The God Equation

by Michio Kaku

  A new picture began to develop that used particle physics to explain the greatest mystery of cosmology, the birth of the universe. Suddenly, two very different fields, quantum mechanics and general relativity, began to gradually turn into one.

  In this new picture, at the instant of the Big Bang, all the four forces were merged into a single superforce that obeyed the master symmetry. This master symmetry could rotate all the particles of the universe into one another. The equation that governed the superforce was the God equation. Its symmetry was the symmetry that had eluded Einstein and physicists ever since.

  After the Big Bang, as the universe expanded, it began to cool and the various forces and symmetries began to break into pieces, leaving the fragmented weak and strong force symmetries of the Standard Model of today. This process is called symmetry breaking. This means that we need a mechanism that can precisely break this original symmetry, leaving us with the Standard Model. That is where the Higgs boson comes in.

  To imagine this, think of a dam. The water in the reservoir also has a symmetry. If you rotate the water, the water looks very much the same. We all know from experience that water runs downhill. This is because, according to Newton, water always seeks out a lower energy state. If the dam were to break, the water would suddenly rush downstream into a lower energy state. So the water behind the dam is in a higher energy state. Physicists call the state of the water behind the dam the false vacuum, because it is unstable until the water in the burst dam reaches the true vacuum, meaning the lowest energy state in the valley below. After the dam bursts, the original symmetry is gone, but the water has reached its true ground state.

  This effect is also found when you analyze water that is beginning to boil. Just before it boils, the water is in the false vacuum. It is unstable but symmetrical—that is, you can rotate the water and the water looks the same. But eventually, tiny bubbles form, where each bubble exists in a lower energy state than the surrounding water. Each bubble starts to expand, until enough bubbles merge and the water boils.

  According to this scenario, the universe was originally in a perfectly symmetrical state. All the subatomic particles were part of the same symmetry, and they all had zero mass. Because they had zero mass, they could be rearranged but the equation would remain the same. However, for some unknown reason, it was unstable; it was in the false vacuum. The field necessary to shift to the true (but broken) vacuum is the Higgs field. Like Faraday’s electric field that permeated all corners of space, the Higgs field also filled up all of space-time.

  But for some reason, the symmetry of the Higgs field began to break.

  Tiny bubbles began to form inside the Higgs field. Outside the bubbles, all particles remained massless and symmetrical. Inside the bubble, some particles had mass. As the Big Bang progressed, the bubble expanded rapidly, the particles began to acquire different masses, and the original symmetry was broken. Eventually, the entire universe exists in the new vacuum state inside a gigantic bubble.

  So by the 1970s, the hard work of scores of physicists began to pay off. After decades of wandering in the wilderness, they were finally beginning to fit all the pieces of the jigsaw puzzle together. They realized that by cobbling together three theories (representing the strong, weak, and electromagnetic forces) they could write a set of equations that truly coincided with the results observed in the laboratory.

  The key was to create a master symmetry by gluing together three distinct smaller symmetries. The first symmetry described the strong nuclear force, which shuffled three quarks among each other. The second symmetry described the weak force, by shuffling electrons and neutrinos. The third symmetry described the original Maxwell field. The final theory was awkward, but it was hard to argue with success.

  Theory of Almost Everything

  Remarkably, the Standard Model could accurately predict the properties of matter all the way back to a fraction of a second after the Big Bang.

  Figure 9. The Standard Model is a strange collection of subatomic particles that accurately describes our quantum universe, with thirty-six quarks and anti-quarks, twelve weakly interacting particles and anti-particles (called leptons), and a large assortment of Yang-Mills fields and Higgs bosons, particles that are created when you excite the Higgs field.

  Although the Standard Model represented our best understanding of the subatomic world, there were numerous glaring holes. First, the Standard Model made no mention of gravity. This was a huge problem, since gravity is the force that controls the large-scale behavior of the universe. And every time physicists tried to add it to the Standard Model, they could not solve the equations. The quantum corrections due to it, instead of being small, turn out to be infinite, just like QED and Yang-Mills particles. So the Standard Model is unable to shed light on some of the stubborn secrets of the universe, such as what happened before the Big Bang and what lies inside a black hole. (We will return to these important questions later.)

  Second, the Standard Model was created by splicing together by hand the theories that described the various forces, so the resulting theory was a patchwork. (One physicist compared it to taping a platypus, an aardvark, and a whale together and declaring it to be nature’s most elegant creature. The resulting animal, it was said, was one only a mother could love.)

  Third, the Standard Model had a number of parameters that were undetermined (such as the masses of the quarks and the strength of the interactions). In fact, there are about twenty constants that had to be put in by hand, with no understanding whatsoever of where these constants came from or what they represented.

  Fourth, it had not just one copy but three identical copies, or generations, of the quarks, gluons, electrons, and neutrinos in the Standard Model. (So altogether, there are thirty-six quarks, with three colors, three generations, along with their corresponding anti-particles, and twenty free parameters.) Physicists found it difficult to believe that anything so clumsy and unwieldy could be the fundamental theory of the universe.

