The God Equation Read online

  When I was a child and read about this, I wondered why we could only see the smallest piece of the EM spectrum. What a waste, I thought. But the reason, I now realize, is that the wavelength of an EM wave is roughly the size of the antenna that produces it. Hence, your cell phone is only a few inches in size because that is the size of the antenna, which is about the wavelength of the EM waves being broadcasted. Similarly, the size of a cell in your retina is about the size of the wavelength of colors you can see. Hence we can see only colors whose wavelengths are the size of our cells. All the other colors of the EM spectrum are invisible because they are either too big or too small to be detected by our retinal cells. So if the cells of our eyes were the size of a house, we might be able to see all the radio and microwave radiation swirling around us.

  Similarly, if the cells of our eyes were the size of atoms, we might be able to see X-rays.

  Yet another application of Maxwell’s equation is the way in which EM energy can power the whole planet. Although oil and coal have to be shipped by boat and train over vast distances, electrical energy can be sent over wires with the flick of a switch, electrifying entire cities.

  This, in turn, led to a famous controversy between two giants of the electric age, Thomas Edison and Nikola Tesla. Edison was the genius behind many electrical inventions including the light bulb, motion pictures, the phonograph, the ticker tape, and hundreds of other marvels. He was also the first to wire a street with electricity, in this case Pearl Street in downtown Manhattan.

  This created the second great revolution in technology, the electric age.

  Edison assumed that direct current, or DC (which always moves in the same direction and never varies in voltage), would be the best way to transmit electricity. Instead of DC power, Tesla, who used to work for Edison and helped lay the groundwork for the telecommunication network of today, advocated AC power (alternating current, so electricity reverses direction about sixty times a second). This resulted in the celebrated battle of the currents, with giant corporations investing millions in rival technologies, with General Electric backing Edison and Westinghouse backing Tesla. The future of the electric revolution would hinge on who won this conflict, Edison’s DC or Tesla’s AC.

  Although Edison was the mastermind behind electricity and one of the architects of the modern world, he did not fully understand Maxwell’s equations. This would be a very costly mistake. In fact, he thumbed his nose at scientists who knew too much mathematics. (In a famous story, he would often ask scientists looking for a job to calculate the volume of a light bulb. He would smile as these scientists tried to use advanced mathematics to tediously calculate the shape of the light bulb and then its volume. Afterward, Edison would simply pour water into an empty light bulb and then pour it into a graduated beaker.)

  Engineers knew that wires strung over many miles lost a significant amount of energy if they carried low voltages, as advocated by Edison. So Tesla’s high-voltage power lines were economically preferred, but high-voltages cables were too dangerous to be introduced into your living room. The trick was to use efficient high-voltage cables from the power plant to your city, and then somehow transform the high voltage to low voltage just before it entered your living room. The key was to use transformers.

  As we recall, Maxwell showed that a moving magnetic field created an electric current, and vice versa. This allows you to create a transformer that can rapidly change the voltage in a wire. For example, the voltage of the electrical cables from a power station may carry thousands of volts. But the transformer located just outside your house can reduce the voltage to 110 volts, which easily powers your microwave oven and refrigerator.

  If these fields are static and do not change, then they cannot be converted into each other. Because it is constantly changing, AC electricity can easily be converted into magnetic fields that are then converted back into electric fields, but at a lower voltage, meaning that AC current can easily change voltage using transformers; but DC current (because it is constant in voltage and not alternating) cannot.

  In the end, Edison lost the battle and the considerable funds he invested in DC technology. That is the price of ignoring Maxwell’s equations.

  The End of Science?

  In addition to explaining the mysteries of nature and bringing in a new era of economic prosperity, a combination of Newton’s and Maxwell’s equations gave us a very convincing theory of everything. Or at least everything then known.

  By 1900 prominent scientists were proclaiming the “end of science.” Thus, the turn of the last century was a heady time to be alive. Everything that could be discovered had already been discovered, or so it seemed.

  Physicists at that time did not realize that the two great pillars of science, Newton’s and Maxwell’s equations, were actually incompatible. They contradicted each other.

  One of these two great pillars had to fall. And a sixteen-year-old boy held the key. That boy would be born the very year that Maxwell died, 1879.

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  EINSTEIN’S QUEST FOR UNIFICATION

  As a teenager, Einstein asked himself a question that would alter the course of the twentieth century. He asked himself:

  Can you outrace a light beam?

  Years later, he would write that this simple question contained the key to his theory of relativity.

  Earlier, he had read a children’s book, Aaron David Bernstein’s Popular Books on Natural Science, that asks you to imagine racing alongside a telegraph wire. Instead, Einstein envisioned running along a light beam, which should look frozen. Racing neck and neck alongside the beam, the light waves should be stationary, he thought, as Newton might have predicted.

  But even as a sixteen-year-old boy, Einstein realized that no one had ever seen a frozen light beam before. Something was missing. He would ponder this question for the next ten years.

