The God Equation Read online

  Halley was stunned.

  He immediately realized that he was witnessing a landmark in science and volunteered to pay for the printing costs of what would eventually become one of the greatest masterpieces in all science, Mathematical Principles of Natural Philosophy, or simply Principia.

  Furthermore, Halley, realizing that Newton was predicting that comets could return at regular intervals, calculated that the comet of 1682 would return in 1758. (Halley’s comet sailed over Europe on Christmas Day, 1758, as predicted, helping to seal Newton’s and Halley’s reputations posthumously.)

  Newton’s theory of motion and gravitation stands as one of the greatest achievements of the human mind, a single principle unifying the known laws of motion. Alexander Pope wrote:

  Nature and Nature’s laws lay hid in night:

  God said, Let Newton be!

  And all was light.

  Even today, it is the laws of Newton that allow NASA engineers to guide our space probes across the solar system.

  What Is Symmetry?

  Newton’s law of gravity is also noteworthy because it possesses a symmetry, so that the equation remains the same if we make a rotation. Imagine a sphere surrounding the Earth. The force of gravity is identical at every point on it. In fact, that is why the Earth is spherical, rather than another shape: because gravity compressed the Earth uniformly. That is why we never see cubical stars or pyramidal planets. (Small asteroids are often shaped irregularly, because the gravitational force on an asteroid is too small to compress it evenly.)

  The concept of symmetry is simple, elegant, and intuitive. Moreover, throughout this book, we will see that symmetry is not just frivolous window dressing to a theory, but in fact is an essential feature that indicates some deep, underlying physical principle about the universe.

  But what do we mean when we say an equation is symmetrical?

  An object is symmetrical if, after you rearrange its parts, it is left the same, or invariant. For example, a sphere is symmetrical because it remains the same after you rotate it. But how can we express this mathematically?

  Think of the Earth revolving around the sun (see figure 2). The radius of the Earth’s orbit is given by R, which remains the same as the Earth moves in its orbit (actually, the Earth’s orbit is elliptical so R varies slightly, but that’s not important for this example). The coordinates of the Earth’s orbit are given by X and Y. As the Earth moves in its orbit, X and Y continually change, but R is invariant—that is it doesn’t change.

  So the equations of Newton maintain this symmetry, meaning that the gravity between the Earth and the sun remains the same as the Earth orbits the sun. As our frame of reference changes, the laws stay constant. No matter what orientation we take looking at a problem, the rules are unchanging, and the results come out the same.

  Figure 2. If the Earth revolves around the sun, its radius R remains the same. The coordinates X and Y of the Earth continually change as it orbits, but R is an invariant. By the Pythagorean theorem, we know that X2 + Y2 = R2. So Newton’s equation has a symmetry when expressed either in terms of R (because R is an invariant) or X and Y (via the Pythagorean theorem).

  We will encounter this concept of symmetry over and over when we discuss the unified field theory. In fact, we will see that symmetry is one of our most powerful tools in unifying all the forces of nature.

  Confirmation of Newton’s Laws

  Over the centuries, numerous confirmations of Newton’s laws have been found, and they had a tremendous impact on science and also society. In the nineteenth century, astronomers noticed a strange anomaly in the heavens. The planet Uranus was deviating from the predictions of Newton’s laws. Its orbit was not a perfect ellipse, but wobbled a bit. Either Newton’s laws were flawed, or there was a planet that was not yet discovered whose gravity was tugging on the orbit of Uranus. Faith in Newton’s laws was so great that physicists like Urbain Le Verrier tediously calculated where this mystery planet might lie. In 1846, on the very first try, astronomers found this planet to within one degree of where it was predicted to be. The new planet was dubbed Neptune. This was a tour de force for Newton’s laws, and the first time in history that pure mathematics was used to detect the presence of a major celestial body.

  As we mentioned earlier, every time scientists decoded one of the four fundamental forces of the universe, it not only revealed the secrets of nature, it also revolutionized society itself. Newton’s laws not only unlocked the secret of the planets and comets, they also laid the foundation of the laws of mechanics, which we use today to design skyscrapers, engines, jet planes, trains, bridges, submarines, and rockets. For example, in the 1800s physicists applied Newton’s laws to explain the nature of heat. At the time, scientists speculated that heat was some form of liquid that spread through a substance. But further investigation showed that heat was actually molecules in motion, resembling tiny steel balls constantly colliding with one another. Newton’s laws allowed us to calculate precisely how two steel balls bounced off each other. Then, by adding trillions upon trillions of molecules, one could calculate the precise properties of heat. (For example, when a gas in a chamber is heated, it expands according to Newton’s laws since the heat increases the velocity of the molecules inside the chamber.)

  Engineers could then use these calculations to perfect the steam engine. They could calculate how much coal was needed to turn water into steam, which could then be used to push gears, pistons, wheels, and levers to power machines. With the coming of the steam engine in the 1800s, the energy available to a worker skyrocketed to hundreds of horsepower. Suddenly, steel rails were connecting distant parts of the world and vastly increasing the flow of goods, knowledge, and people.

