by Michio Kaku
In 1913, Einstein was elected to the Prussian Academy of Sciences and later offered a position in Berlin at the university. He would be made director of the Kaiser Wilhelm Institute for Physics. But beyond the titles, which meant little to him, what made the offer especially attractive was the fact that there were no teaching obligations. (Although Einstein was a popular lecturer among the students, noted for treating his students with respect and kindness, teaching detracted from his main interest, general relativity.)
In 1914, Einstein arrived in Berlin to meet the faculty. He felt a bit nervous as they looked him over. Einstein would write, “The gentlemen in Berlin are gambling on me as if I were a prize hen. As for myself, I don’t even know whether I’m going to lay another egg.” The thirty-five-year-old rebel, with strange politics and stranger attire, soon had to adjust to the stiff, upper-crust ways of the Prussian Academy of Sciences, where members addressed each other as “Privy Councillor” or “Your Excellency.” Einstein would muse, “It seems that most members confine themselves to displaying a peacock-like grandeur in writing; otherwise they are quite human.”
Einstein’s triumphant march from the patent office in Bern to the top ranks of German research was not without its personal toll. As his fame began to rise within the scientific community, his personal life began to unravel. These were Einstein’s most productive years, bearing fruit that would eventually reshape human history, and almost impossible demands were being placed on his time, estranging him from his wife and children.
Einstein wrote that living with Mileva was like living in a cemetery, and when alone he tried to avoid being in the same room with her. His friends split on the question of who was mainly to blame. Many believed that Mileva was becoming increasingly isolated and resentful of her famous husband. Even Mileva’s friends were distressed that she had aged considerably in those years and had let her appearance deteriorate noticeably. She was becoming increasingly shrill and cold, jealous even of the time he spent with his colleagues. When she discovered a letter of congratulations sent to Einstein by Anna Schmid (who knew Einstein during his brief time in Aarau and had since married), she blew her top, precipitating perhaps one of the angriest rifts in their already shaky marriage.
On the other hand, others believed that Einstein was hardly the perfect husband, constantly on the road, leaving Mileva to raise two children mainly by herself. Travel at the turn of the century was notoriously difficult, and extensive travels were taking him away for days and weeks. Like passing ships in the night, when he was home, they would only meet briefly at night for dinner or the theater. He was so immersed in the abstract world of mathematics that he had little emotional energy to connect with his wife. Worse, the more she complained to him about his absences, the more he withdrew into the world of physics.
It is probably safe to say that there is some truth in both allegations and it is pointless to assign blame. In retrospect, it was probably inevitable that the marriage would experience enormous strains. Perhaps their friends were right years ago when they said that the two were incompatible.
But the final break was precipitated by his acceptance of the offer from Berlin. Mileva was reluctant about going to Berlin. Perhaps being a Slav in the center of a Teutonic culture was too intimidating to her; more importantly, many of Einstein’s relatives lived in Berlin, and Mileva feared being under their harsh, disapproving gaze. It was no secret that her in-laws hated her. At first, Mileva and the children made the trip to Berlin with Einstein, but then suddenly she left for Zurich, taking the children with her. They would never be united again. Einstein, who cherished his children more than anyone, was devastated. From that point on, he was forced to maintain a long-distance relationship with his sons, making the grueling ten-hour trip from Berlin to Zurich for visits. (When Mileva was eventually awarded custody of the children, Einstein’s secretary, Helen Dukas, wrote that he cried all the way home.)
But what probably also precipitated the rupture was the growing presence of a certain cousin of Einstein’s in Berlin. He would confess, “I live a very withdrawn life but not a lonely one, thanks to the care of a female cousin who actually drew me to Berlin in the first place.”
Elsa Lowenthal was a double cousin; her mother and Einstein’s mother were sisters, and their grandfathers were brothers. She was divorced, living with her two daughters, Margot and Ilse, just upstairs from her parents (Einstein’s aunt and uncle). She and Einstein met briefly in 1912 when he visited Berlin. By then, Einstein had apparently decided that his marriage to Mileva was finished and that divorce was inevitable. However, he feared the repercussions a divorce would have on his young sons.