  Unfortunately, many people considered him to be a failure. Although he was a brilliant student, his professors hated his freewheeling, bohemian lifestyle. Because he already knew most of the material, he often cut classes, so his professors wrote unflattering letters of recommendation; and every time he applied for a job he was turned down. Unemployed and desperate, he took a job tutoring (from which he got fired for arguing with his employer). He once considered selling insurance to support his girlfriend and child. (Can you imagine opening your door one day and seeing Einstein trying to sell you insurance?) Unemployable, he considered himself to be a drain on his family. In one letter, he wrote despondently, “I am nothing but a burden to my relatives….It would surely be better if I did not live at all.”

  He finally managed to get a job as a clerk, third class, at the patent office in Bern. It was humiliating but actually a blessing in disguise. In the quiet of the patent office Einstein could return to the old question that had haunted him since he was a child. From there, he would launch a revolution that turned physics and the world upside down.

  As a student at the famed École Polytechnique in Switzerland, he had come across Maxwell’s equations for light. He asked himself, what happens to Maxwell’s equations if you travel at the speed of light? Remarkably, no one had ever asked that question before. Using Maxwell’s theory, Einstein calculated the speed of a light beam in a moving object, such as a train. He expected that the speed of the light beam, as seen by a stationary outside observer, would simply be its usual speed plus the speed of the train. According to Newtonian mechanics, velocities can simply add. For example, if you throw a baseball while riding on a train, a stationary observer would say that the speed of the ball is just the speed of the train plus the speed of the ball relative to the train. Likewise, velocities can also subtract. So, if you traveled neck and neck alongside a light beam, it should look stationary.

  To his shock, he found that the light beam was not frozen at all but sped away at the same velocity. But this was impossible,
he thought. According to Newton you can always catch up with anything if you move fast enough. That was common sense. But Maxwell’s equations said that you could never catch up to light, which always moved at the same velocity, no matter how fast you traveled.

  To Einstein, this insight was monumental. Either Newton or Maxwell was correct. The other had to be wrong. But how could it be that you could never catch up to light? At the patent office, he had plenty of time to ponder this question. One day, in the spring of 1905, it struck him while riding the train in Bern. “A storm broke loose in my mind,” he would recall.

  His brilliant insight was that since the speed of light is measured by clocks and metersticks, and since the speed of light is constant no matter how fast you move, space and time must be distorted in order to keep the speed of light constant!

  It meant that if you are on a fast-moving spaceship, then clocks inside the ship beat slower than clocks on the Earth. Time slows down the faster you move—this phenomenon is described by Einstein’s special relativity. So the question What time is it? depends on how fast you have been moving. If the rocket ship is traveling near the speed of light, and we observe it from the ground using a telescope, everyone in the rocket ship seems to move in slow motion. Also, everything in the ship seems to be compressed. Finally, everything in the rocket ship is heavier. Surprisingly, to someone in the rocket ship, everything appears normal.

  Einstein would later recall, “I owe more to Maxwell than to anyone.” Today, this experiment can be done routinely. If you place an atomic clock on an airplane, and compare it with a clock on the Earth, you can see that the clock on the airplane has slowed down (by a small factor, one part in a trillion).

  But if space and time can vary, then everything you can measure must also vary, including matter and energy. And the faster you move, the heavier you become. But where does the extra mass come from? It comes from the energy of motion. This means that some of the energy of motion is turned into mass.

  The precise relationship between matter and energy was E = mc2. This equation, as we shall see, answered one of the deepest questions in all of science: Why does the sun shine? The sun shines because when you compress hydrogen atoms at great temperatures, some of the mass of the hydrogen gets converted to energy.

  The key to understanding the universe is unification. For relativity, it was the unification of space and time and matter and energy. But how is this unification accomplished?

  Symmetry and Beauty

  To poets and artists, beauty is an ethereal, aesthetic quality that evokes great emotion and passion.

  To a physicist, beauty is symmetry. Equations are beautiful because they have a symmetry—that is, if you rearrange or reshuffle the components, the equation remains the same. It is invariant under this transformation. Think of a kaleidoscope. It takes a random jumble of colored shapes and, with mirrors, makes numerous copies and then arranges these images symmetrically in a circle. So something that is chaotic suddenly becomes ordered and beautiful because of symmetry.

  Similarly, a snowflake is beautiful because, if we rotate it by 60 degrees, it remains the same. A sphere has even more symmetry. You can rotate it by any amount around its center, and the sphere looks identical. To a physicist, an equation is beautiful if we rearrange its various particles and components inside the equation and find the result does not change—in other words, if we find it has symmetry among its parts. The mathematician G. H. Hardy once wrote, “A mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics.” And that beauty is symmetry.

  We saw earlier that if you take Newton’s gravitational force for the Earth going around the sun, the radius of the Earth’s orbit is constant. The coordinates X and Y change, but R does not. This can also be generalized to three dimensions.

