by Michio Kaku
In Planck’s mind, these two constants, Planck’s constant and the speed of light, laid down the limits of common sense and Newtonian physics. We cannot see the fundamentally weird nature of physical reality because of the smallness of Planck’s constant and the immensity of the speed of light. If relativity and the quantum theory violated common sense, it was only because we live our entire life in a tiny corner of the universe, in a sheltered world where velocities are low compared to the speed of light and objects are so large we never encounter Planck’s constant. Nature, however, does not care about our common sense, but created a universe based on subatomic particles that routinely go near the speed of light and obey Planck’s formula.
In the summer of 1906, Planck sent his assistant, Max von Laue, to meet this obscure civil servant who appeared out of nowhere to challenge the legacy of Isaac Newton. They were supposed to meet in the waiting room of the patent office but comically walked right past each other because von Laue expected to see an imposing, authoritative figure. When Einstein finally introduced himself, von Laue was surprised to find someone completely different, a surprisingly young and casually dressed civil servant. They became lifelong friends. (However, von Laue knew a bad cigar when he saw one. When Einstein offered him a cigar, von Laue discretely threw it into the Aare River when Einstein wasn’t looking as they talked and crossed a bridge.)
With the blessing of Max Planck, the work of Einstein gradually began to attract the attention of other physicists. Ironically, one of Einstein’s old professors at the Polytechnic, who had called him a “lazy dog” for cutting his classes, took a particular interest in the work of his former student. The mathematician Hermann Minkowski plunged ahead and developed the equations of relativity even further, trying to reformulate Einstein’s observation that space turns into time and vice versa the faster you move. Minkowski put this into mathematical language and concluded that space and time formed a four-dimensional unity. Suddenly, everyone was talking about the fourth dimension.
For example, on a map, two coordinates (length and width) are required to locate any point. If you add a third dimension, height, then you can locate any object in space, from the tip of your nose to the ends of the universe. The visible world around us is thus three-dimensional. Writers like H. G. Wells had conjectured that perhaps time could be viewed as a fourth dimension, such that any event could be located by giving its three-dimensional coordinates and the time at which it occurs. Thus, if you want to meet someone in New York City, you might say, “Meet me at the corner of 42nd Street and Fifth Avenue, on the twentieth floor, at noon.” Four numbers uniquely specify the event. But Wells’s fourth dimension was only an idea without any mathematical or physical content.
Minkowski then rewrote Einstein’s equations to reveal this beautiful four-dimensional structure, forever linking space and time into a four-dimensional fabric. Minkowski wrote, “From now on, space and time separately have vanished into the merest shadows, and only a kind of union of the two will preserve any independent reality.”
At first, Einstein was not impressed. In fact, he even wrote derisively, “The main thing is the content, not the mathematics. With mathematics, you can prove anything.” Einstein believed that at the core of relativity lay basic physical principles, not pretty but meaningless four-dimensional mathematics, which he called “superfluous erudition.” To him, the essential thing was to have a clear and simple picture (e.g., trains, falling elevators, rockets), and the mathematics would come later. In fact, at this point he thought that mathematics represented only the bookkeeping necessary to track what was happening in the picture.
Einstein would write, half in jest, “Since the mathematicians have attacked the relativity theory, I myself no longer understand it anymore.” With time, however, he began to appreciate the full power of Minkowski’s work and its deep philosophical implications. What Minkowski had shown was that it was possible to unify two seemingly different concepts by using the power of symmetry. Space and time were now to be viewed as different states of the same object. Similarly, energy and matter, as well as electricity and magnetism, could be related via the fourth dimension. Unification through symmetry became one of Einstein’s guiding principles for the rest of his life.
Think of a snowflake, for example. If you rotate the snowflake by 60 degrees, the snowflake remains the same. Mathematically, we say that objects that maintain their form after a rotation are called “covariant.” Minkowski showed that Einstein’s equations, like a snowflake, remain covariant when space and time are rotated as four-dimensional objects.
In other words, a new principle of physics was being born, and it further refined the work of Einstein: The equations of physics must be Lorentz covariant (i.e., maintain the same form under a Lorentz transformation). Einstein would later admit that without the four-dimensional mathematics of Minkowski, relativity “might have remained stuck in its diapers.” What was remarkable was that this new four-dimensional physics allowed physicists to compress all the equations of relativity into a remarkably compact form. For example, every electrical engineering student and physicist, when first studying Maxwell’s series of eight partial differential equations, finds that they are fiendishly difficult. But Minkowski’s new mathematics collapsed Maxwell’s equations down to just two. (In fact, one can prove using four-dimensional mathematics that Maxwell’s equations are the simplest possible equations describing light.) For the first time, physicists appreciated the power of symmetry in their equations. When a physicist talks about “beauty and elegance” in physics, what he or she often really means is that symmetry allows one to unify a large number of diverse phenomena and concepts into a remarkably compact form. The more beautiful an equation is, the more symmetry it possesses, and the more phenomena it can explain in the shortest amount of space.