  LHC

  Because there is so much at stake, nations are willing to spend billions to create the next-generation particle accelerators. Currently, the headlines have been dominated by the Large Hadron Collider outside Geneva, Switzerland, the largest machine of science ever built, costing more than $12 billion and stretching almost seventeen miles in circumference.

  LHC looks like a huge doughnut that straddles the border between Switzerland and France. Inside the tube, protons are accelerated until they reach extremely high energy. Then they collide with another high-energy beam of protons heading in the opposite direction, releasing fourteen trillion electron volts of energy and creating an enormous shower of subatomic particles. The world’s most advanced computers are then used to make sense out of this cloud of particles.

  The goal of the LHC is to duplicate the conditions found shortly after the Big Bang and thereby to create these unstable particles. Finally, in 2012, the Higgs boson, the last piece of the Standard Model, was found.

  Although this was a great day for high-energy physics, physicists realized there was still a long way to go. On one hand, the Standard Model does describe all particle interactions, from deep inside the proton to the very edges of the visible universe. The problem is that the theory is ungainly. In the past, every time physicists probed the fundamental nature of matter, new and elegant symmetries began to emerge, so physicists found it problematic that, at the most fundamental level, nature seemed to prefer a slapdash theory.

  In spite of its practical successes, it is obvious to everyone that the Standard Model is just a warm-up act for the final theory, yet to come.

  Meanwhile, physicists, buoyed by the astonishing successes of the quantum theory when applied to subatomic particles, began to reexamine the theory of general relativity, which had languished for decades. Now physicists set their eyes on a more ambitious goal—to combine the Sta
ndard Model with gravity, meaning that one would need a quantum theory of gravity itself. This would truly be a theory of everything, where all quantum corrections to both the Standard Model and general relativity could be calculated.

  Previously, renormalization theory was a clever sleight of hand that canceled all the quantum corrections of QED and the Standard Model. The key was to represent the electromagnetic and the nuclear forces as particles, called photons and Yang-Mills particles, and then magically wave your hand to make the infinities disappear by reabsorbing them elsewhere. All the unpleasant infinities were brushed under the rug.

  Naively, physicists followed this time-honored tradition and took Einstein’s theory of gravity and introduced a new point particle of gravity, called the graviton. So the smooth surface introduced by Einstein to represent the fabric of space-time was now surrounded by a cloud of trillions of tiny graviton particles.

  Sadly, the bag of tricks painfully accumulated by physicists for the past seventy years to eliminate these infinities failed for the graviton. The quantum corrections created by gravitons were infinite and could not be reabsorbed somewhere else. Here, physicists hit a brick wall. Their winning streak came to an abrupt end.

  Frustrated, physicists then began to try a more modest goal. Unable to create a complete quantum theory of gravity, they tried to calculate what happens when ordinary matter is quantized, leaving gravity alone. This meant calculating the quantum corrections due to stars and galaxies but keeping gravity untouched. By only quantizing the atom, it was hoped to create a stepping-stone and gain insight into the larger goal of formulating a quantum theory of gravity.

  This was a more modest goal, but it opened the floodgates to an astonishing array of new, fascinating physical phenomena that would challenge the way we view the universe. Suddenly, quantum physicists encountered the most bizarre phenomena in the universe: black holes, wormholes, dark matter and dark energy, time travel, and even the creation of the universe itself.

  But the discovery of these strange cosmic phenomena was also a challenge for the theory of everything that must now explain not only the familiar subatomic particles of the Standard Model but all these strange phenomena that stretch the human imagination.

  5

  THE DARK UNIVERSE

  In 2019, newspapers and websites across the planet splashed sensational news on the front page: astronomers had just taken the first photograph of a black hole. Billions of people saw the stark image, a red ball of hot fiery gas with a black, round silhouette in the middle. This mysterious object captured the public’s imagination and dominated the news. Not only have black holes intrigued and fascinated physicists, but they have also entered into the public’s consciousness, being featured in numerous science specials and a plethora of movies.

  The black hole that was photographed by the Event Horizon Telescope lies inside the galaxy M87, 53 million light-years from Earth. The black hole is truly a monster, weighing in at a staggering five billion times the mass of the sun. Our entire solar system, even past Pluto, could easily fit inside the black silhouette in the photograph.

  To accomplish this stunning achievement, astronomers created a super telescope. Normally, a radio telescope is not large enough to take in enough faint radio signals to create an image of an object so distant. But astronomers were able to photograph this black hole by lashing together the signals from five individual ones scattered around the world. By using supercomputers to carefully combine these diverse signals, they effectively created a single giant radio telescope the size of planet Earth itself. This composite was so powerful that it could, in principle, detect an orange sitting on the surface of moon from the Earth.

  A host of new, remarkable astronomical discoveries like this have rejuvenated interest in Einstein’s theory of gravity. Sadly, for the past fifty years, research in Einstein’s general relativity was relatively stagnant. The equations were fiendishly difficult, often involving hundreds of variables; and experiments on gravity were simply too expensive, involving detectors that were miles across.