Ever since they were children, Elsa had taken a liking to Einstein. She confessed to having fallen in love with him as a child when she heard him play Mozart. But what apparently most attracted her was his rising stardom in the academic world, his respect by physicists around the world. In fact, she made it no secret that she loved to bask in this fame. Like Mileva, she was older, four years older than Einstein. But that is where the resemblance ended. In fact, they were like polar opposites. Einstein, in fleeing Mileva, was apparently going overboard in the other direction. While Mileva was often uncaring of her appearance and looked continually harassed, Elsa was highly bourgeois and conscious of class ranking. She was always trying to cultivate acquaintances in intellectual circles in Berlin and would proudly show off Einstein to all her friends in high society. Unlike Mileva, who was laconic, withdrawn, and moody, Elsa was a social butterfly, fluttering between dinner parties and theater openings. And unlike Mileva, who gave up trying to reform her husband, Elsa was more of a mother, continually correcting his manners while devoting her full energies to helping him fulfill his destiny. A Russian journalist later summed up the relationship between Einstein and Elsa: “She is all love for her great husband, always ready to shield him from the harsh intrusions of life and to ensure the peace of mind necessary for his great ideas to mature. She is filled with the realization of his great purpose as a thinker and with the tenderest feelings of companion, wife, and mother towards a remarkable, exquisite, grown-up child.”
After Mileva stormed out of Berlin in 1915, taking the children with her, Einstein and Elsa got even closer. What consumed Einstein during this important period, however, was not love, but the universe itself.
PART II
SECOND PICTURE
Warped Space-Time
CHAPTER 4
General Relativity and “the Happiest Thought of My Life”
Einstein was still not satisfied. He was already ranked among the top physicists of his time, yet he was restless. He realized that there were at least two glaring holes in his theory of relativity. First, it was based entirely on inertial motions. In nature, however, almost nothing is inertial. Everything is in a state of constant acceleration: the jostling of trains, the zigzags of falling leaves, the rotation of the earth around the sun, the motion of heavenly bodies. Relativity theory failed to account for even the commonest acceleration found on the earth.
Second, the theory said nothing about gravity. It made the sweeping claim that it was a universal symmetry of nature, applying to all sectors of the universe, yet gravity seemed beyond its reach. This was also quite embarrassing, because gravity is everywhere. The deficiencies of relativity were obvious. Since the speed of light was the ultimate speed of the universe, relativity theory said that it would take eight minutes for any disturbance on the sun to reach the earth. This, however, contradicted Newton’s theory of gravity, which stated that gravitational effects were instantaneous. (The speed of Newton’s gravity was infinite, since the speed of light does not appear anywhere in Newton’s equations.) Einstein therefore needed to completely overhaul Newton’s equations to incorporate the speed of light.
In short, Einstein realized the immensity of the problem of generalizing his relativity theory to include accelerations and gravity. He began to call his earlier theory of 1905 the “special theory of relativi
ty,” to differentiate it from the more powerful “general theory of relativity” that was needed to describe gravity. When he told Max Planck of his ambitious program, Planck warned him, “As an older friend, I must advise you against it for in the first place you will not succeed, and even if you succeed, no one will believe you.” But Planck also realized the importance of the problem when he said, “If you are successful, you will be called the next Copernicus.”
The key insight into a new theory of gravity took place while Einstein was still slaving over patent applications as a lowly civil servant back in 1907. He would recall, “I was sitting in a chair in the patent office at Bern when all of a sudden, a thought occurred to me: If a person falls freely, he will not feel his own weight. I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.”
In an instant, Einstein realized that if he had fallen over off his chair, he would be momentarily weightless. For example, if you are in an elevator and the cable suddenly breaks, you would be in free fall; you would fall at the same rate as the elevator floor. Since both you and the elevator are now falling at the same speed, it would appear that you are weightless, floating in air. Similarly, Einstein realized that if he fell over off his chair, he would be in free fall and the effect of gravity would be canceled perfectly by his acceleration, making him appear weightless.