Thus, the power of symmetry allows us to unify disparate pieces into their harmonious, integral whole. Rotations of a snowflake, for example, allow us to see the unity that exists between each point on the snowflake. Rotating in four-dimensional space unifies the concept of space and time, turning one into the other as the velocity is increased. This beautiful, elegant concept, that symmetry unifies seemingly dissimilar entities into a pleasing, harmonious whole, guided Einstein for the next fifty years.
Paradoxically, as soon as Einstein completed the theory of special relativity, he began to lose interest, preferring to contemplate another, deeper question, the problem of gravity and acceleration, which seemed beyond the reach of special relativity. Einstein had given birth to relativity theory, but like any loving parent, he immediately realized its potential faults and tried to correct them. (More will be said about this later.)
Meanwhile, experimental evidence began to confirm some of his ideas, raising his visibility within the physics community. The Michelson-Morley experiment was repeated, each time yielding the same negative result and casting doubt on the entire aether theory. Meanwhile, experiments on the photoelectric effect confirmed Einstein’s equations. Furthermore, in 1908 experiments on high-speed electrons seemed to prove that the mass of the electron increased the faster it moved. Buoyed by the experimental successes slowly piling up for his theories, he applied for a lectureship (privatdozent) position at the nearby University of Bern. This position was below that of a professor, but it offered the advantage that he could simultaneously continue with his job at the patent office. He submitted his relativity thesis as well as other published works. At first, he was rejected by the department head, Aime Foster, who declared that relativity theory was incomprehensible. His second try was successful.
In 1908, with evidence mounting that Einstein had made major breakthroughs in physics, he was seriously considered for a more prestigious position at the University of Zurich. He faced stiff competition, however, from an old acquaintance, Friedrich Adler. Both top candidates for the position were Jewish, which was a handicap, but Adler was the son of the founder of the Austrian Social Democratic Party, to which many fac
ulty members were sympathetic, so it appeared as if Einstein would be passed over for the position. It was surprising, therefore, that Adler himself argued strongly for Einstein to take the position. He was a shrewd observer of character and sized up Einstein correctly. He wrote eloquently of Einstein’s outstanding abilities as a physicist, but noted, “As a student he was treated contemptuously by the professors…. He has no understanding of how to get on with important people.” Due to Adler’s extraordinary self-sacrifice, Einstein was awarded the position and began his meteoric climb up the academic ladder. He now returned to Zurich, but this time as a professor, not as a failed, unemployed physicist and misfit. When he found an apartment in Zurich, he was delighted to learn that Adler lived one floor just below his, and they became good friends.
In 1909, Einstein gave his maiden lecture at his first major physics conference in Salzburg, where many luminaries, including Max Planck, were in attendance. In his talk, “The Development of Our Views on the Nature and Constitution of Radiation,” he forcefully brought the equation E=mc2 to the world. Einstein, used to scrimping on funds for his lunch, marveled at the opulence of this conference. He recalled, “The festivities ended in the Hotel National, with the most opulent banquet I have ever attended in my life. It encouraged me to say to the Genevan patrician sitting next to me: Do you know what Calvin would have done had he been here?…He would have erected an enormous stake and had us burnt for sinful extravagance. The man never addressed another word to me.”
Einstein’s talk was the first time in history that anyone had clearly presented the concept of “duality” in physics, the concept that light can have dual properties, either as a wave, as Maxwell had suggested in the previous century, or as a particle, as Newton had suggested. Whether one saw light as a particle or as a wave depended on the experiment. For low-energy experiments, where the wavelength of the light beam is large, the wave picture was more useful. For high-energy beams, where the wavelength of light is extremely small, the particle picture was more suitable. This concept (which decades later would be attributed to Danish physicist Niels Bohr) proved to be a fundamental observation of the nature of matter and energy and one of the richest sources of research on the quantum theory.
Although now a professor, Einstein remained just as bohemian as ever. One student vividly recalled his maiden lecture at the University of Zurich: “He appeared in class in somewhat shabby attire, wearing pants that were too short and carrying with him a slip of paper the size of a visiting card on which he had sketched his lecture notes.”
In 1910, Einstein’s second son, Eduard, was born. Einstein, ever the restless wanderer, was already looking for a new position, apparently because some professors wanted to remove him from the university. The next year, he was offered a position at the German University of Prague’s Institute of Theoretical Physics at an increased salary. Ironically, his office was next to an insane asylum. Pondering the mysteries of physics, he often wondered if the inmates were the sane ones.
That same year, 1911, also marked the meeting of the First Solvay Conference in Brussels, funded by a wealthy Belgian industrialist, Ernest Solvay, which would highlight the work of the world’s leading scientists. This was the most important conference of its time, and it gave Einstein a chance to meet the giants of physics and exchange ideas with them. He met Marie Curie, the two-time Nobel laureate, and formed a lifelong relationship. The theory of relativity and his photon theory were the center of attention. The theme of the conference was “The Theory of Radiation and the Quanta.”