  The irony is that, although Einstein had reservations about the quantum theory, the current renaissance in relativity research has been fueled by the merger of the two, by the application of the quantum theory to general relativity. As we mentioned, a complete understanding of the graviton and how to eliminate its quantum corrections is considered too difficult, but a more modest application of the quantum theory to stars (neglecting graviton corrections) has opened the heavens to a wave of startling scientific breakthroughs.

  What Is a Black Hole?

  The basic idea of a black hole actually can be traced back to Newton’s discovery of the laws of gravity. His Principia gave us a simple picture: if you fire a cannonball with enough energy, it will completely circle the Earth, then return to its original point.

  But what happens if you aim the cannonball straight up? Newton realized that the cannonball will eventually reach a maximum height and then fall back to Earth. But with enough energy, the cannonball would reach escape velocity—that is, the speed necessary to escape the Earth’s gravity and soar into space, never to return.

  It is a simple exercise, using Newton’s laws to calculate the escape velocity of the Earth, which turns out to be 25,000 miles per hour. This is the velocity that our astronauts had to attain to reach the moon in 1969. If you do not reach escape velocity, then you will either enter orbit or fall back to Earth.

  In 1783, an astronomer named John Michell asked himself a deceptively simple question: What happens if the escape velocity is the speed of light? If a light beam is emitted from a giant star so massive that its escape velocity is the speed of light, then perhaps even its light cannot escape. All light emitted from this star will eventually fall back into the star. Michell called these dark stars, celestial bodies that appeared black because light could not escape their immense gravity. Back in the 1700s, scientists knew little about the physics of stars and did not know the correct value for the speed of light, and hence this idea languished for several centuries.

  In 1916, during World War I, German physicist Karl Schwarzschild was stationed on the Russian front as an artilleryman. While fighting in the middle of a bloody war, he found time to read and digest Einstein’s famous 1915 paper introducing general relativity. In a brilliant stroke of mathematical insight, Schwarzschild somehow found an exact solution of Einstein’s equations. Instead of solving the equations for a galaxy or the universe, which was too difficult, he started with the simplest of all possible objects, a tiny point particle. This object, in turn, would approximate the gravity field of a spherical star as seen from a distance. One could then compare Einstein’s theory with experiment.

  Einstein’s reaction to Schwarzschild’s paper was ecstatic. Einstein realized that this solution of his equations would allow him to make more precise calculations with his theory, such as the bending of starlight around the sun and the wobbling of the planet Mercury. So instead of making crude approximations to his equations, he could calculate exact results from his theory. This was a monumental breakthrough that would prove important for understanding black holes. (Schwarzschild died shortly after his remarkable discovery. Saddened, Einstein wrote a moving eulogy for him.)

  But despite the enormous impact of Schwarzschild’s solution, it also raised some bewildering questions. From the start, his solution had weird properties that pushed the boundaries of our understanding of space and time. Surrounding a supermassive star was an imaginary sphere (which he called the magic sphere and today is called the event horizon). Far outside this sphere, the gravity field resembled an ordinary Newtonian star’s, so Schwarzschild’s solution could be used to approximate its gravity. But if you were unfortunate enough to approach the star and pass through the event horizon, you would be trapped forever and would be crushed to death. The event horizon is the point of no return: anything that falls in never comes o
ut.

  But as you approached the event horizon, even more bizarre things would begin to happen. For example, you would encounter light beams that had been trapped for perhaps billions of years and are still orbiting the star. The gravity pulling on your feet would be greater than the gravity pulling on your head, so you would be stretched like spaghetti. In fact, this spaghettification becomes so severe that even the atoms of your body get pulled apart and eventually disintegrate.

  To someone watching this remarkable event from a great distance, it would appear as if time inside the rocket ship on the edge of the event horizon had gradually slowed down. In fact, to an outsider, it appears as if time has stopped as the ship hits the event horizon. What is remarkable is that, to the astronauts in the ship, everything seems to be normal as they pass through the event horizon—normal, that is, until they are torn apart.

  This concept was so bizarre that, for many decades, it was considered science fiction, a strange by-product of Einstein’s equations that didn’t exist in the real world. Astronomer Arthur Eddington once wrote that “there should be a law of Nature to prevent a star from behaving in this absurd way!”

  Einstein even wrote a paper arguing that, under normal conditions, black holes could never form. In 1939, he showed that a whirling ball of gas could never be compressed by gravity to within the event horizon.

  Ironically, that very same year, Robert Oppenheimer and his student Hartland Snyder showed that black holes could indeed form from natural processes that Einstein did not foresee. If you start with a giant star ten to fifty times more massive than our sun, when it uses up its nuclear fuel, it can eventually explode as a supernova. If the remnant of the explosion is a star that is compressed by gravity to its event horizon, then it can collapse into a black hole. (Our sun is not massive enough to undergo a supernova explosion, and its event horizon is about four miles across. No known natural process can squeeze our sun down to two miles, and hence our sun will not become a black hole.)