  Before the Industrial Revolution, goods were made by tiny, exclusive guilds of skilled craftsmen who toiled to create even the simplest household items. They also jealously guarded the secrets of their handicraft. Hence, goods were often scarce and expensive. With the coming of the steam engine and the powerful machines it made possible, goods could be stamped out at a fraction of the original cost, vastly increasing the collective wealth of nations and raising our standard of living.

  When I teach Newton’s laws to promising engineering students, I try to emphasize that these laws are not just dry, boring equations, but they have changed the course of modern civilization, creating the wealth and prosperity we see all around us. We sometimes even show our students the catastrophic collapse of the Tacoma Narrows Bridge in Washington State in 1940, recorded on film, as a stunning example of what happens when we misapply Newton’s laws.

  Newton’s laws, based on unifying the physics of the heavens with the physics of the Earth, helped to usher in the first great revolution in technology.

  Mystery of Electricity and Magnetism

  It would take another two hundred years for the next big breakthrough, which came from the study of electricity and magnetism.

  The ancients had known that magnetism could be tamed; the invention of the compass by the Chinese harnessed the power of magnetism and helped launch an age of discovery. But the ancients feared the power of electricity. Lightning bolts were thought to express the anger of the gods.

  The man who finally laid the foundation for this field was Michael Faraday, a poor but industrious youth who lacked any formal education. As a child, he managed to get a job working as an assistant at the Royal Institution in London. Normally, someone of his low social standing would forever sweep the floor, wash bottles, and hide in the shadows. But this young man was so tireless and inquisitive that his supervisors allowed him to perform experiments.

  Faraday would go on to make some of the greatest discoveries in electricity and magnetism. He showed that if you take a magnet and move it inside a hoop of wire, then electricity is generated in the wire. This was an amazing and important observation, since th
e relationship between electricity and magnetism was then totally unknown. One could also show the reverse, that a moving electric field can create a magnetic one.

  It gradually dawned on Faraday that these two phenomena were actually two sides of the same coin. This simple observation would help to open up the electric age, in which giant hydroelectric dams would light up entire cities. (In a hydroelectric dam, the river pushes against a wheel that spins a magnet that then pushes electrons inside a wire that sends the electricity to the sockets in your home. The opposite effect, turning electric fields into magnetic ones, is the reason why your vacuum cleaner works. Electricity from the wall socket causes a magnet to spin, which drives a pump creating suction and causes the rollers of the vacuum cleaner to spin as well.)

  But because Faraday had no formal education, he did not have the command of the mathematics that would allow him to describe his remarkable discoveries. Instead, he filled up notebooks with strange diagrams showing lines of force that look like the web of lines iron filings make when surrounding a magnet. He also invented the concept of a field, one of the most important concepts in all of physics. A field consists of these lines of force spread throughout space. Magnetic lines surround every magnet, and the magnetic field of the Earth emanates from the north pole, spreads through space, and then returns to the south pole. Even Newton’s theory of gravity can be expressed in terms of fields, so that the Earth moves around the sun because it moves in the sun’s gravitational field.

  Faraday’s discovery helped to explain the origin of the magnetic field surrounding the Earth. Since the Earth spins, the electric charges inside the Earth also spin. This constant motion moving inside the Earth is responsible for the magnetic field. (But this still left open a mystery: Where does the magnetic field of a bar magnet come from, since there is nothing moving or spinning in it? We will return to this mystery later.) Today, all the known forces of the universe are expressed in the language of fields first introduced by Faraday.

  Given Faraday’s immense contribution to initiating the electric age, physicist Ernest Rutherford declared him the “greatest scientific discoverer of all time.”

  Faraday was also unusual at least for his time because he loved to engage the public, and even children, in his discoveries. He was famous for his Christmas Lectures, where he would invite everyone to the Royal Institution in London to witness dazzling displays of electrical wizardry. He would enter a large room whose walls were covered with metal foil (which today is called a Faraday cage), and then electrify it. Although the metal was clearly electrified, he was totally safe because the electric field spread out over the entire surface of the room, so the electric field inside remained zero. Today, this effect is commonly used to shield microwave ovens and delicate equipment from stray electric fields, or to protect jet planes, which are often struck by lightning bolts. (For a Science Channel program I once hosted, I went inside a Faraday cage at the Boston Museum of Science. Huge bolts of electricity, up to two million volts, bombarded the cage, filling the auditorium with a loud crackling sound. But I did not feel a thing.)

  Maxwell’s Equations

  Newton had shown that objects move because they were pushed by forces, which could be described by calculus. Faraday showed that electricity moved because it was pushed by a field. But the study of fields required a new branch of mathematics, which was eventually codified by Cambridge mathematician James Clerk Maxwell and called vector calculus. So in the same way that Kepler and Galileo laid the foundation for Newtonian physics, Faraday paved the way for Maxwell’s equations.