This concept is an old one. It was known to Galileo, who in an apocryphal story dropped a small rock and a large cannonball from the Leaning Tower of Pisa. He was the first to show that all objects on Earth accelerate at precisely the same rate under gravity (32 feet per second squared). Newton also knew this fact when he realized that the planets and the moon were actually in a state of free fall in their orbit around the sun or the earth. Every astronaut who has ever been shot into outer space also realizes that gravity can be canceled by acceleration. In a rocket ship, everything inside, including the floor, the instruments, and you, falls at the same rate. Thus, when you look around, everything is floating. Your feet drift above the floor, giving the illusion that gravity has vanished, because the floor is falling along with your body. And if an astronaut takes a space walk outside the ship, he or she does not suddenly drop to the earth, but instead floats gently alongside the rocket because both the rocket and the astronaut are falling in unison even as they orbit the earth. (Gravity has not actually disappeared in outer space, as many science books erroneously claim. The sun’s gravity is powerful enough to whip the planet Pluto in its orbit billions of miles from Earth. Gravity has not disappeared; it has just been canceled by the falling of the rocket ship beneath your feet.)
This is called the “equivalence principle,” in which all masses fall at the same rate under gravity (more precisely, the inertial mass is the same as the gravitational mass). This was indeed an old idea, almost a curiosity to Galileo and Newton, but in the hands of a seasoned physicist like Einstein, it was to become the foundation of a new relativistic theory of gravity. Einstein went one giant step further than Galileo or Newton. He formulated his next postulate, the postulate behind general relativity: The laws of physics in an accelerating frame or a gravitating frame are indistinguishable. Remarkably, this simple statement, in the hands of Einstein, became the basis of a theory that would give us warped space, black holes, and the creation of the universe.
After this brilliant insight of 1907 in the patent office, it took years for Einstein’s new theory of gravity to gestate. A new picture of gravity was emerging from the equivalence principle, but it wouldn’t be until 1911 that he began to publish the fruits of his thoughts. The first consequence of the equivalence principle is the fact that light must bend under gravity. The idea that gravity might influence light beams is an old one, dating back at least to the time of Isaac Newton. In his book Opticks, he asked whether or not gravity can influence starlight: “Do not Bodies act upon light at a Distance, and by their Action bend its Rays; and is not this Action strongest at the least Distance?” Unfortunately, given the technology of the seventeenth century, he could give no answer.
But now Einstein, after more than two hundred years, returned to this question. Consider turning on a flashlight inside a rocket ship that is accelerating in outer space. Because the rocket is accelerating upward, the light beam droops downward. Now invoke the equivalence principle. Since the physics inside the spaceship must be indistinguishable from the physics on Earth, it means that gravity must also bend light. In a few brief steps, Einstein was led to a new physical phenomenon, the bending of light due to gravity. He immediately realized that such an effect was calculable.
The largest gravitational field in the solar system is generated by the sun, so Einstein asked himself whether the sun was sufficient to bend starlight from distant stars. This could be tested by taking two photographs of the same collection of stars in the sky at two different seasons. The first photo of these stars would be taken at night when starlight is undisturbed; the second photo would be taken several months later when the sun is positioned directly in front of this same collection of stars. By comparing the two photographs, one might be able to measure how the stars have shifted slightly in the sun’s vicinity due to the sun’s gravity. Because the sun overwhelms the light coming from the stars, any experiment on the bending of starlight would have to be performed during a solar eclipse, when the moon blocks out the light from the sun and the stars become visible during the daytime. Einstein reasoned that photographs of the day sky taken during an eclipse, compared to photographs taken of the same sky at night, should show a slight distortion in the location of the stars in the vicinity of the sun. (The presence of the moon also bends starlight a bit because of the moon’s gravity, but this is a very tiny amount compared to the bending of starlight caused by the sun, which is much larger. Thus, the bending of starlight during an eclipse is not affected by the presence of the moon.)