One question debated at the conference was the famous “twin paradox.” Einstein had already made mention of the strange paradoxes concerning the slowing down of time. The twin paradox was proposed by physicist Paul Langevin, who announced a simple thought experiment that probed some of the seeming contradictions in relativity theory. (At the time, the newspapers were filled with lurid stories about Langevin, who was unhappily married and involved in a scandalous romance with Marie Curie, a widow.) Langevin considered two twins living on Earth. One twin is transported near the speed of light and then returns back to Earth. Fifty years, say, may have transpired on Earth, but since time slows down on the rocket, the rocket twin has aged by only ten years. When the twins finally meet, there is a mismatch in their ages, with the rocket twin being forty years younger.
Now view the situation from the point of view of the rocket twin. From his perspective, he is at rest, and it is Earth that has blasted off, so the earth twin’s clocks become slower. When the two twins finally meet, the earth twin should be younger, not the rocket twin. But since motions should be relative, the question is, which twin is really younger? Since the two situations seem symmetrical, this puzzle even today remains a thorn in the side of any student who has tried to tackle relativity.
The resolution of the puzzle, as Einstein pointed out, is that the rocket twin, not the earth twin, has accelerated. The rocket has to slow down, stop, and then reverse, which clearly causes great stress on the rocket twin. In other words, the situations are not symmetrical because accelerations, which are not covered by the assumptions behind special relativity, only occur for the rocket twin, who is really younger.
(However, the situation becomes more puzzling if the rocket twin never returns. In this scenario, each sees by telescope the other twin slowing down in time. Since the situations are now perfectly symmetrical, then each twin is convinced that the other one is younger. Likewise, each twin is convinced that the other is compressed. So which twin is younger and thinner? As paradoxical as it seems, in relativity theory it is possible to have two twins, each younger than the other, each thinner than the other. The simplest way to determine who is really thinner or younger in all these paradoxes is to bring the two twins together, which requires yanking one of the twins around, which in turn determines which twin is “really” moving.
Although these mind-bending paradoxes were indirectly resolved in Einstein’s favor at the atomic level with studies of cosmic rays and atom smashers, this effect is so small that it was not directly seen in the laboratory until 1971, when airplanes carrying atomic clocks were sent into the air at great speeds. Because these atomic clocks can measure the passing of time with astronomical precision, scientists, by comparing the two clocks, could verify that time beat slower the faster you moved, exactly as Einstein had predicted.)
Another paradox involves two objects, each shorter than the other. Imagine a big-game hunter trying to trap a tiger about 10 feet long with a cage that is only 1 foot wide. Normally, this is impossible. Now imagine that the tiger is moving so fast that it shrinks to only 1 foot, so the cage can drop and capture the tiger. As the tiger screeches to a halt, it expands. If the cage is made of webbing, the tiger breaks the webbing. If the cage is made of concrete, the poor tiger is crushed to death.
But now look at the situation from the point of view of the tiger. If the tiger is at rest, the cage is now moving and has shrunk down to only one-tenth of a foot. How can a cage that small catch a tiger 10 feet long? The answer is that as the cage drops, it shrinks in the direction of motion, so it becomes a parallelogram, a squashed square. The two ends of the cage therefore do not necessarily hit the tiger simultaneously. What is simultaneous to the big-game hunter is not simultaneous to the tiger. If the cage is made of webbing, then the front part of the cage hits the tiger’s nose first and begins to rip. As the cage drops, it continues to rip along the tiger’s body, until the back end of the cage finally hits the tail. If the cage is made of concrete, then the tiger’s nose is crushed first. As the cage descends, it continues to crush the length of the tiger’s body, until the back end of the cage finally captures the tail.
These paradoxes even seized the public’s imagination, with the following limerick running in the humor magazine Punch:
There once was a young lady named Bright
Who could travel much faster than light
She set out one day, in a relative way
And came bac
k the previous night.
By this time, his good friend Marcel Grossman was a professor at the Polytechnic, and he sounded out Einstein to see if he wanted a position at his old school, this time as a full professor. Letters of recommendation spoke of Einstein in the highest terms. Marie Curie wrote that “mathematical physicists are unanimous in considering his work as being of the first rank.”
So, just sixteen months after arriving in Prague, he returned to Zurich and the old Polytechnic. Returning to the Polytechnic (since 1911 called the Swiss Federal Institute of Technology, or ETH), this time as a famous professor, marked a personal victory for Einstein. He left the university with his name clouded in disgrace, with professors like Weber actively sabotaging his career. He returned as the leader of the new revolution in physics. That year, he received his first nomination for the Nobel Prize in physics. His ideas were still considered too radical for the Swedish academy, and there were dissident voices among Nobel laureates who wanted to sabotage his nomination. In 1912, the Nobel Prize did not go to Einstein, but to Nils Gustaf Dalén, for his work on improving lighthouses. (Ironically, lighthouses today have been made largely obsolete by the introduction of the global positioning satellite system, which depends crucially on Einstein’s theory of relativity.)
Within another year, Einstein’s reputation was growing so rapidly that he began to get inquiries from Berlin. Max Planck was eager to capture this rising star in physics, and Germany was the unquestioned leader of the world’s research in physics, the crown jewel of German research being in Berlin. Einstein hesitated at first, since he had renounced his German citizenship and still nursed some bitter memories from his youth, but the offer was too tempting.