  Maxwell was a virtuoso in mathematics who made astonishing breakthroughs in physics. He realized that the behavior of electricity and magnetism, as discovered by Faraday and others, could be summarized in precise mathematical language. One law stated that a moving magnetic field could create an electric field. Another law stated the opposite, that a moving electric field could create a magnetic field.

  Then Maxwell had an idea for the ages. What if a changing electric field created a magnetic one that then created another electric field that then created another magnetic field, etc.? He had the brilliant insight that the end product of this rapid back-and-forth motion would be a moving wave, where electric and magnetic fields were constantly turning into each other. This infinite sequence of transformations has a life of its own, creating a moving wave of vibrating electric and magnetic fields.

  Using vector calculus, he calculated the speed of this moving wave, and he found it to be 310,740 kilometers per second. He was shocked beyond belief. To within experimental error, this speed was remarkably close to the speed of light (which is now known to be 299,792 kilometers per second). He then made the next bold step to claim that this was light! Light is an electromagnetic wave.

  Maxwell then wrote prophetically, “We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”

  Today, every physics student and electrical engineer has to memorize Maxwell’s equations. They are the basis for TVs, lasers, dynamos, generators, etc.

  Figure 3. Electrical and magnetic fields are two sides of the same coin. Oscillating electric and magnetic fields turn into each other and move like a wave. Light is one manifestation of an electromagnetic wave.

  Faraday and Maxwell unified electricity and magnetism. And the key to unification is symmetry. Maxwell’s equations contain the symmetry called duality. If the electric within a light beam is represented by E and the magnetic field by B, then the equations for electricity and magnetism remain the same when we switch E and B. This duality implies that electricity and magnetism are two manifestations of the same force. So the symmetry between E and B allows us to unify electricity and magnetism, thereby creating one of the greatest breakthroughs of the nineteenth century.

  Physicists were fascinated by this discovery. The Berlin Prize was offered to anyone who could actually reproduce these Maxwell waves in the laboratory. In 1886, physicist Heinrich Hertz performed the historic test.

  First, Hertz created an electric spark in one corner of his laboratory. Several feet away, he had a coil of wire. Hertz showed that by turning on the spark, he could generate an electrical current in the coil, thereby proving that a new, mysterious wave traveled wirelessly from one place to another. This heralded the creation of a new kind of phenomenon, called radio. In 1894, Guglielmo Marconi introduced this new form of communication to the public. He showed that you could send wireless messages across the Atlantic Ocean at the speed of light.

  With the introduction of radio, we now had a superfast, convenient, and wireless way of communicating over long distances. Historically, the lack of a fast and reliable communication system was one of the great obstacles to the march of history. (In 490 BCE, after the Battle of Marathon between the Greeks and the Persians, a poor runner was ordered to spread the news of the Greek victory as fast as he could. Bravely, he ran 26 miles to Athens after previously running 147 miles to Sparta, and then, according to legend, dropped dead of sheer exhaustion. His heroism, in the age before telecommunication, is now celebrated in the modern marathon.)

  Today, we take for granted that we can send messages and information effortlessly across the globe, utilizing the fact that energy can be transformed in many ways. For example, when speaking on a cell phone, the energy of the sound of your voice converts to mechanical energy in a vibrating diaphragm. The diaphragm is attached to a magnet that relies on the interchangeability of electricity and magnetism to create an electrical impulse, the kind that can be transported and read by a computer. This electrical impulse is then translated into electromagnetic waves that are picked up by a nearby microwave tower. There, the message is amplified and sent across the globe.

  But Maxwell’s equations not only gave us nearly instantaneous communication via radio, cell phone, and fiber-optic
cables, they also opened up the entire electromagnetic spectrum, of which visible light and radio were just two members. In the 1660s, Newton had shown that white light, when sent through a prism, can be broken up into the colors of the rainbow. In 1800, William Herschel had asked himself a simple question: What lies beyond the colors of the rainbow, which extend from red to violet? He took a prism, which created a rainbow in his lab, and placed a thermometer below the color red, where there was no color at all. Much to his surprise, the temperature of this blank area began to rise. In other words, there was a “color” below red that was invisible to the naked eye but contained energy. It was called infrared light.

  Today, we realize that there is an entire spectrum of electromagnetic radiation, most of which is invisible, and each has a distinct wavelength. The wavelength of radio and TV, for example, is longer than that of visible light. The wavelength of the colors of the rainbow, in turn, is longer than that of ultraviolet and X-rays.

  This also meant that the reality we see all around us is only the tiniest sliver of the complete EM spectrum, the smallest approximation of a much larger universe of EM colors. Some animals can see more than we can. For example, bees can see ultraviolet light, which is invisible to us but essential for them to find the sun even on a cloudy day. Since flowers evolved their gorgeous colors in order to attract insects like bees to pollinate them, this means that flowers are often even more spectacular when viewed using UV light.

  Figure 4. Most of the “colors” of the EM spectrum, extending from radio to gamma rays, are invisible to our eyes. Our eyes can only see the tiniest sliver of the entire EM spectrum, due to the size of the cells in our retinas.