99% the speed of light. In fact, Einstein could show that no matter how hard you tried, you could never boost yourself beyond the speed of light. Light velocity was the ultimate speed limit in the universe.
We never see these bizarre distortions in our experience because we never travel near the speed of light. For everyday velocities, Newton’s laws are perfectly fine. This is the fundamental reason why it took over two hundred years to discover the first correction to Newton’s laws. But now imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps 1 inch in length, for example, although its height would remain the same. Because the passengers in the car are compressed down to 1 inch, we might expect them to yell and scream as their bones are crushed. In fact, the passengers see nothing wrong, since everything inside the car, including the atoms in their bodies, is squeezed as well.
As the car slows down to a stop, it would slowly expand from 1 inch to about 10 feet, and the passengers would walk out as if nothing happened. Who is really compressed? You or the car? According to relativity, you cannot tell, since the concept of length has no absolute meaning.
In retrospect, one can see that others came tantalizingly close to discovering relativity. Lorentz and FitzGerald obtained the same contraction, but they had the totally wrong understanding of the result, thinking it was an electromechanical deformation of the atoms, rather than a subtle transformation of space and time itself. Henri Poincaré, recognized as the greatest French mathematician of his time, came close. He understood that the speed of light must be a constant in all inertial frames, and even showed that Maxwell’s equations retained the same form under a Lorentz transformation. However, he too refused to abandon the Newtonian framework of the aether and thought that these distortions were strictly a phenomenon of electricity and magnetism.
Einstein then pushed further and made the next fateful leap. He wrote a small paper, almost a footnote, late in 1905 that would change world history. If meter sticks and clocks became distorted the faster you moved, then everything you can measure with meter sticks and clocks must also change, including matter and energy. In fact, matter and energy could change into each other. For example, Einstein could show that the mass of an object increased the faster it moved. (Its mass would in fact become infinite if you hit the speed of light—which is impossible, which proves the unattainability of the speed of light.) This meant that the energy of motion was somehow being transformed into increasing the mass of
the object. Thus, matter and energy are interchangeable. If you calculated precisely how much energy was being converted into mass, in a few simple lines you could show that E=mc2, the most celebrated equation of all time. Since the speed of light was a fantastically large number, and its square was even larger, this meant that even a tiny amount of matter could release a fabulous amount of energy. A few teaspoons of matter, for example, has the energy of several hydrogen bombs. In fact, a piece of matter the size of a house might be enough to crack the earth in half.
Einstein’s formula was not simply an academic exercise, because he believed that it might explain a curious fact discovered by Marie Curie, that just an ounce of radium emitted 4,000 calories of heat per hour indefinitely, seemingly violating the first law of thermodynamics (which states that the total amount of energy is always constant or conserved). He concluded that there should be a slight decrease in its mass as radium radiated away energy (an amount too small to be measured using the equipment of 1905). “The idea is amusing and enticing; but whether the Almighty is laughing at it and is leading me up the garden path—that I cannot know,” he wrote. He concluded that a direct verification of his conjecture “for the time being probably lies beyond the realm of possible experience.”
Why hadn’t this untapped energy been noticed before? He compared this to a fabulously rich man who kept his wealth secret by never spending a cent.
Banesh Hoffman, a former student, wrote, “Imagine the audacity of such a step…. Every clod of earth, every feather, every speck of dust becoming a prodigious reservoir of untapped energy. There was no way of verifying this at the time. Yet in presenting his equation in 1907 Einstein spoke of it as the most important consequence of his theory of relativity. His extraordinary ability to see far ahead is shown by the fact that his equation was not verified…until some twenty-five years later.”
Once again, the relativity principle forced a major revision in classical physics. Before, physicists believed in the conservation of energy, the first law of thermodynamics, which states that the total amount of energy can never be created or destroyed. Now, physicists considered the total combined amount of matter and energy as being conserved.
Einstein’s restless mind tackled yet another problem that same year, the photoelectric effect. Heinrich Hertz, back in 1887, noticed that if a light beam struck a metal, under certain circumstances a small electric current could be created. This is the same principle that underlies much of modern electronics. Solar cells convert ordinary sunlight into electrical power, which can be used to energize our calculators. TV cameras take light beams from the subject and convert them into electric currents, which eventually wind up on our TV screen.
However, at the turn of the century, this was still a total mystery. Somehow, the light beam was knocking electrons out of the metal, but how? Newton had believed that light consisted of tiny particles that he called “corpuscles,” but physicists were convinced that light was a wave, and according to classical wave theory its energy was independent of its frequency. For example, although red and green light have different frequencies, they should have the same energy, and hence, when they hit a piece of metal, the energy of the ejected electrons should be the same as well. Similarly, classical wave theory said that if one turned up the intensity of the light beam by adding more lamps, then the energy of the ejected electrons should increase. The work of Philipp Lenard, however, demonstrated that the energy of the ejected electrons was strictly dependent on the frequency or color of the light beam, not its intensity, which was opposite the prediction of the wave theory.
