The God Equation

by Michio Kaku

  Now replace the merry-go-round with the solar system. The Earth goes around the sun, so we Earthlings have the illusion that the sun exerts a force of attraction, called gravity, on the Earth. But to someone outside the solar system, they would not see a force at all; they would observe that the space around the Earth has curved, so that empty space is pushing the Earth so that it goes in a circle around the sun.

  Einstein had the brilliant observation that gravitational attraction was actually an illusion. Objects moved not because they are pulled by gravity or the centrifugal force but because they are pushed by the curvature of space around it. That’s worth repeating: gravity does not pull; space pushes.

  Shakespeare once said that all the world is a stage, and we are actors making our entrances and exits. This was the picture adopted by Newton. The world is static, and we move on this flat surface, obeying Newton’s laws.

  But Einstein abandoned this picture. The stage, he said, is curved and warped. If you walk on it, you cannot walk in a straight line. You are constantly being pushed because the floor beneath your feet is curved, and you stagger like a drunk.

  Gravitational attraction is an illusion. For example, you might be sitting in a chair right now, reading this book. Normally, you would say that gravity is pulling you down into your chair, and that is why you don’t fly off into space. But Einstein would say that you are sitting in your chair because the Earth’s mass warps the space above your head, and this warping pushes you into your chair.

  Imagine putting a heavy shot put on a large mattress. It sinks into the bed, causing it to warp. If you shoot a marble along the mattress, it moves in a curved line. In fact, it will circle the shot put. From a distance, an observer may say that there is an invisible force pulling on the marble, forcing it to orbit. But close up, you see that there is no invisible force at all. The marble does not move in a straight line because the mattress is curved, making the most direct path an ellipse.

  Figure 6. A heavy shot put on a mattress sinks into the fabric. A marble circles around the depression it creates. From a distance, it appears that a force from the shot put grabs the marble and forces it into an orbit. Actually, the marble is orbiting the shot put because the mattress is warped. In the same way, the sun’s gravity warps the starlight from distant stars, which can be measured by telescopes during an eclipse of the sun.

  Now replace the marble with the Earth, the shot put with the sun, and the mattress with space-time. Then we see that the Earth goes around the sun because the sun has warped the space around it, and the space Earth is traveling in is not flat.

  Also, think of ants moving on a crumpled sheet of paper. They cannot move in a straight line. They might feel as if a force is continually tugging on them. But to us, looking down on the ants, we see that there is no force all. This is the insight of what Einstein called general relativity: space-time is warped by heavy masses, causing the illusion of gravitational force.

  This means that general relativity is much more powerful and symmetrical than special relativity, since it describes gravity, which affects all things in space-time. Special relativity, on the other hand, only worked for objects moving smoothly in space and time in a straight line. But in our universe, almost everything is accelerating. From racing cars to helicopters to rockets, we see that they are all accelerating. General relativity works for accelerations that are continually changing at every point in space-time.

  Solar Eclipse and Gravity

  Any theory, no matter how beautiful, must eventually confront experimental verification. So Einstein seized upon several possible experiments. The first was the erratic orbit of Mercury. When calculating its orbit, astronomers found a slight anomaly. Instead of moving in a perfect ellipse, as predicted by Newton’s equations, it wobbled a bit, making a flowerlike pattern.

  To protect Newton’s laws, astronomers posited the existence of a new planet, called Vulcan, inside the orbit of Mercury. The gravity of Vulcan would tug on Mercury, causing the aberration. Earlier, we saw that this strategy allowed astronomers to discover the planet Neptune. But astronomers failed to find any observational evidence for Vulcan.

  So when Einstein recalculated the perihelion of Mercury, the spot where it is closest to the sun, using his theory of gravity, he found a slight deviation from Newton’s laws. He was ecstatic to find a perfect match with his own calculations. He found the difference from a perfect ellipse in its orbit to be 42.9 seconds of arc per century, well within the experimental result. He would recall fondly, “For some days, I was beyond myself with excitement. My boldest dreams have now come true.”

  He also realized that according to his theories light should be deflected by the sun.

  Einstein realized that the sun’s gravity would be powerful enough to bend the starlight of nearby stars. Since these stars could only be seen during a solar eclipse, Einstein proposed that an expedition be sent to witness the solar eclipse of 1919 to test his theory. (Astronomers would have to take two pictures of the night sky, one where the sun was absent and another during a solar eclipse. By comparing these two photographs, the position of the stars during the eclipse would have to move due to the sun’s gravity.) He was certain his theory would be shown to be correct. When he was asked what he would think if the experiment disproved his theory, he said that God must have made a mistake. He was convinced he was correct, he wrote to his colleagues, because it had superb mathematical beauty and symmetry.

  When this epic experiment was finally performed by astronomer Arthur Eddington, there was remarkable agreement between Einstein’s prediction and the actual result. (Today, the bending of starlight due to gravity is routinely used by astronomers. When starlight passes near a distant galaxy, light is bent, giving the appearance of a lens bending the light. These are called gravity lenses or Einstein lenses.)