The equivalence principle could help him to calculate the approximate motion of light beams as they were pulled by gravity, but it still did not tell him anything about gravity itself. What was lacking was a field theory of gravity. Recall that Maxwell’s equations describe a genuine field theory, in which lines of force are like a spider web that could vibrate and support waves traveling along the lines of force. Einstein sought a gravitational field whose lines of force could support gravitational vibrations that traveled at the speed of light.
Around 1912, after years of concentrated thought, he slowly began to realize that he needed to overhaul our understanding of space and time; to do so required new geometries beyond those inherited from the ancient Greeks. The key observation that sent him on the road to curved space-time was a paradox, sometimes referred to as “Ehrenfest’s paradox,” that his friend Paul Ehrenfest once posed to Einstein. Consider a simple merry-go-round or a spinning disk. At rest, we know that its circumference is equal to p times the diameter. Once the merry-go-round is set into motion, however, the outer rim travels faster than the interior and hence, according to relativity, should shrink more than the interior, distorting the shape of the merry-go-round. This means that the circumference has shrunk and is now less than p times the diameter; that is, the surface is no longer flat. Space is curved. The surface of the merry-go-round can be compared to the area within the Arctic Circle. We can measure the diameter of the Arctic Circle by walking from one point on the circle, directly across the North Pole, to the opposite point on the circle. Then we can measure the circumference of the Arctic Circle. If we compare the two, we also find that the circumference is less than p times the diameter because the earth’s surface is curved. But for the last two thousand years, physicists and mathematicians relied on Euclidean geometry, which is based on flat surfaces. What would happen if they imagined a geometry based on curved surfaces?
Once we realize that space can be curved, a startling new picture emerges. Think of a heavy rock placed on a bed. The rock, of course, will sink into the bed. Now shoot a tiny marble over the bed. The marble w
ill not move in a straight line but in a curved line around the rock. There are two ways to analyze this effect. From a distance, a Newtonian may say that there is a mysterious “force” that emanates from the rock to the marble, forcing the marble to change its path. This force, although invisible, reaches out and pulls on the marble. However, a relativist may see an entirely different picture. To a relativist looking at the bed close up, there is no force that pulls the marble. There is just the depression in the bed, which dictates the motion of the marble. As the marble moves, the surface of the bed “pushes” the marble until it moves in a circular motion.
Now replace the rock with the sun, the marble with the earth, and the bed with space and time. Newton would say that an invisible force called “gravity” pulls the earth around the sun. Einstein would reply that there is no gravitational pull at all. The earth is deflected around the sun because the curvature of space itself is pushing the earth. In a sense, gravity does not pull, but space pushes.
In this picture, Einstein could explain why it would take eight minutes for any disturbance on the sun to reach the earth. For example, if we suddenly remove the rock, the bed will spring back to normal, creating ripples that travel at a definite speed across the bed. Similarly, if the sun were to disappear, it would create a shock wave of warped space that would travel at the speed of light. This picture was so simple and elegant that he could explain the essential idea to his second son, Eduard, who asked him why he was so famous. Einstein replied, “When a blind beetle crawls over the surface of a curved branch, it doesn’t notice that the track it has covered is indeed curved. I was lucky enough to notice what the beetle didn’t notice.”
Newton, in his landmark Philosophiae Naturalis Principia Mathematica, confessed that he was unable to explain the origin of this mysterious pull, which acted instantly throughout the universe. He coined his famous phrase hypotheses non fingo (I frame no hypotheses) because of his inability to explain where gravity came from. With Einstein, we see that gravity is caused by the bending of space and time. “Force” is now revealed to be an illusion, a by-product of geometry. In this picture, the reason why we are standing on the earth is not because the earth’s gravity pulls us down. According to Einstein, there is no gravitational pull. The earth warps the space-time continuum around our bodies, so space itself pushes us down to the floor. Thus, it is the presence of matter that warps space around it, giving us the illusion that there is a gravitational force pulling on neighboring objects.