  Imagine sitting on the surface of the Earth, where your location is given by three dimensions: X, Y, and Z are your coordinates (see figure 5). As you travel anywhere along the surface of the Earth, the radius of the Earth, R, remains the same, where R2 = X2 + Y2 + Z2. This is a three-dimensional version of the Pythagorean theorem.*

  FIgure 5. As you wander over the surface of the Earth, the radius, R, of Earth is a constant, an invariant, although your coordinates X, Y, and Z constantly change into one another. So the three-dimensional Pythagorean theorem is the mathematical expression of this symmetry.

  Now, if we take Einstein’s equations and then rotate space into time and time into space, the equations remain the same. This means that the three dimensions of space are now joined with the dimension of time, T, which becomes the fourth dimension. Einstein showed that the quantity X2 + Y2 + Z2 − T2 (with time expressed in certain units) remains the same, which is a modified version of the Pythagorean theorem in four dimensions. (Notice that the time coordinate has an additional minus sign. This means that although relativity is invariant under rotations in four dimensions, the time dimension is treated slightly differently from the other three spatial dimensions.) So Einstein’s equations are symmetric in four dimensions.

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  Maxwell’s equations were first written down around 1861, the year the American Civil War began. Earlier, we noted that they possess a symmetry that turns electric and magnetic fields into each other. But Maxwell’s equations possess an additional hidden symmetry. If we alter Maxwell’s equations in four dimensions by interchanging X, Y, Z, and T among themselves as Einstein did in the 1910s, they remain the same. This means that, if physicists had not been blinded by the success of Newtonian physics, then relativity might have been discovered during the Civil War!

  Gravity as Curved Space

  Although Einstein showed that space, time, matter, and energy were all part of a larger four-dimensional symmetry, there was one glaring gap in his equation: they said nothing about gravity and accelerations. He was not satisfied. He wanted to generalize his earlier theory, which he called special relativity, so that it included gravity and accelerated motions, creating a more powerful general theory of relativity.

  His colleague physicist Max Planck, however, warned him of the difficulty of creating a theory that combined relativity and gravity. He said, “As an older friend, I must advise against it. For in the first place, you won’t succeed, and even if you do, no one will believe you.” But then he added, “If you are successful, you will be called the next Copernicus.”

  It was obvious to any physicist that Newton’s theory of gravity and Einstein’s theory were at odds. If the sun were to suddenly disappear without a trace, then Einstein claimed that it would take eight minutes for the Earth to feel the absence. Newton’s famous equation for gravity does not mention the speed of light. Hence gravity travels instantaneously, violating relativity, so the Earth should immediately feel the effect of the missing sun.

  Einstein had pondered the question of light for ten years, from when he was sixteen to twenty-six. He would spend the next ten years, until he was thirty-six, concentrating on the theory of gravity. The key to the whole puzzle occurred to him one day while leaning back on his chair, almost causing him to fall over. In that brief instant, he realized that if he had fallen over, he would be weightless. Then he realized that this might be the key to a theory of gravity. He would fondly recall that it was “the happiest thought of his life.”

  Galileo realized that if you fell off a building, you would momentarily be weightless, but only Einstein realized how to exploit this fact to reveal the secret of gravity. Imagine for a moment being in an elevator and the cable is cut. You would fall, but the floor falls at the same rate, so inside the elevator, you begin to float, as if there is no gravity (at least until the elevator hits the ground). Inside the elevator, gravity was precisely canceled by
the acceleration of a falling elevator. This is called the equivalence principle, that acceleration in one frame is indistinguishable from gravity in another frame.

  When our astronauts in space appear weightless on TV, it is not because gravity has disappeared from space. There is plenty of gravity throughout the entire solar system. The reason is because their rocket is falling at exactly the same rate as they are. Like Newton’s imaginary cannonball shot from a mountaintop, they and their capsule are both in free fall around the Earth. Thus, inside the ship, it is an optical illusion that they are weightless, since everything, including your body and the ship itself, are falling at the same rate.

  Einstein then applied this to a children’s merry-go-round. According to relativity, the faster you move, the flatter you become because space compresses. As it spins, the outer rim of the ride moves faster than the interior. This means that, because of relativity’s effect on space-time, the rim contracts more than the interior since the rim is moving faster. But as the merry-go-round approaches the speed of light, the floor is distorted. It is no longer just a flat disc. The rim has shrunk while the center remains the same, so the surface is curved like an upside-down bowl.

  Now imagine trying to walk on the curved floor of the merry-go-round—you cannot walk in a straight line. At first you might think there is an invisible force that tries to throw you off because the surface is warped or curved. So someone on the merry-go-round says that there is a centrifugal force pushing everything off it. But to someone outside, there is no external force at all, just the curvature of the floor.

  Einstein put it all together. The force that causes you to fall on a merry-go-round is actually caused by the warping of the merry-go-round. The centrifugal force you feel is equivalent to gravity—that is, it is a fictitious force created by being on an accelerating frame. In other words, acceleration in one frame is identical to the effect of gravity in another, which is due to space being curved.