  Einstein would go on to win the Nobel Prize in 1921.

  Soon, he became one of the most recognized figures on the planet, even more than most movie stars and politicians. (In 1933, he appeared with Charlie Chaplin at a movie premiere. When they were mobbed by autograph seekers, Einstein asked Chaplin, “What does all this mean?” Chaplin replied, “Nothing, absolutely nothing.” Then he said, “They cheer me because everyone understands me. They cheer you because no one understands you.”)

  Of course, a theory that would overthrow 250 years of Newtonian physics would also be met with fierce criticism. One of the skeptics leading the charge was Columbia professor Charles Lane Poor. After reading about relativity, he fumed, “I feel as if I had been wandering with Alice in Wonderland and had tea with the Mad Hatter.”

  But Planck would always reassure Einstein. He would write, “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because the opponents eventually die and a new generation grows up that is familiar with it.”

  Over the decades, there have been many challenges to relativity, but each time Einstein’s theory has been verified. In fact, as we shall see in later chapters, Einstein’s theory of relativity has reshaped the entire discipline of physics, revolutionizing our conception of the universe, its origin, and its evolution and changing the way we live.

  One easy way to confirm Einstein’s theory is to use the GPS system on your cell phone. The GPS system consists of thirty-one satellites orbiting the Earth. At any time, your cell phone can receive signals from three of them. Each of these three satellites is moving in a slightly different trajectory and angle. The computer in your cell phone then analyzes this data from the three satellites and triangulates your precise position.

  The GPS system is so accurate that it has to take tiny corrections from both special and general relativity into account.

  Since the satellites are moving at roughly 17,000 miles per hour, a clock in the GPS satellites beats slightly slower those than on Earth due to speci
al relativity, which states that higher speeds result in slower time—the phenomenon demonstrated in Einstein’s thought experiment of outracing a light beam. But since gravity is weaker the farther you move into outer space, time actually speeds up a bit due to general relativity, which states that space-time can be warped by gravitational pull—the weaker the gravitational pull, the faster time moves. This means that special and general relativity work in opposite directions, with special relativity causing the signals to slow down, while general relativity causes the signals to speed up. Your cell phone then factors in both competing effects and tells you precisely where you are located. So without special and general relativity working in tandem, you would be lost.

  Newton and Einstein: Polar Opposites

  Einstein was heralded as the next Newton, but Einstein and Newton were polar opposites in personality. Newton was a loner, reticent to the point of being antisocial. He had no lifelong friends and was incapable of everyday conversation.

  Physicist Jeremy Bernstein once said, “Everyone who had any substantial contact with Einstein came away with an overwhelming sense of the nobility of the man. A descriptive term for him that recurs again and again is ‘humanitarian’—a reference to the simple, lovable quality of his character.”

  But both Newton and Einstein shared certain key characteristics. The first was the ability to concentrate and focus tremendous mental energy. Newton could forget to eat or sleep for days when concentrating on a single problem. He would stop in the middle of a conversation and scribble on whatever was available, sometimes a napkin or the wall. Similarly, Einstein could focus on a problem for years, even decades. He even suffered a near breakdown while working on the general theory.

  Another characteristic they shared was the ability to visualize a problem in pictures. Although Newton could have written Principia entirely in terms of algebraic symbols, instead he filled the masterpiece with geometric diagrams. To use calculus with abstract symbols is relatively easy; but deriving them from triangles and squares can only be done by a master. Similarly, Einstein’s theory is filled with diagrams of trains, metersticks, and clocks.

  Search for a Unified Theory

  In the end, Einstein created two major theories. The first was special relativity, which governed the properties of light beams and space-time. It introduced a symmetry based on rotations in four dimensions. The second was general relativity, where gravity is unveiled as the bending of space-time.

  But after these two monumental achievements, he tried to reach for a third, even greater achievement. He wanted a theory that would unify all the forces of the universe into a single equation. He wanted to use the language of field theory to create an equation that could combine Maxwell’s theory of electricity and magnetism with his own theory of gravity. He tried for decades to unify these two, and failed. (Michael Faraday was actually the first to propose a unification of gravity with electromagnetism. Faraday used to go to the London Bridge and drop magnets, hoping to find some measurable effect of gravity on the magnet. He found none.)

  One reason why Einstein failed was that, in the 1920s, there was a huge hole in our understanding of the world. It would take advances in a new theory, the quantum theory, for physicists to realize that there was a missing piece of the puzzle: the nuclear force.

  But Einstein, although he was one of the founders of the quantum theory, ironically would become the quantum’s greatest adversary. He would unleash a barrage of criticisms against the quantum theory. Over the decades, the theory has met every experimental challenge and has given us a deluge of wondrous electrical appliances that fill up our lives and workplaces. However, as we shall see, his deep, subtle philosophical objections to it resonate even now.