Einstein sought to explain the photoelectric effect by using the new “quantum theory” recently discovered by Max Planck in Berlin in 1900. Planck made one of the most radical departures from classical physics by assuming that energy was not a smooth quantity, like a liquid, but occurred in definite, discrete packets, called “quanta.” The energy of each quantum was proportional to its frequency. The proportionality constant was a new constant of nature, now called “Planck’s constant.” One reason why the world of the atom and the quantum seems so bizarre is the fact that Planck’s constant is a very small number. Einstein reasoned that if energy occurred in discrete packets, then light itself must be quantized. (Einstein’s packet of “light quanta” was later christened the “photon,” a particle of light, by chemist Gilbert Lewis in 1926.) Einstein reasoned that if the energy of the photon was proportional to its frequency, then the energy of the ejected electron should also be proportional to its frequency, contrary to classical physics. (It’s amusing to note that on the popular TV series Star Trek, the crew of the Enterprise fires “photon torpedoes” at its enemies. In reality, the simplest launcher of photon torpedoes is a flashlight.)
Einstein’s new picture, a quantum theory of light, made a direct prediction that could be tested experimentally. By turning up the frequency of the incoming light beam, one should be able to measure a smooth rise in the voltage generated in the metal. This historic paper (which would eventually win him the Nobel Prize in physics) was published on June 9, 1905, with the title “On a Heuristic Point of View Concerning the Production and Transformation of Light.” With it, the photon was born, as well as the quantum theory of light.
In yet another article written during the “miracle year” of 1905, Einstein tackled the problem of the atom. Although the atomic theory was remarkably successful in determining the properties of gases and chemical reactions, there was no direct proof of their existence, as Mach and other critics were fond of pointing out. Einstein reasoned that one might be able to prove the existence of atoms by noticing their effect on small particles in a liquid. “Brownian motion,” for example, refers to tiny, random motions of small particles suspended in a liquid. This property was discovered in 1828 by the botanist Robert Brown, who observed tiny pollen grains under a microscope exhibiting strange random motions. At first, he thought that these zigzag movements were like the motion of male sperm cells. But he found that this strange aberrant behavior was also exhibited by tiny grains of glass and granite.
Some had speculated that Brownian motion might be due to the random impacts of molecules, but no one could formulate a reasonable theory of this. Einstein, however, took the next decisive step. He reasoned that although atoms were too small to be observed, one could estimate their size and behavior by calculating their cumulative impact on large objects. If one seriously believed in the theory of atoms, then the atomic theory should be able to calculate the physical dimensions of atoms by analyzing Brownian motion. By assuming that the random collisions of trillions upon trillions of molecules of water were causing random motions of a dust particle, he was able to compute the size and weight of atoms, thereby giving experimental evidence of the existence of atoms.
It was nothing short of amazing that by peering into a simple microscope, Einstein could calculate that a gram of hydrogen contained 3.03 × 1023 atoms, which is close to the correct value. The title was “On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat” (July 18). This simple paper, in effect, gave the first experimental proof of the existence of atoms. (Ironically, just a year after Einstein calculated the size of atoms, the physicist Ludwig Boltzmann, who had pioneered the theory of atoms, committed suicide, in part because of the incessant ridicule he received for advancing the atomic theory.) After Einstein wrote these four historic papers, he also submitted an earlier paper of his, on the size of molecules, to his advisor, Professor Alfred Kleiner, as his dissertation. That night, he got drunk with Mileva.
At first, his dissertation was refused. But on January 15, 1906, Einstein was finally granted a Ph.D. from the University of Zurich. He could now call himself Dr. Einstein. The birth of the new physics all took place at the Einstein residence on Kramgasse 49 in Bern. (Today, it is called the “Einstein House.” As you gaze through its beautiful bay window facing the street, you can read a plaque which states that through this window,
the theory of relativity was created. On the other wall, you can see a picture of the atomic bomb.)
Thus, 1905 was truly an annus mirabilis in the history of science. If we are to find another comparable miracle year, we would have to look back to 1666, when Isaac Newton, twenty-three years old, hit upon the universal law of gravitation, the integral and differential calculus, the binomial theorem, and his theory of color.
Einstein had finished the year 1905 laying down the photon theory, providing evidence for the existence of atoms, and toppling the framework of Newtonian physics, each of them worthy of international acclaim. He was disappointed, however, in the deafening silence that ensued. His work, it seemed, was totally ignored. Disheartened, Einstein went about his personal life, raising his child and toiling by himself at the patent office. Perhaps the thought of pioneering new worlds in physics was all a pipe dream.
In early 1906, however, the first glimmer of response riveted Einstein’s attention. He got just a single letter, but it was from perhaps the most important physicist of the time, Max Planck, who immediately understood the radical implications of Einstein’s work. What drew Planck to the theory of relativity was that it elevated a quantity, the speed of light, into a fundamental constant of nature. Planck’s constant, for example, demarcated the classical world from the subatomic world of the quantum. We are shielded from the strange properties of atoms because of the smallness of Planck’s constant. Similarly, felt Planck, Einstein was raising the speed of light into a new constant of nature. We were shielded from the equally bizarre world of cosmic physics by the huge value of the speed of light.