  Skip Notes

  * To see this, let us take Z = 0. Then the sphere reduces down to a circle in the X and Y plane, just as before. We saw that as you move around this circle, we have X2 + Y2 = R2. Now, let us gradually increase Z. The circle gets smaller as we rise in the Z direction. (The circle corresponds to the lines of equal latitude on a globe.) R remains the same, but the equation for the small circle becomes X2 + Y2 + Z2 = R2, for a fixed value of Z. Now, if we let Z vary, we see that any point on the sphere has coordinates given by X, Y, and Z, such that the three-dimensional Pythagorean theorem holds. So in summary, the points on a sphere can all be described by the Pythagorean theorem in three dimensions, such that R remains the same, but X, Y, and Z all vary as you move around the sphere. Einstein’s great insight was to generalize this to four dimensions, with the fourth dimension being time.

  3

  RISE OF THE QUANTUM

  While Einstein was single-handedly creating this vast new theory based on space and time and matter and energy, a parallel development in physics was unraveling this age-old question: What is matter made of? This would lead to the next great theory of physics, the quantum theory.

  After Newton had finished his theory of gravity, he performed numerous experiments in alchemy, trying to understand the nature of matter. His bouts of depression, it is theorized, were because of his experiments with mercury, a poison known to cause neurological symptoms. However, little was known about the fundamental properties of matter, and little was learned from the work of these early alchemists, who spent much of their time and energy trying to convert lead into gold.

  It would take several centuries to slowly reveal the secrets of matter. By the 1800s, chemists began to find and isolate the basic elements of nature—elements that, in turn, could not be decomposed into anything simpler. While the stunning advances in physics were pioneered by mathematics, the breakthroughs in chemistry came mainly from tedious hours toiling in a laboratory.

  In 1869, Dmitry Mendeleyev had a dream, in which all the elements of nature fell into a table. Upon awakening, he quickly began to arrange the known elements into a regular table, showing that there was a pattern to the elements. Out of the chaos of chemistry suddenly came order and predictability. The sixty or so known elements could be arranged into this simple table, but there were gaps, and Mendeleyev was able to predict the properties of these missing elements. When these elements were actually found in the laboratory, as predicted, it sealed the reputation of Mendeleyev.

  But why were the elements arranged in such a regular pattern?

  The next development occurred in 1898, when Marie and Pierre Curie isolated a new series of unstable elements, never before seen. Without any power source, radium glowed brightly in the laboratory, violating one of the cherished principles of physics, the conservation of energy (that energy can never be created or destroyed). The energy of these radium rays seemed to come from nowhere. Clearly, a new theory would be necessary.

  Until then, chemists believed that the fundamental ingredients of matter, the elements, were eternal, that elements like hydrogen or oxygen were stable for all time. But in their laboratories, chemists could see that elements like radium were decaying into other elements, releasing radiation in the process.

  It was also possible to calculate how quickly these unstable elements were decaying, which could be measured in thousands or even billions of years. The Curies’ discoveries helped settle a long-standing debate. Geologists, amazed at the glacial pace of rock formations, realized that the Earth must be billions of years old. But Lord Kelvin, one of the giants of classical Victorian physics, calculated that a molten Earth would cool down in a matter of a few million years. Who was right?

  As it turns out, it was the geologists. Lord Kelvin did not understand that a new force of nature, the one being discovered by the Curies called the nuclear force, could add to the Earth’s heat. Since radioactive decay could take place over billions of years, it meant that the Earth’s core could be heated by the decay of uranium, thorium, and other radioactive elements. So the enormous power of shattering earthquakes, thundering volcanoes, and slow, grinding continental drift all originate from the nuclear forc
e.

  In 1910, Ernest Rutherford put a piece of glowing radium in a lead box with a minuscule hole. A tiny beam of radiation emerged from the hole, aimed at a thin sheet of gold. It was expected that the atoms of gold would absorb the radiation. To his shock, he found that the beam from the radium went right through the sheet, as if it weren’t there.

  This was an astonishing result: it meant that atoms were composed primarily of empty space. We sometimes demonstrate this to students. We put a piece of harmless uranium in their hand and a Geiger counter beneath it, which can detect radiation. Students are shocked to hear the Geiger counter clicking away because their body is hollow.

  In the early 1900s, the standard picture of the atom was the raisin pie model—that is, the atom was like a pie of positive charge, with raisins of electrons sprinkled inside. Gradually, a radically new picture of the atom began to emerge. The atom was basically hollow, consisting of a swarm of electrons circling a tiny dense core, called the nucleus. Rutherford’s experiment helped prove this because his radioactive beam would occasionally be deflected by the tightly packed particles in the nucleus. By analyzing the number, frequency, and angles of deflection, he was able to estimate the size of the nucleus of the atom. It was one hundred thousand times smaller than the atom itself.

  Later, scientists determined that the nucleus was, in turn, made of even tinier subatomic particles: protons (which carried positive charge) and neutrons (which carried no charge). The entire Mendeleyev table, it seemed, could be created using only three subatomic particles: the electron, proton, and neutron. But what equation did these particles obey?